KURT MAHLER 26 JULY 1903 ? 26 FEBRUARY 1988 ELECTED ...

KURT MAHLER
26 JULY 1903 — 26 FEBRUARY 1988
ELECTED F.R.S. 1948
By J. H. Coates F.R.S. and A. J. van der Poorten
Kurt Mahler was born on 26 July, 1903 at Krefeld am Rhein in Germany; he died
in his 85th year on 26 February, 1988 in Canberra, Australia. From 1933 onwards
most of his life was spent outside of Germany, but his mathematical roots remained
in the great school of mathematics that existed in Germany between the two world
wars. Above all Mahler lived for mathematics; he took great pleasure in lecturing,
researching and writing. It was no surprise that he remained active in research until
the last days of his life. He was never a narrow specialist and had a remarkably
broad and thorough knowledge of large parts of current and past mathematical
research. At the same time he was oblivious to mathematical fashion, and very
much followed his own path through the world of mathematics, uncovering new
and simple ideas in many directions. In this way he made major contributions to
transcendental number theory, diophantine approximation, p-adic analysis, and the
geometry of numbers. Towards the end of his life, Kurt Mahler wrote a considerable
amount about his own experiences; see ‘Fifty years as a mathematician’ [209],
‘How I became a mathematician’ [192], ‘Warum ich eine besondere Vorliebe für die
Mathematik habe’ [214], ‘Fifty years as a mathematician II’ [222]. There is also a
recent excellent account of his life and work by Cassels (1991). In preparing this
Memoir we have freely used these sources. We have also drawn on our knowledge of
and conversations with Mahler, whom we first met when we were undergraduates
in Australia in the early 1960s.
Krefeld, where Mahler spent the first twenty years of his life, was a town of some
100,000 inhabitants in a predominantly Catholic part of the Prussian Rhineland.
His family was Jewish, and had lived in the Rhineland for several generations. His
father and several of his uncles worked in the printing and bookbinding trade, beginning
as apprentices and slowly saving enough money to start small firms of their
own. Kurt and his twin sister Hilde (1903-1934) were the youngest of eight children
born to Hermann Mahler (1858-1941) and his wife Henriette, née Stern (1860-1942).
Four of the children died young. An elder sister Lydia (who died in 1984) married
a printer who was also a musician, and lived in the Netherlands. An elder brother
Josef, who joined and eventually took over his father’s firm, disappeared together
with his wife in a concentration camp during the Second World War.
Work of the second author supported in part by grants from the Australian Research Council and
by aresearch agreement with Digital Equipment Corporation.
Typeset by AMS-TEX

2 John Coates and Alf van der Poorten
The family had no academic traditions. None of Kurt’s four grandparents went to
more than elementary school (Volksschule). However, the four children acquired a
love of reading from their father. At the age of 5, Kurt contracted tuberculosis,
which severely affected his right knee. The knee was subsequently operated on
several times, but it did not heal until he was 35 and left him with a stiff leg, which
very much hindered his walking throughout his life. Because of this illness, Kurt
only attended school for a total of four years up till the age of 14, but he had some
private tuition at home for two additional years. At Easter 1917, shortly before he
turned 14, he left elementary school, and attended technical schools for the next
two years, with the intention of becoming a precision tool and instrument maker.
Mahler always retained a fascination with technical drawing and calligraphy. Most
important, these technical schools gave him his first training in algebra and geometry.
He very quickly decided that mathematics was what he really liked doing.
Already, from the summer vacation of 1917, he began teaching himself logarithms
(the arithmetic properties of which turned out to be one of his abiding interests
in transcendental number theory), plane and spherical trigonometry, analytic geometry
and calculus. In 1918, he became an apprentice in a machine factory in
Krefeld, working for one year in the drawing office and then for almost two years
in the factory itself. Later, the drafting skills he acquired would be useful; see
the papers of L. J. Mordell in the period 1941–45. Mahler said himself that his
aim in taking the apprenticeship was that it might eventually allow him to study
mathematics at a technical university (Technische Hochschule), thereby avoiding
the difficult entrance examination required to enter a traditional university. He did
learn a little more elementary mathematics as part of evening classes, but quickly
progressed with his mathematical self-education. How successful he was as an autodidact
is illustrated by the fact that he soon acquired and began reading, without
any expert guidance, such sophisticated books as Bachmann’s Zahlentheorie, Landau’s
Primzahlen, Knopp’s Funktionentheorie, Klein and Fricke’s Modulfunktionen
and Automorphe Funktionen, and Hilbert’s Grundlagen der Geometrie.
In Mahler’s own words: ‘The great day came in 1921’. He was in the habit of
writing little articles about the mathematics he had read. Without his knowledge
his father had sent some of Kurt’s work to the director of the local grammar school
(Realschule). Dr Junker was a mathematician, having written a doctoral thesis in
invariant theory under Christoffel. He was evidently impressed by the apprentice’s
efforts, and sent some of Mahler’s work to Klein in Göttingen, who passed it on to his
young Assistant, C. L. Siegel. Thus began a lifelong association between Siegel and
Mahler: Siegel urged that Mahler should be helped to pass the university entrance
examination. Mahler left the factory and spent two years at home, preparing for
the entrance examination (he cites preparation for papers in German, French, and
English) with the assistance of teachers at the Realschule, as well as continuing his
own reading in mathematics. He passed the examination (he says ‘I just scraped
through’) in the fall of 1923, amidst the political turmoil of German hyper-inflation
and the occupation of the Ruhr. Mahler’s 1927 Frankfurt doctoral dissertation is
dedicated to Dr Josef Junker.
Siegel had moved to the University of Frankfurt am Main and, following his suggestion,
Mahler went to study there in 1923, at the age of 20. Frankfurt was then

Kurt Mahler 1903–1988 3
avery stimulating place for study with Dehn, Hellinger, Epstein, Szass and Siegel
making up the Mathematics Faculty (see Siegel’s lecture (1964) on this period at
Frankfurt). Mahler was an unusual freshman. In his first semester, he speaks of
attending lectures on calculus by Siegel, topology by Dehn and elliptic functions by
Hellinger, a seminar on cyclotomy (in which he gave several lectures), and a seminar
on the history of mathematics. Mahler was clearly greatly influenced during this
period by Siegel, who was the only person whom he recognized as his teacher in
mathematical research. In the summer of 1925, when Siegel left for a period of leave
overseas, Mahler moved to Göttingen, where he remained until 1933. Göttingen
was then still the world’s leading mathematical centre, but was going through a
period of change because the great era of Hilbert and Klein was almost at an end.
Landau seems to have been kind to Mahler, but took little active interest in his
work. From Emmy Noether’s lectures, he learnt of p-adic numbers, whose study
grew to be one of the main themes of his mathematical research. (A few years later,
Mahler proudly reports lecturing on his own work on p-adic numbers at Marburg
to Hensel). Perhaps most importantly, in Göttingen Mahler met a galaxy of young
mathematicians from Europe and the United States, many of whom became leading
figures in later years. These included Alexandroff, Hopf, Koksma, Mordell, Popken,
van der Waerden, Weil and Wiener. In 1927, Mahler submitted his doctoral dissertation,
on the zeroes of the incomplete gamma function, to Frankfurt (he reports
that Ostrowski was not very impressed with the dissertation, and advised him ‘to
do less easy mathematics’.)
For most of his time at Göttingen, Mahler was wholly supported in his studies
by his parents and other members of the Jewish community in Krefeld. However,
shortly before he was 30, he was awarded a two-year research fellowship by the
Notgemeinschaft der Deutschen Wissenschaft, and records that he was even able
to save some of the stipend. In the Göttingen years, all the main themes of his
later research, with the exception of the geometry of numbers, appeared in his
papers (which are the first twenty or so papers in his list of publications). Mahler
invented a new transcendence method, he discovered his celebrated classification of
transcendental numbers, extended the ideas of Hermite’s original work in his studies
of the approximation properties of e, pioneered diophantine approximation in padic
fields, and applied his results on p-adic diophantine approximation to prove his
well known generalization of Siegel’s theorem on integer points on curves of genus
1. Mahler undoubtedly realized that his method could be extended to curves of
genus greater than 1, but it was typical of his outlook that he did not have the
patience to work through his generalization of Siegel’s method. Mahler mentions
that his idea of extending the Thue-Siegel theorem to p-adic algebraic numbers
came to him on a small island in the North Sea during the Whitsun holidays of
1930, when bad weather had forced him to stay inside!
Mahler had been appointed to his first post, an assistantship in the University of
Königsberg, but had not yet taken it up, when Hitler came to power in 1933. He
seems to have realized immediately that he must leave Germany. In the summer
of 1933, Mahler spent six weeks in Amsterdam with van der Corput and his two
pupils Koksma and Popken, whom Mahler had met in Göttingen; they were to
remain his lifelong friends. Mahler moved to Manchester for the academic year

4 John Coates and Alf van der Poorten
1933-34, where Mordell had secured him a small research fellowship called the
Bishop Harvey Goodwin Fellowship. Mahler often spoke in later years of Mordell’s
kindness to him on this and many subsequent occasions, including in helping him
to learn English. It seems that the first English lesson Mahler had in Manchester
consisted of being put in front of a blackboard immediately on his arrival and told
to give a seminar! The next two academic years were spent in Groningen in the
Netherlands, supported by a stipend obtained by van der Corput from a Dutch
Jewish group. Here a new theme, the geometry of numbers, began to emerge in
Mahler’s work.
In 1936, he was run into by a bicycle in Groningen, and this accident reactivated
the tuberculosis in his right knee. He was unable to walk for some time and had to
return to Krefeld, where he had several operations culminating in the removal of the
kneecap. These operations together with two three-month periods in a sanatorium
at Montana, Valis, Switzerland during the summers of 1937 and 1938, finally cured
the tuberculosis, but he was left with a permanent limp. Mahler speaks of having
to take morphine to lessen the pain after his last operation, and being relieved to
find that he could still do mathematical research when he proved that the decimal
expansion 0.123456789101112 ... is a transcendental number. (In later years,
Mahler often stated the view that twentieth century mathematicians had greatly
neglected the study of the arithmetic properties of decimal expansions.) Needless
to say, there were other difficulties during these years, which Mahler rarely talked
about and certainly did not record in his own written memories. In one incident
(which one of us learnt of from Popken, and which Mahler subsequently confirmed
in conversation), Mahler was refused entry at the Dutch border and was about to
be sent back to Nazi Germany. Fortunately, Koksma had a colleague at the Free
University of Amsterdam, G. H. A. Grosheide, who was related to a senior member
of the Dutch government. An urgent intercession was made on Mahler’s behalf via
this channel, and he was finally allowed to enter the Netherlands.
In 1937, Mahler returned to Manchester. He thoroughly enjoyed the lively intellectual
atmosphere in number theory that Mordell had fostered in the Department.
While his own research flourished, the practical side of life could not always have
been easy for him. In the period 1937–41, he had two short appointments as a
temporary assistant lecturer and a little support from fellowship stipends, but for
over two of these years he lived on his own savings. In 1939, he had planned to take
up an appointment at the University of Szechuan in China, where his friend Chao
Ko was teaching, but he was forced to abandon the idea because of the outbreak of
war. However, Mahler had begun to learn Chinese and that study was to remain
an important interest and hobby. In 1940, he was interned for three months as an
‘enemy alien’, first in a tent city near the Welsh border and then in boarding houses
on the Isle of Man. Here he lectured to the other internees on the construction of
the real numbers by means of Cauchy sequences of rational numbers, as part of a
university set up in the internment camp. Mahler records that he later found the
same material very suitable for the beginning of first year honours courses in analysis
at Manchester. While interned, he was awarded a ScD degree by the University
of Manchester.
In 1941, Mahler was appointed to the Assistant Lectureship at Manchester, which

Kurt Mahler 1903–1988 5
Davenport had vacated when he moved to take a chair at Bangor. In the next few
years, Mahler developed a geometry of numbers of general sets in n-dimensional
space, including his celebrated compactness theorem. His future was now assured.
He was promoted to Lecturer (1944), Senior Lecturer (1947) and Reader (1949),
and in 1952 the first personal chair in the history of the University was created for
him. He became a British subject in 1946 and was elected a Fellow of the Royal
Society in 1948. He made his first visit to the United States in 1949, spending most
of the time at the Institute for Advanced Study in Princeton. At Christmas 1949,
he contracted diphtheria and had another severe bout of illness for three months,
but recovered in time to spend the summer lecturing in Colorado, and taking part
in the International Congress of Mathematicians at Harvard University.
At Manchester, he lived from 1938 until 1958 at Donner House, a hostel where
some 25-30 single staff lived in bedsitting rooms and dined communally. When the
hostel was pulled down to make way for more extensive student dormitories, Mahler
bought a small house in suburban Manchester, and lived there until his departure for
Canberra. However, in later life he complained that he found the burden of looking
after his own house rather onerous, and one senses that the fact that he could live
at University House (a collegiate institution for postgraduate students and research
workers in the Institute of Advanced Studies of the Australian National University)
was one of the factors which made him decide to move to Canberra in 1963. Of
course, there were many other reasons for this move. Most of the mathematicians
he had known at Manchester had moved on to positions around the world, and he
was clearly feeling a little isolated there.
In the early 1960s, B. H. Neumann, a colleague of Mahler at Manchester, was
invited to set up a new Department of Mathematics in the purely research side of
the Australian National University, the Institute of Advanced Studies. Mahler was
one of the first of many visitors whom Bernhard Neumann quickly invited. There is
no doubt that Mahler immediately liked the warm and stimulating atmosphere in
the new department, as well as the beautiful climate of Canberra and the delightful
setting of the ANU campus on the edge of what was then a large country town.
Mahler himself says he was very happy to accept the offer of a research professorship,
which he took up in September 1963. The position gave him great freedom to travel
and to pursue his own research, both of which he did with energy and enthusiasm.
However, Mahler was also very concerned with sowing the seeds of his own mathematical
knowledge in his new country. As in his own mathematical research, he
instinctively felt that the best way to do this was to go back to first principles, and
to begin by teaching beginners in the subject. The ANU had begun to award undergraduate
degrees only a few years before Mahler arrived, and Hannah Neumann
was appointed to head the new Department of Mathematics in the teaching side of
the University (the School of General Studies) at about the same time that Mahler
took up his chair. Between them, they arranged for Mahler to give two courses
to the small number of undergraduates reading mathematics, one in 1963 on elementary
number theory, and the second in 1964 on the elliptic modular function
j(z). One of us had the good fortune to attend these courses. Mahler started and
finished each lecture with extraordinary punctuality; in between, the audience was
given a rare insight into his understanding of and enthusiasm for the material of

6 John Coates and Alf van der Poorten
the lecture. As he spoke, he would produce a beautiful written exposition on the
blackboard of the key points, which were neatly placed in order in his characteristic
rectangular boxes. Although he seemed at first so different and forbidding, we soon
discovered that he was very willing to talk about his knowledge of mathematics in
general, and to lend us his own mathematical books when we could not find them in
the library. Mahler gave lectures at various summer schools in Canberra and elsewhere
around Australia, as well as a number of advanced courses on transcendental
number theory in the Institute of Advanced Studies. In the end the fascination of
what he was doing beguiled us both into research in number theory, and we made
our first steps in mathematical research on problems suggested by him.
In 1968, Mahler reached the statutory retiring age for professors, and was forced
to retire from the ANU. He then moved to a chair at the Ohio State University
in Columbus, Ohio, where the chairman was an old friend, Arnold Ross (whose
summer schools for gifted high school students have attracted many young people
into mathematical research in Australia, the USA and Germany). In 1972, Mahler
returned to Canberra for his ‘final retirement’, living once more in University House.
But his mathematical activity never abated, as is shown by the publication of
some forty papers from 1972 until his death. He left the bulk of his estate to the
Australian Mathematical Society, which has already used part of it to establish a
lectureship in his memory.
Kurt Mahler never married. Indeed, he affirms in notes left with the Royal Society
that on his part that was a deliberate decision made on grounds of his poor health.
In the event, he outlived his contemporaries. That was of course a source of sadness
for him, but also one of wry pride.
Mahler was an excellent photographer; many of his pictures adorn University House
at the ANU where he lived for more than twenty years. He remained fascinated
by Chinese and was exceptionally proud of having written the paper [96]. His
non-mathematical reading comprised mostly science fiction and history.
Mahler received many distinctions during his lifetime. He was elected a Fellow of
the Australian Academy of Science in 1965 and received its Lyle Medal in 1977.
The London Mathematical Society awarded him its Senior Berwick Prize in 1950,
and its De Morgan medal in 1971. In November 1977, he received a diploma at
aspecial ceremony in Frankfurt to mark the golden jubilee of his doctorate. Het
Wiskundig Genootschap (the Dutch Mathematical Society) made him an honorary
member in 1957, as did the Australian Mathematical Society in 1986.
In a letter to one of us dated 24 February 1988 — received after hearing of his
death — Kurt Mahler sets the following problem:
Let f(x) be apolynomial in x with integral coefficients which is positive
for positive x. Study the integers x for which the representation of f(x)
to the base g ≥ 3 has only digits 0 and 1.
Concerning his life’s work he also happened to write, in that letter: ‘When my old
papers first appeared, they produced little interest in the mathematical world, and
it was only in recent times that they have been rediscovered and found useful ... ’.
That grossly underrates the impact of his work in the past, but correctly notices
the richness of even his minor remarks.

Kurt Mahler 1903–1988 7
Mathematical Work
There are a number of interlinked themes that run through Mahler’s work. The
primary one, and in any case the one on which we are best able to comment and
therefore will concentrate, is diophantine approximation and transcendence theory.
We recall Mahler saying disappointedly that he had never proved a ‘major result’,
that his main contribution had been to prove mere lemmas. In that sense his ‘near
proof’ [169] of the transcendence of Euler’s constant γ must have been a major
disappointment. Yet even there he proves such results as the transcendence of
πY0(2)
2J0(2)
− γ ;
γ does not separate from the Bessel functions because of a nonlinear algebraic
relation relating the Bessel functions appearing in the auxiliary function featuring
in the proof.
The future will tell which of his papers have been the most inappropriately neglected.
On a number of occasions Mahler mentioned his paper [43] (and [152]) on
periodic algorithms as well warranting further study. He was disappointed by the
apparent lack of response to his paper [157] on ideal bases.
Mahler’s Method. Of course Mahler introduced entire new subjects. Chronologically,
the first is ‘Mahler’s method’, so named by Loxton and van der Poorten (1977)
which is introduced in Mahler’s papers [4,7,8] but was then long-neglected, except
for Mahler’s foray into Chinese [96] and its translation [115], until revived by him
in [170]. The method yields transcendence and algebraic independence results for
the values at algebraic points of ‘Mahler functions’, to wit power series f satisfying
functional equations with simplest example the so-called Fredholm series
f(z) =
∞
z 2h
with f(z 2 )=f(z) − z.
h=0
Multivariable examples include the remarkable series
Fω(z1,z2) =
∞
h1=1 1≤h2≤h1ω
which satisfy a chain of functional equations
Fωk (zak
1 z2,z1) =−Fωk−1 (z1,z2)+
z h1
1 zh2 2
z ak+1
1 z2
(1 − z ak
1 z2)(1 − z1) ;
here 0

8 John Coates and Alf van der Poorten
When ω is a quadratic irrational the periodicity of the continued fraction expansion
allows one to compact the chain of functional equations to a single functional
equation, yielding the celebrated result that
∞
⌊hα⌋z h
h=1
is transcendental for all quadratic irrational α and algebraic z satisfying 0

Kurt Mahler 1903–1988 9
transcendence theory, that of LeVeque, has a chapter which is essentially a translation
of [11]). Mahler achieves that by returning to and generalizing the formulae
that allowed Hermite to prove the transcendence of e and π and in effect gives
avery explicit example applying the transcendence method developed by Siegel.
The Padé approximations of [10] foreshadow present important work on effective
approximation of algebraic numbers.
The underlying principle [168] is the following: Let f1,...,fm be power series linearly
independent over C(z). Then for every choice of ρ1,...,ρm non-negative integers
with sum σ , there are (by elementary linear algebra) polynomials a1,...,am ,
not all zero, of respective degrees not exceeding ρ1 −1,...,ρm−1 sothat the linear
form
a1(z)f1(z)+a2(z)f2(z)+···+ am(z)fm(z)
has a zero of order σ − 1atz =0. Ifα1,...,αm are distinct complex numbers
then
a1(z)e α1z + a2(z)e α2z ···+ am(z)e αmz
has a zero of order at most σ − 1atz =0.InMahler’s felicitous terminology [168]
the vector e α1z ,e α2z ,... ,e αmz is a perfect system of functions. It is now a relatively
easy matter to construct m linearly independent polynomial ‘approximations’ to
the vector. Eventually, Shidlovskii showed — this is the essence of the celebrated
Shidlovskii’s lemma — that if f1,...,fm are the solutions of a linear differential
equation with rational function coefficients then there is a defect δ = δ(f), independent
of the ρi ,sothat in any case the order of the zero at z =0 of the linear
form does not exceed σ − 1+δ . Mahler’s paper [168], though written in Groningen
in the mid-1930s, was not published until 1968. In the meantime, his handwritten
manuscript had been the basis of work by Jager in Amsterdam, of the honours
essay of Coates and an important part of van der Poorten’s doctoral thesis; see also
Baker (1966). Other instances of exact constructions include [119] and [164].
Mahler’s celebrated inequality: for integers p, q>0
|π − p/q| >q −42
is proved in [119]. The index 42 was subsequently decreased by various authors
employing minor refinements of the method employed by Mahler. It was a real
surprise when Apéry showed by an elementary method that
see van der Poorten’s story (1979).
|π 2 − p/q| >q −11.851... ;
Geometry of numbers. The term ‘Geometry of Numbers’ was first used by Minkowski
to describe arguments based on considerations of packing and covering. In simple
situations this leads to striking proofs, but it required Mahler’s compactness theorem
of 1946 to systematize and simplify the intuitive considerations of the subject.
The theorem states that the set of lattices in n-dimensional space satisfying
some clearly necessary conditions is compact with respect to a natural topology.
Mahler explores the consequences of his compactness theorem in [82,83,84,87,88].

10 John Coates and Alf van der Poorten
The geometry of numbers of some nonconvex bodies had already been considered
by Davenport, Mordell and others but Mahler gives a systematic general theory,
especially for star bodies. In particular Mahler makes a special study of the consequences
of the existence of an infinite group of automorphisms of a star body. A
recent important result in which Mahler’s compactness theorem plays an essential
role is that of Margulis (1987).
Chabauty (1950) provides a treatment of Mahler’s compactness theorem which is
independent of successive minima and which generalizes to certain algebraic groups;
see also Mumford (1971).
In [79] Mahler proved, independently and almost simultaneously with Hlawka, a
form of the Minkowski-Hlawka theorem. In [126,127,128,129] Mahler introduces the
notion of the p-th compound of a convex body. Any choice of p points from the
body determines a point of the appropriate grassmannian. The p-th compound is
the convex closure. At first this was considered a useless, if interesting, abstraction.
It turned out to be a vital tool in the generalization by Schmidt (1970) of Roth’s
theorem. The special case of polar convex bodies [57] serves for the case n =3 of
Schmidt’s theorem and is applied by Davenport to his study of indefinite quadratic
forms; see also Schmidt (1977).
Mahler’s work [44,53,56] and particularly [57] on Khintchine’s transference theorems
systematizes the observed relationships and shows their natural setting to be convex
bodies, their distance functions and their duals.
Minkowski developed his theory of the reduction of definite quadratic forms in the
context of the geometry of numbers. It seems likely that [55] led Mahler to his work
on transference theorems and also to his compactness theorem. Minkowski had
shown that given a reduced quadratic form aijxixj in n variables and of determinant
D there is a constant λn depending only on n such that λna11 ···ann ≤ D .
Bieberbach and Schur (1928) gave the first weak estimate; it was improved by Remak
(1938) in a difficult paper. In [55] Mahler gives an estimate which applies
to all convex distance-functions. The best possible value of λ3 already appears in
Gauss. In [70] Mahler gives an alternative derivation and in [92] he obtains the best
possible λ4 . The discussion by van der Waerden (1956) shows in particular that
Remak’s estimate for λn can be obtained by a modification of Mahler’s argument.
For more on the matters of this section see the books by Cassels (1959), Rogers
(1964) and Gruber and Lekkerkerker (1987).
Polynomials. Several of Mahler’s papers (for example [143,144,148,150,153,154,156])
are concerned with measures for the size of a polynomial
f(x) =a0X n + ···+ an = a0(X − α1) ···(X − αn)
and thence with measures for the numbers αi defined by f .Aprimary concern is
the study of inequalities, important in transcendence theory, between such quanti-
ties as the classical height
the length of f
max |aj| ,
j
L(f) =|a0| + ···+ |an| ,

Kurt Mahler 1903–1988 11
and the size (spoken of as their ‘house’ by Mahler) of its zeros
α = max |αi| .
i
To that end Mahler introduces [143] the measure, now known as Mahler’s measure,
M(f) =|a0|
1
max(1, |αi|) =exp log |f(e
0
2πit )| dt .
i
One has, congenially,
M(fg)=M(f) M(g) .
Nowadays, if f ∈ Z[X] is irreducible (and a0,...,an are relatively prime) one
defines the absolute logarithmic height h(α) =log H(α) ofazero α of f by
(deg f) h(α) =log M(f) =
max(0, log |α|v) ,
where v runs over the appropriately normalised absolute values of the number
field Q(α). This definition is efficient inter alia because it provides an equitable
treatment of all absolute values, quite eliminating the mystique once possessed
by p-adic generalizations of classical diophantine results. It is a measure of the
conservatism of transcendence theory that the absolute height is still not in universal
use.
Mahler’s measure probably first appears in work of Landau (1905). It is central
to Lehmer’s celebrated question : Suppose f(X) ∈ Z[X]. Then, by a theorem
of Kronecker, M(f) =1if and only if f is a cyclotomic polynomial. Is M(f)
uniformly bounded away from 1 if f is not cyclotomic? The best result known is
Dobrowolski’s inequality
log M(f) ≫
v
log log deg f
log deg f
David Boyd’s work, see for example his survey (1981), is a major contribution
towards this important problem.
p-Adic numbers and p-adic Diophantine Approximation. The p-adic numbers had
been introduced by Hensel at the beginning of the century and Hasse had made
vital use of them in his study of quadratic forms in the early ’twenties. Mahler’s
work helped p-adic numbers to become part of the general mathematical culture.
Amongst Mahler’s lasting contributions is his characterization [139] of continuous
functions f on Zp as sums
x
an with an → 0 .
n
This has become part of the general mathematical culture and plays a fundamental
role in the theory of p-adic L-functions and Iwasawa theory; as an example see
Iwasawa (1972).
3
.

12 John Coates and Alf van der Poorten
Suppose
f(X, Y )=a0X n +a1X n−1 Y +···+an−1XY n−1 +anY n = a0(X−α1Y ) ···(X−αnY )
is a binary form of degree n ≥ 3 defined over Z and irreducible in Z[X, Y ]. Then
the diophantine equation f(X, Y )=m, m some given integer, has only finitely
many solutions (X, Y ) ∈ Z 2 . This is essentially because a solution entails that
some factor X − αY is small:
|X − αY |≪f |m|/Y n−1 ,
and that is impossible for infinitely many Y by Thue’s theorem on the approximation
of algebraic numbers by rationals.
Mahler’s p-adic generalization (in fact of Siegel’s sharpening of Thue’s result) allows
one to replace the given m by an integer composed from the primes of some given
finite set S ; alternatively X , Y may be S -integers: they are allowed denominators
whose factors come from S . This entails results on the greatest prime factor of
integers represented by binary forms [17,20]; or that there are only finitely many
S -integral points on a curve of genus 1 [21]. One refers to the generalized equation
as the Thue-Mahler equation. Indeed, generally, the suffix ‘-Mahler’ signals the
generalization of a diophantine problem to one in S -integers. In the present spirit,
Mahler and Lewis [147] provide bounds for the number of solutions (X, Y ) ∈ Z2 of f(X, Y )=m as m varies in terms of the number of prime factors of m. This
problem has been studied more recently by Bombieri and Schmidt (1987).
In an interesting application, Mahler [135] employs a 2-adic argument to obtain a
lower bound for the fractional part of (3/2) k —aresult settling the value of g(k)
in Waring’s problem. However, Mahler’s solution is ineffective in k . More recent
effective arguments, using p-adic versions of Baker’s inequalities, are too weak to
apply to Waring’s problem.
Mahler initiated the now burgeoning study of transcendence properties of p-adic
analytic functions with his paper [14] on the p-adic exponential function and his
proof [30] of the p-adic analogue of the Gelfond-Schneider theorem on the transcendence
of αβ for algebraic α = 0, 1 and irrational algebraic β .
In 1933 Skolem deduced results about the number of solutions of certain diophantine
problems from a new kind of series expansion. Mahler concluded that in essence
the method was p-adic and applied similar techniques to prove the beautiful result
[25,31] that the zero Taylor coefficients of a rational function defined over an algebraic
number field occur periodically (from some point on). Eventually Lech and
independently Mahler [131,131a] generalized the result to arbitrary fields of definition
of characteristic zero. Cassels (1976) gives an elegant simplified treatment. For
results placing the Lech-Mahler theorem in context, see van der Poorten’s survey
(1989).
In [139,145] Mahler proves a criterion for a function defined on the positive integers
to have an interpolation continuing it to a continuous function on the p-adic integers.
This result is basic for the modern theory of p-adic L-functions. Mahler’s

Kurt Mahler 1903–1988 13
book (an extended second edition of [184]) studies the elementary analysis of p-adic
functions defined in this way.
The paper [17] includes a proof that if S0 , S1 , S2 are disjoint nonempty finite sets
of rational primes then an equation
z0 + z1 + z2 =0,
where zi is only divisible by primes in Si , has only finitely many solutions in integers
z0,z1,z2 . This is the genesis of the important and presently fashionable study
of S -unit equations initiated by remarks of van der Poorten and Schlickewei, and
of Evertse. Dubois and Rhin, and Schlickewei had applied p-adic generalizations of
Schmidt’s subspace theorem to extend Mahler’s result to general S -unit equations
z0 + z1 + ···+ zn =0,
settling a conjecture of Mahler. The more recent results eliminate the requirement
that the sets of primes be pairwise disjoint by asking for primitive solutions (in an
appropriate sense) of the S -unit equation. There are numerous applications; see
the survey of Evertse, Győry, Stewart and Tijdeman.
Mahler regarded p-adic numbers as a special case of g -adic numbers defined by
a pseudo-valuation [34,37,38,39]. Here g is any positive integer and, it turns out,
the g -adic completion is the product of the p-adic completions over the primes p
dividing g .
Decimal expansions. Mahler regretted that, apart from his own work, little interest
had been shown by twentieth century mathematicians in the study of arithmetic
properties of decimal expansions. Reaction to his papers [191, 203, 207] is beginning
to repair that neglect. It appears that Mahler was the first to conjecture that
an irrational algebraic number has a normal decimal expansion. Other than for
experimental support little is as yet known on this beyond the result of Loxton
and van der Poorten op cit to the effect that such expansions are at any rate not
generated by a finite automaton.
Unexpectedly, perhaps, Mahler’s p-adic generalization of the Thue-Siegel theorem
allowed him to prove the following amusing but striking result [46]: Suppose f is
a non-constant polynomial taking integer values at the nonnegative integers. Then
the concatenated decimal
φ =0.f(1)f(2)f(3) ...
is transcendental. In particular Champerknowne’s normal number
0.12345678910111213 ...
is transcendental. Mahler’s argument relies on the observation that one readily
obtains rational approximations to φ with denominators high powers of the base
10, thus composed of the primes 2 and 5 alone. Perhaps disappointingly, Roth’s
definitive form of the Thue-Siegel inequalities permits a more immediate argument
obviating the need for an appeal to the p-adic results.
In his ‘post-retirement’ years Mahler made extensive use of his TI-calculator to
study digital patterns [218,221]. His desk was covered with detailed such calculations
at the time of his death.

Kurt Mahler
Bibliography
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(3) 1928 Über einen Satz von Mellin. Math. Ann. 100, 384–398
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(6) Zur Fortsetzbarkeit gewisser Dirichletscher Reihen. Math. Ann. 102, 30–48
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(8) Arithmetische Eigenschaften einer Klasse transzendental-transzendente Funktionen.
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(9) Über Beziehungen zwischen der Zahl e und Liouvilleschen Zahlen. Math. Z. 31,
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(10) 1931 Ein Beweis des Thue-Siegelschen Satzes über die Approximation algebraischer
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(11) Zur Approximation der Exponentialfunktion und des Logarithmus. Teil I. J. für
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(12) 1932 Über das Maß der Menge aller S -Zahlen. Math. Ann. 106, 131–139
(13) Zur Approximation der Exponentialfunktion und des Logarithmus. Teil II. J. für
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(14) Ein Beweis der Transzendenz der P -adischen Exponentialfunktion. J. für Math.
169, 61–66
(15) Einige Sätze über Diophantische Approximationen. Jahresbericht d. Deutschen
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(16) Über die Darstellung von Zahlen durch Binärformen höheren Grades, Kongressbericht,
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(17) 1933 Zur Approximation algebraischer Zahlen. I ( Über den grössten Primteiler binärer
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(18) Zur Approximation algebraischer Zahlen II ( Über die Anzahl der Darstellungen
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(19) Zur Approximation algebraischer Zahlen III ( Über die mittlere Anzahl der Darstellungen
grosser Zahlen durch binäre Formen). Acta Math. 62, 91–166

– 15 –
(20) 1933 Über den grössten Primteiler der Polynome X2 ∓1.Archiv Math. og. Naturv. 41,
1–8
(21) Über die rationalen Punkte auf Kurven vom Geschlecht Eins. J. für Math. 170,
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(22) 1934 Zur Approximation P -adischer Irrationalzahlen. Nieuw Arch. Wisk. 18, 22–34
(23) Über die Darstellungen einer Zahl als Summe von drei Biquadraten. Mathematica
(Zutphen) 3, 69–72
(24) Über Diophantische Approximationen in Gebiete der p-adische Zahlen. Jahresbericht
d. Deutschen Math. Verein. 44, 250–255
(25) Eine arithmetische Eigenschaft der recurrierenden Reihen. Mathematica (Zutphen)
3, 153–156
(26) 1935 On Hecke’s theorem on the real zeros of the L-functions and the class number of
quadratic fields. J. London Math. Soc. 9, 298–302
(27) Über eine Klasseneinteilung der P -adischen Zahlen. Mathematica (Zutphen) 3B,
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(28) On the lattice points on curves of genus 1. Proc. London Math. Soc. 39, 431–466
(29) On the division-values of Weierstrass’s ℘-function. Quart. J. Math. Oxford
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(30) Über transzendente P -adische Zahlen. Compositio Math. 2, 259–275; A Correction,
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(31) Eine arithmetische Eigenschsaft der Taylor-Koeffizienten rationaler Funktionen.
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(32) Über den grössten Primteiler spezieller Polynome zweiten Grades. Archiv Math.
og. Naturv. 41, Nr 6, 3–26
(33) (With J. Popken) Ein neues Prinzip für Transzendenzbeweise. Proc. Kon. Nederlandsche
Akad. v. Wetenschappen 38, 864–871
(34) Über Pseudobewertungen, I. Acta Math. 66, 79–119
(35) 1936 Eine arithmetische Eigenschaft der kubischen Binärformen. Nieuw Arch. Wisk.
18, 1–9
(36) Ueber Polygone mit Um– oder Inkreis. Mathematica (Zutphen) 4A, 33–42
(37) Über Pseudobewertungen II. Acta Math. 67, 51–80
(38) Über Pseudobewertungen III. Acta Math. 67, 283–328
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(40) Note on Hypothesis K of Hardy and Littlewood. J. London Math. Soc. 11, 136–
138
(41) Ein Analog zu einem Schneiderschen Satz, I. Proc. Kon. Nederlandsche Akad. v.
Wetenschappen 39, 633–640
(42) Ein Analog zu einem Schneiderschen Satz, II. Proc. Kon. Nederlandsche Akad. v.
Wetenschappen 39, 729–737

Kurt Mahler
(42a) 1936 Pseudobewertungen. Proc. International Congress of Mathematicians, Oslo, 1936
1p.
(43) 1937 Über die Annäherung algebraischer Zahlen durch periodische Algorithmen. Acta
Math. 68, 109–144
(44) Neuer Beweis eines Satzes von A. Khintchine. Mat. Sb. I, 43, 961–963
(45) Über die Dezimalbruchentwicklung gewisser Irrationalzahlen. Mathematica (Zutphen)
B6, 22–26
(46) Arithmetische Eigenschaften einer Klasse von Dezimalbrüchen. Proc. Kon. Nederlandsche
Akad. v. Wetenschappen 40, 421–428
(47) 1938 (With P. Erdős) On the number of integers which can be represented by a
(48)
binary form. J. London Math. Soc. 13, 134–139
On a special class of Diophantine equations, I. J. London Math. Soc. 13, 169–173
(49) On a special class of Diophantine equations, II. J. London Math. Soc. 13, 173–
177
(50) Ein P -adisches Analogon zu einem Satz von Tchebycheff. Mathematica (Zutphen)
B7, 2–6
(51) On the fractional parts of the powers of a rational number. Acta Arith. 3, 89–93
(52) Über einen Satz von Th. Schneider. Acta Arith. 3, 94–101
(53) A theorem on inhomogenous Diophantine inequalities. Proc. Kon. Nederlandsche
Akad. v. Wetenschappen 41, 634–637
(54) Eine Bemerkung zum Beweis der Eulerschen Summenformel. Mathematica (Zutphen)
B7, 33–42
(55) On Minkowski’s theory of reduction of positive definite quadratic forms. Quart. J.
Math. Oxford 9, 259–262
(56) 1939 Ein Übertragungsprinzip für lineare Ungleichungen. Math. Časopis 68, 85–92
(57) Ein Übertragungsprinzip für konvexe Körper. Math. Časopis 68, 93–102
(58) (With P. Erdős) Some arithmetical properties of the convergents of a continued
fraction. J. London Math. Soc. 14, 12–18
(59) Ein Minimalproblem für convexe Polygone. Mathematica (Zutphen) B7, 118–127
(60) On a the solutions of algebraic differential equations. Proc. Kon. Nederlandsche
Akad. v. Wetenschappen 42, 61–63
(61) A proof of Hurwitz’s theorem. Mathematica (Zutphen) 8B, 57–61
(62) Bemerkungen über die Diophantischen Eigenschaften der reellen Zahlen. Mathematica
(Zutphen) 8B, 11–16
(63) 1939 On the minimum of positive definite Hermitian forms. J. London Math. Soc. 14,
137–143
(64) 1940 On a geometrical representation of p-adic numbers. Ann. of Math. (2) 41, 8–56
(65) On the product of two complex linear polynomials in two variables. J. London
Math. Soc. 15, 213–236 [Mahler writes “This is an extension of an earlier paper

– 17 –
which was accepted for publication by Acta Arith. in February 1939 of which I
had just received the first proofs when war broke out”.]
(66) 1940 Note on the sequence √ n (mod 1). Nieuw Arch. Wisk. 20, 176–178
(67) (With G. Billing) On exceptional points on cubic curves. J. London Math. Soc.
15, 32–43
(68) On a special functional equation. J. London Math. Soc. 15, 115–123
(69) Über Polynome mit ganzen rationalen Koeffizienten. Mathematica (Zutphen) 8B,
173–182
(70) On reduced positive definite ternary quadratic forms. J. London Math. Soc. 15,
193–195
(71) On a property of positive definite ternary quadratic forms. J. London Math. Soc.
15, 305–320
(72) 1941 An analogue to Minkowski’s Geometry of Numbers in a field of series. Ann. of
Math. (2), 42, 488–522
(73) 1942 On ideals in the Cayley-Dickson algebra. Proc. Royal Irish Academy 48, 123–133
(74) Remarks on ternary Diophantine equations. Amer. Math. Monthly 49, 372–378
(75) Note on lattice points in star domains. J. London Math. Soc. 17, 130–133
(76) 1943 On lattice points in an infinite star domain. J. London Math. Soc. 18, 233–238
(77) 1944 A problem of Diophantine approximation in quaternions. Proc. London Math.
Soc.(2) 48, 435–466
(78) (With B. Segre) On the densest packing of circles. Amer. Math. Monthly
261–270
51,
(79) On a theorem of Minkowski on lattice points in non-convex point sets. J. London
Math. Soc. 19, 201–205
(80) On lattice points in the domain |xy| ≤1, |x + y| ≤ √ 5, and applications to
asymptotic formulae in lattice point theory, I. Proc. Cambridge Phil. Soc. 40,
(81)
107–116
On lattice points in the domain |xy| ≤1, |x + y| ≤ √ 5, and applications to
asymptotic formulae in lattice point theory, II. Proc. Cambridge Phil. Soc.
116–120
40,
(82) 1945 A theorem of B. Segre. Duke Math. J. 12, 367–371
(83) 1946 Lattice points in two-dimensional star domains, I. Proc. London Math. Soc.
128–157
49,
(84) Lattice points in two-dimensional star domains, II. Proc. London Math. Soc. 49,
158–167
(85) Lattice points in two-dimensional star domains, III. Proc. London Math. Soc. 49,
168–183
(86) On lattice points in a cylinder. Quart. J. Math. Oxford 17, 16–18
(87) On lattice points in n-dimensional star bodies, I, Existence theorems. Philos.
Trans. Roy. Soc. London Ser. A 187, 151–187

Kurt Mahler
(88) 1946 Lattice points in n-dimensional star bodies. II. Reducibility theorems. I, II, III,
IV. Proc. Kon. Nederlandsche Akad. v. Wetenschappen 49, 331–343, 444–454,
(89)
524–532, 622–631 = Indag. Math. 8, 200–212, 299–309, 343–351, 381–390
(With H. Davenport) Simultaneous Diophantine approximation. Duke Math. J.
13, 105–111
(90) Lattice points in n-dimensional star bodies. Rev. Univ. Nac. Tucumán, A
113–124
5,
(91) The theorem of Minkowski-Hlawka. Duke Math. J. 13, 611–621
(92) On reduced positive definite quarternary quadratic forms. Nieuw Arch. Wisk.
207–212
(93) 1947 A remark on the continued fractions of conjugate algebraic numbers. Simon Stevin
25, 45–48
(94) On irreducible convex domains. Proc. Kon. Nederlandsche Akad. v. Wetenschappen
50, 98–107 = Indag. Math. 9, 73–82
(95) On the area and the densest packing of convex domains. Proc. Kon. Nederlandsche
Akad. v. Wetenschappen 50, 108–118 = Indag. Math. 9, 83–93
(96) On the generating functions of integers with a missing digit [in Chinese]. K’o Hsüeh
Science 29, 265–267
(97) On the adjoint of a reduced positive definite ternary quadratic form. Sci. Rec.
Academia Sinica 2, 21–31
(98) On the minimum determinant and the circumscribed hexagons of a convex domain.
Proc. Kon. Nederlandsche Akad. v. Wetenschappen
9, 326–337
50, 692–703= Indag. Math.
(99) 1948 (With K.C. Hallum) On the minimum of a pair of positive definite Hermitean
forms. Nieuw Arch. Wisk. 22, 324–354
(100) On the admissible lattices of automorphic star bodies. Sci. Rec. Academia Sinica
2, 146–148
(101) On lattice points in polar reciprocal convex domains. Proc. Kon. Nederlandsche
Akad. v. Wetenschappen 51, 176–179 = Indag. Math. 10, 482–485
(102) Sui determinante minimi delle sezione di un corpo convesso. Atti Accad. Naz.
Lincei Cl. Sci. Fis. Mat. Natur. (8) 5, 251–252
(103) On the successive minima of a bounded star domain. Ann. Mat. Pura Appl. (4)
27, 153–163
(104) 1949 On the critical lattices of arbitrary point sets. Canad. J. Math. 1, 78–87
(105) On the minimum determinant of a special point set. Proc. Kon. Nederlandsche
Akad. v. Wetenschappen 52, 633–642 = Indag. Math. 11, 959–968
(106) On a theorem of Liouville in fields of positive characteristic. Canad. J. Math. 1,
397–400
(107) On Dyson’s improvement of the Thue-Siegel theorem. Proc. Kon. Nederlandsche
Akad. v. Wetenschappen 52, 449–458 = Indag. Math. 11, 1175–1184
(108) (With W. Ledermann) On lattice points in a convex decagon. Acta Math. Acad.
Sci. Hungar. 81, 319–351

– 19 –
(109) 1949 On the continued fractions of quadratic and cubic irrationals. Ann. Mat. Pura
Appl. (4) 30, 147–172
(109a) A correction to “On the continued fractions of quadratic and cubic irrationals”.
Compositio Math. 8, p112
(110) On the arithmetic on algebraic curves, in Seminar on Algebra and Number Theory,
Chicago. 28–32
(111) On algebraic relations between two units of an algebraic field, in “Algèbre et théorie
des nombres”. Colloques internationaux du CNRS, CNRS, Paris, 1950 24, 47–55
(112) 1950 On a theorem of Dyson [in Russian]. Mat. Sb. 26, 457–462
(113) Geometry of numbers. [Lectures given at the University of Colorado in the
(114) 1951
summer of 1950 (mimeographed notes).]
(With J.W.S. Cassels and W. Ledermann) Farey section in k(i) and k(ρ).
Philos. Trans. Roy. Soc. London Ser. A 243, 585–626
(114a) 1952 Farey section in the fields of Gauß and Eisenstein. Proc. International Congress
Cambridge, Mass. 1950 1, 281–285
(115) 1951 On the generating function of the integers with a missing digit. J. Indian Math.
Soc. 15A, 34–40 [Translation of (96)]
(116) On a question in elementary geometry. Simon Stevin 28, 90–97
(117) 1953 On the lattice determinants of two particular point sets. J. London Math. Soc.
28, 229–232
(118) On the approximation of logarithms of algebraic numbers. Philos. Trans. Roy.
Soc. London Ser. A 245, 371–398
(119) On the approximation of π . Proc. Kon. Nederlandsche Akad. v. Wetenschappen
Ser. A 56, 30–42 = Indag. Math. 15, 30–42
(120) (With J. Popken) Over een Maximumprobleem uit de Rekenkunde. Nieuw Arch.
Wisk. (3) 1, 1–15
(121) On the greatest prime factor of ax m + by n . Nieuw Arch. Wisk. (3) 1, 113–122
(122) (With P. M. Cohn) On the composition of pseudo-valuations. Nieuw Arch. Wisk.
(3) 1, 161–198
(123) 1954 A problem in elementary geometry. Math. Gaz. 37, 241–243
(124) On a problem in the geometry of numbers. Rendi. Mat. Appl. (5) 14, 38–41
(125) 1955 On a problem in Diophantine approximations. Archiv Math. og. Naturv. 6, 208–
214
(126) On compound convex bodies, I. Proc. London Math. Soc. (3) 5, 358–379
(127) On compound convex bodies, II. Proc. London Math. Soc. (3) 5, 380–384
(128) The pth compound of a sphere. Proc. London Math. Soc. (3) 5, 385–391
(129) On the minima of compound quadratic forms. Czech. Math. J. (5) 80, 180–193
(130) A remark on Siegel’s theorem on algebraic curves. Mathematika 2, 116–127
(131) 1956 On the Taylor coefficients of rational functions. Proc. Cambridge Phil. Soc. 52,
39–48

Kurt Mahler
(131a) 1957 Addendum to “On the Taylor coefficients of rational functions”. Proc. Cambridge
Phil. Soc. 53, p544
(132) 1956 Invariant matrices and the geometry of numbers. Proc. Roy. Soc. Edinburgh A
64, 223–238
(133) A property of the star domain |xy| ≤1. Mathematika 3, p80
(134) 1957 Über die konvexen Körper, die sich einem Sternkörper einbeschrieben lassen. Math.
Z. 66, 25–33
(135) On the fractional parts of the powers of a rational number, II. Mathematika
122–124
4,
(136) A matrix representation of the primitive residue classes
Math. Soc. 8, 525–531
(mod 2n). Proc. Amer.
(137) 1958 A factorial series for the rational multiples of e. Math. Gaz. 42, 13–16
(138) On the Chinese remainder theorem. Math. Nachr. 18, 120–122
(139) An interpolation series for continuous functions of a p-adic variable. J. für Math.
199, 23–34
(140) 1959 An arithmetic property of groups of linear transformations. Acta Arith. 5, 197–
203
(141) On a theorem of Shidlovski, lectures given at the Math. Centrum, Amsterdam,
December 1959.
(142) 1960 On a theorem by E. Bombieri. Proc. Kon. Nederlandsche Akad. v. Wetenschappen
Ser. A 63, 245–253 = Indag. Math. 22, 245–253
(142a) On a theorem by E. Bombieri. Proc. Kon. Nederlandsche Akad. v. Wetenschappen
Ser. A 64, 141= Indag. Math. 23, 141
(143) An application of Jensen’s formula to polynomials. Mathematika 7, 98–100
(144) 1961 On the zeros of the derivative of a polynomial. Philos. Trans. Proc. Roy. Soc.
London Ser. A 264, 145–154
(145) A correction the the paper “An interpolation series for continuous functions of a
p-adic variable”, [Seq. No. 139]. J. für Math. 208, 70–72
(146) 1961 Lectures on Diophantine Approximations I : g -adic Numbers and Roth’s Theorem,
Lectures given at the University of Notre Dame in 1957, University of Notre Dame
Press, Notre Dame, Indiana, 188 pages.
(147) (With D.J. Lewis) On the representation of integers by binary forms. Acta Arith.
6, 333–363
(148) 1962 On some inequalities for polynomials in several variables. J. London Math. Soc.
37, 341–344
(149) Geometric number theory, Lectures given at the University of Notre Dame in 1962
(bound mimeographed notes). 151 pp
(150) 1963 On two extremum properties of polynomials. Illinois J. Math. 7, 681–701
(151) On the approximation of algebraic numbers by algebraic integers. J. Austral. Math.
Soc. 3, 408–434
(152) 1964 Periodic algorithms for algebraic number fields (mimeographed lecture notes). 4th
Summer Research Institute of the Austral. Math. Soc. University of Sydney, January
1964 19pp

– 21 –
(153) 1964 A remark on a paper of mine on polynomials. Illinois J. Math. 8, 1–4
(154) An inequality for the discriminant of a polynomial. Michigan Math. J. 11, 257–
262
(155) Transcendental numbers. J. Austral. Math. Soc. 4, 393–396
(156) An inequality for a pair of polynomials that are relatively prime. J. Austral. Math.
Soc. 4, 418–420
(157) Inequalities for ideal bases in algebraic number fields. J. Austral. Math. Soc.
425–448
4,
(158) 1965 (With G. Baumschlag) Equations in free metabelian groups. Michigan Math. J.
12, 417–419
(159) Arithmetic properties of lacunary power series with integral coefficients. J. Austral.
Math. Soc. 5, 56–64
(160) 1966 A remark on recursive sequences. J. Math. Sci. Delhi 1, 12–17
(161) Transcendental numbers. Encyclopaedia Brittannica, 1 column
(162) A remark on Kronecker’s theorem. Enseign Math. 12, 183–189
(163) 1967 (With G. Szekeres) On the approximation of real numbers by roots of integers.
Acta Arith. 12, 315–320
(164) Applications of some formulae by Hermite to the approximation of exponentials
and logarithms. Math. Ann. 168, 200–227
(165) On a class of entire functions. Acta Math. Acad. Sci. Hungar. 18, 83–96
(166) On a lemma by A. B. Shidl’ovski [in Russian]. Math. Zametki Acad. Nauk SSR 2,
25–32
(167) 1968 An unsolved problem on the powers of 3/2. J. Austral. Math. Soc. 8, 313–321
(168) Perfect systems. Compositio Math. 19, 95–166 [First publication of a manuscript
written in the 1930s]
(169) Applications of a theorem by A. B. Shidlovski. Philos. Trans. Roy. Soc. London
Ser. A 305, 149–173
(170) 1969 Remarks on a paper by W. Schwarz. J. Number Theory 1, 512–521
(171) On algebraic differential equations satisfied by automorphic functions. J. Austral.
Math. Soc. 10, 445–450
(172) Lectures on transcendental numbers (Summer Institute on Number Theory at
Stony Brook, 1969). Proc. Symp. Pure Math. (Amer. Math. Soc.) XX, 248–274
(173) 1971 A lecture on the geometry of numbers of convex bodies. Bull. Amer. Math. Soc.
77, 319–325
(174) (With H. Brown) A generalization of Farey sequences. J. Number Theory
364–370
3,
(175) On formal power series as integrals of algebraic differential equations. Rend. Accad.
Naz. dei Lincei (8) 50, 36–49
(176) A remark on algebraic differential equations. Rend. Accad. Naz. dei Lincei (8) 50,
174–184

Kurt Mahler
(177) 1971 An arithmetical remark on entire functions. Bull. Austral. Math. Soc. 5, 191–195
(178) An elementary existence theorem for entire functions. Bull. Austral. Math. Soc.
5, 415–419
(179) On the order function of a transcendental number. Acta Arith. 18, 63–76
(180) Transcendental numbers. I. Existence and chief characteristics of transcendental
numbers [in Hungarian]. Mat. Lapok 22, 31–50
(181) 1972 Transcendental numbers. II. Convergence of Laurent series and formal Laurent
series [in Hungarian]. Mat. Lapok 23, 13–23
(182) On the coefficients of the 2n -th transformation polynomial for j(ω). Acta Arith.
21, 89–97
(182a) Obituary, L. J. Mordell. J. Number Theory 4, iii–iv
(183) 1973 The classification of transcendental numbers. Proc. Symp. Pure Math. (Amer.
Math. Soc.) XXIV, 175–179
(184) Introduction to p-adic Numbers and their Functions, Cambridge Tracts in Mathematics
64, Cambridge University Press, London/New York. 89pp
(185) Arithmetical properties of the digits of the multiples of an irrational number. Bull.
Austral. Math. Soc. 8, 191–203
(186) On a class of Diophantine inequalities. Bull. Austral. Math. Soc. 8, 247–259
(187) A p-adic analogue of a theorem by J. Popken. J. Austral. Math. Soc. 16, 176–184
(188) 1974 On the coefficients of the transformation polynomials for the modular function.
Bull. Austral. Math. Soc. 10, 197–218
(189) On rational approximations of the exponential function at rational points. Bull.
Austral. Math. Soc. 10, 325–335
(190) Polar analogues of two theorems by Minkowski. Bull. Austral. Math. Soc.
121–129
11,
(191) On the digits of the multiples of an irrational p-adic number. Proc. Cambridge
Phil. Soc. 76, 417–422
(192) How I became a mathematician. Amer. Math. Monthly 81, 981–983
(193) 1975 On a paper by A. Baker on the approximation of rational powers of e. Acta Arith.
27, 61–87
(194) A necessary and sufficient condition for transcendency. Math. Comp. 29, 145–153
(195) On the transcendency of the solutions of a special class of functional equations.
Bull. Austral. Math. Soc. 13, 389–410
(196) 1976 On certain non-archimedean functions analogous to complex analytic functions.
Bull. Austral. Math. Soc. 14, 23–36
(197) An addition to a note of mine. Bull. Austral. Math. Soc. 14, 397–398
(198) A theorem on Diophantine approximations. Bull. Austral. Math. Soc.
465
14, 463–
(199) Corrigendum to “On the transcendency of the solutions of a special class of functional
equations”. Bull. Austral. Math. Soc. 15, 477–478

– 23 –
(200) 1976 Lectures on Transcendental Numbers, Edited and completed by B. Divis and W. J.
LeVeque, Lecture Notes in Mathematics 546 , Springer-Verlag, Berlin/New York.
254pp
(201) On a class of non-linear functional equations connected with modular functions.
J. Austral. Math. Soc. Ser. A 22, 65–118
(202) On a class of transcendental decimal fractions. Comm. Pure Appl. Math.
717–725
29,
(203) 1977 On some special decimal fractions, in Hans Zassenhaus ed., Number Theory and
Algebra, Academic Press, New York, 1977. 209–214
(204) 1980 On a special function. J. Number Theory 12, 20–26
(205) 1981 p-Adic Numbers and Their Functions (), Cambridge Tracts in Mathematics 76,
Cambridge University Press, London/New York.
duction to p-adic Numbers and Their Functions]
320pp [2nd edition of Intro-
(206) 1980 On two definitions of the integral of a p-adic function. Acta Arith. 47, 105–109
(207) 1981 On some irrational decimal fractions. J. Number Theory 13, 268–269
(208) On a special non-linear functional equation. Philos. Trans. Roy. Soc. London Ser.
A 378, 155–178
(209) 1982 Fifty years as a mathematician. J. Number Theory 14, 121–155 [The present
publication list is a correction and extension of the one appearing here.]
(210) On a special transcendental number. Arithmétix
of number theorists in France]
5, 18–32 [Information bulletin
(211) On the zeros of a special sequence of polynomials. Math. Comp. 39, 207–212
(212) 1983 On a theorem in the geometry of numbers in a space of Laurent series. J. Number
Theory 17, 403–416
(213) On the analytic solution of certain functional and difference equations. Proc. Roy.
Soc. London Ser. A 389, 1–13
(214) Warum ich eine besondere Vorliebe für die Mathematik habe. Jber. d. Deutschen
Math.-Verein. 85, 50–53 [Essay originally written in 1923]
(215) 1984 On Thue’s theorem. Math. Scand. 55, 188–200
(216) Some suggestions for further research. Bull. Austral. Math. Soc. 29, 101-108
(217) 1986 (With D. H. Lehmer and A. J. van der Poorten) Integers with digits 0 or
1. Math. Comp. 46, 683–689
(218) A new transfer principle in the geometry of numbers. J. Number Theory
20–34
24,
(219) The successive minima in the geometry of numbers and the distinction between
algebraic and transcendental numbers. J. Number Theory 22, 147–160
(220) 1987 On two analytic functions. Acta. Arith. XLIX, 15–20
(221) 1989 The representation of squares to the base 3. Acta Arith. 53, 99–106
(222) 1991 Fifty years as a mathematician II. J. Austral. Math. Soc. (Series A) 51, 366–380
[Appendix 2 of A. J. van der Poorten, Obituary of Kurt Mahler. ibid., 343–380]

John Coates and Alf van der Poorten
References to other authors
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Amsterdam.

– 25 –
C. Hermite (1873), ‘Sur la fonction exponentielle’, Oeuvres, t.III, 151–181.
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John H. Coates
Department of Pure Mathematics and Mathematical Statistics
16 Mill Lane
Cambridge CB2 1SB
jhc13@dpmms.cambridge.ac.uk
Alfred J. van der Poorten
Centre for Number Theory Research
Macquarie University NSW 2109
Australia
alf@macadam.mpce.mq.edu.au