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Your disproof essentially relies on computer computations. What do you
think, which role does and should the computer play in mathematics?
I have written on this topic before (for example in ”
Applications of symbolic
mathematics to mathematics,“ pp. 95–111 in ”
Applications of Computer
Algebra,“ R. Pavelle, ed., Kluwer, 1985, and available on my home page
at http://www.research.att.com/∼amo). The role of computers in mathematics
is increasing rapidly. They can be used to help in proving theorems,
as in the case of Herman te Riele’s and my work on the Mertens conjecture.
However, it is worth keeping in mind that the main purpose of computing in
mathematics should be insight, not numbers (to quote Richard Hamming).
Symbolic algebra allows us to avoid laborious manipulations, by performing
routine integrations and algebraic operations. Numerical routines offer similar
assistance in solving linear algebra or optimization problems. This frees
the human mind to do more creative work. Further, computers allow us to
explore mathematical structures much more thoroughly than we can mentally
or with paper and pencil. This allows us to develop our intuition, guess
what the underlying principles are, and prove theorems about them.
What is your guess for the Riemann conjecture? Do you think that this
problem can be solved in the near future?
It is either true or false, and I do not hazard to guess which it is. I also do
not venture to guess when it might be solved. I have not yet seen any ideas
that seemed likely to lead to a proof of the Riemann Hypothesis in the near
future, but that does not mean that one will not be found tomorrow. On the
other side, if there are any counterexamples, my guess is they are located
far above the regions where we are able to compute the zeta function.
Of course, it is difficult to give ”
prophetic answers“ but do you think
that a positive solution of the Riemann conjecture would have a big impact
on analytic number theory?
It would have a profound effect on all of mathematics. It certainly would
revolutionize analytic number theory, and could also have a huge impact on
areas such as computational number theory, although how that would work
out would depend very much on the nature of the proof.
Prof. Odlyzko, you are working in so many different fields, analytic
and computational number theory, analytic combinatorics, etc. You combine
theory and practice. Do you think that there is a fruitful balance of theory
and applications in modern mathematics or should there be a change?
Overall, I suspect the balance between theory and applications (although
those are not the ideal terms) will have to shift towards applications. The
rapidly developing Information Society offers many opportunities for mathematics,
and if mathematicians do not seize them, others, whether engineers
or physicists, will do so.
Thank you very much for this interview.
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