Modifications of the Harmonic Series - Department of Mathematics

th
where p i is the i prime. Now define k ( ) n λ as the number of integers less than n
whose prime factors are all less than or equal to
p . We also define μ ( ) as the
number of integers less than whose prime factors contain at least one
Then by definition,
k
k n
n i
λ ( n) + μ ( n) = n
k k
p , i> k.
Note that any of the numbers enumerated by λ ( ) can be represented as
A1Ak2 p1 ⋅⋅⋅ p ⋅s , A ∈{0,1} . There are 2 ways to choose the square free part, so
k
k i
( ) 2 k
λ ≤ n. Now the number of integers less than that are divisible by
k n
⎢ n ⎥
⎢ ⎥.
Then
p
⎣ i ⎦
Combining equations (1) and (2) we get,
But values of n greater than or equal to
n n
μk
( n)
≤ ∑ < .
p 2
i> k i
k n
n
k
< λk
( n) ≤2 n ∀n∈ .
2
3.2 General Conditions for Convergence
n
n i
2 2
2 k +
= give a contradiction.
(1)
p is
In his 1941 paper, I. E. Perlin established sufficient conditions for a
modification of any series to converge [7]. We will present his results here, along
with some of his applications to the harmonic series, and further, we will apply the
results to several other modifications of the harmonic series.
15
(2)