Reciprocal Sums and Counting Functions

Abstract

Motivated by the gentle exploration of the distribution of prime numbers typical of an undergraduate number theory course, as well as by a recent breakthrough result in arithmetic combinatorics, we explore connections between the counting function, which counts the number of elements up to a threshold x, and the reciprocal sum function, which adds the reciprocals of the elements up to a threshold x, for subsets of the natural numbers.

MSC::

• Primary 11N05
• Secondary 11B30

Acknowledgment

Proposition 7 was inspired by Chapter 10 of [4], and also appears as an exercise in Chapter 3 of [12]. In the special case where $S(x)=\hspace{0.17em}\text{log}\hspace{0.17em}\hspace{0.17em}\text{log}\hspace{0.17em}x-C$, exercises in Chapter 3 of [10] are similar to Propositions 4 and 5.