Cotangent Sums Related to the Riemann Hypothesis for Various Shifts of the Argument

Helmut Maier,Michael Th Rassias

doi:10.4153/s0008439519000602

Abstract

AbstractOne of the approaches to the Riemann Hypothesis is the Nyman–Beurling criterion. Cotangent sums play a significant role in this criterion. Here we investigate the values of these cotangent sums for various shifts of the argument.

Highlights

In several papers ([10], [11], [12], [13], [14]) the authors have investigated the distribution of the cotangent sums c0 r b := −b−1 m b cot πmr b, m=1 where r, b ∈ N, b ≥ 2, 1 ≤ r ≤ b and (r, b) = 1
M=1 where r, b ∈ N, b ≥ 2, 1 ≤ r ≤ b and (r, b) = 1. They could establish a link with the function g(α), defined by
In [15] the authors considered the distribution of the values of c0 for rational numbers with primes as numerator and a fixed prime as denominator and could prove a result analogous to (1.1) (Theorem 1.3 of [15])

Summary

Introduction

In several papers ([10], [11], [12], [13], [14]) the authors have investigated the distribution of the cotangent sums c0 r b. The second author in his thesis [16] (see [10], Theorem 1.2) could establish the following result: Let A0, A1 be fixed constants, such that 1/2 < A0 < A1 < 1. In [15] the authors considered the distribution of the values of c0 for rational numbers with primes as numerator and a fixed prime as denominator and could prove a result analogous to (1.1) (Theorem 1.3 of [15]). They proved that for all f ∈ C0(R), the following holds true: lim q→+∞ q prime log q (A1 − A0)q f p : A0q≤p≤A1q. We give only a detailed proof of (i) and sketch the changes needed for the proof of (ii)

Outline of the proof

Exponential sums in finite fields

Decomposition of the sums Q

Comparison of Q0 and g

Conclusion of the proof and concluding remarks