Estimates of Newman Sum over Numbers Multiple of a Fixed Integer

TL;DR

This paper investigates the asymptotic behavior of Newman sums over multiples of fixed integers, revealing a ratio tending to zero for certain cases, and explores connections to a prime digit conjecture.

Contribution

It proves that the ratio of Newman sums over multiples of fixed integers not divisible by 3 to those divisible by 3 tends to zero as the limit increases, linking to a prime digit conjecture.

Findings

Ratio of Newman sums tends to zero for certain fixed integers

Establishes a connection between Newman sums and a prime digit conjecture

Provides asymptotic analysis of Newman sums over multiples of fixed integers

Abstract

We prove that the ratio of the Newman sum over numbers multiple of a fixed integer which is not multiple of 3 and the Newman sum over numbers multiple of a fixed integer divisible by 3 is o(1) when the upper limit of summing tends to infinity.We also discuss a connection of our results with a digit conjecture on primes.