Uniform distribution of polynomially-defined additive function to varying moduli

TL;DR

The paper investigates the uniform distribution of polynomially-defined additive functions $f$ with a varying modulus $q$, where $f(p)=F(p)$ for primes and $F\in \mathbb{Z}[x]$ has degree $d$. It extends Delange's criterion to the setting of varying moduli by adopting the Pollack-Singha Roy framework and introducing the exponential-sum device $\mathcal{V}_{F,q,J}(w)$, together with strong bounds for additive-character sums from Cochrane and Loh. The main result identifies explicit obstructions in terms of $F\bmod p$ and parity conditions that determine when $f$ is UD modulo $q$, and proves that for $q$ in $\mathcal{S}_f \cap [1,(\log x)^K]$ the counting function $\sum_{n\le x,f(n)\equiv a\pmod q}1$ is asymptotic to $x/q$ for all residues $a$, provided either $d=1$ or $q$ is not too large relative to $\log x$ (specifically, $q\le (\log x)^{(1-\epsilon)(1-1/d)^{-1}}$ when $d\ge 2$). The results offer a near-optimal range for varying moduli and connect the study to the distribution of additive functions in coprime residue classes, using a combination of a convenient-number decomposition and exponential-sum techniques.

Abstract

In 1969, Delange has proved a general criterion for uniform distribution of additive functions. In this paper, we study the uniform distribution of a special class of polynomially-defined additive functions where the moduli is allowed to vary. This paper takes inspiration from a similar paper that studies the distribution of some special multiplicative functions by Pollack and Singha Roy. The approach would require techniques in the paper mentioned and bound on exponential sums from Cochrane and Loh.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• proof
• Lemma 3.1
• Lemma 3.2
• Corollary 3.3
• Lemma 4.1
• proof
• Lemma 4.2