Sums of values of non-principal characters over shifted primes

TL;DR

The paper addresses estimating sums of a nonprincipal Dirichlet character over shifted primes with composite moduli: $T(\chi)=\sum_{n\le x}\Lambda(n)\chi(n-l)$. It extends Vinogradov–Karatsuba type methods to composite $D$ by combining Heath-Brown's identity, Burgess-type short-sum bounds, and bilinear (double-sum) estimates to control both short and bilinear structures. The main result is a nontrivial bound $T(\chi) \ll x \exp(-0.6\sqrt{\ln D})$ under $x\ge D^{5/6+\varepsilon}$ and $q>\exp\sqrt{2\ln D}$, with a careful decomposition into primitive lifts $\chi_q$ and auxiliary divisors $\nu|q_1$ to manage imprimitive characters. The work advances understanding of shifted-prime character sums for composite moduli and has potential implications for Goldbach-type problems and distribution questions in arithmetic progressions where composite moduli arise.

Abstract

For a nonprincipal character $χ$ modulo $D$, when $x\ge D^{\frac56+\varepsilon}$, $(l,D) = 1$, we prove a nontrivial estimate of the form $\sum_{n\le x}Λ(n)χ(n-l)\ll x\exp\left(-0.6\sqrt{\ln D}\right)$ for the sum of values of $χ$ over a sequence of shifted primes. Bibliography: 41 references.

Theorems & Definitions (15)

• Theorem 1.1
• Lemma 1.1
• Lemma 1.2
• Lemma 1.3
• Lemma 1.4
• Lemma 1.5
• Lemma 1.6
• Lemma 2.1
• Lemma 2.2
• Lemma 3.1