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Hybrid estimation of single exponential sums, exceptional characters and primes in short intervals

Runbo Li

Abstract

We provide a new hybrid estimation of single exponential sums, combining Van der Corput, Huxley and Bourgain's result. We also focus on primes in short intervals $(x-x^α,x]$ under the assumption of the existence of exceptional Dirichlet characters and get a small improvement of a 2004 result of Friedlander and Iwaniec. By using our new estimation of exponential sums, we extend the previous admissible range $0.4937 \leqslant α\leqslant 1$ to $0.4923 \leqslant α\leqslant 1$.

Hybrid estimation of single exponential sums, exceptional characters and primes in short intervals

Abstract

We provide a new hybrid estimation of single exponential sums, combining Van der Corput, Huxley and Bourgain's result. We also focus on primes in short intervals under the assumption of the existence of exceptional Dirichlet characters and get a small improvement of a 2004 result of Friedlander and Iwaniec. By using our new estimation of exponential sums, we extend the previous admissible range to .
Paper Structure (4 sections, 11 theorems, 45 equations)

This paper contains 4 sections, 11 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Higher order derivative tests for single exponential sums
  3. A problem considered by Friedlander, Iwaniec and Nowak
  4. A new application: Primes in Short Intervals

Key Result

Theorem 1.1

([FI2004, Theorem 1.1]). Let $\chi=\chi_{D}$ denotes the real primitive character of conductor D, $x \geqslant D^{r}$ with $r=18289$ and $\frac{39}{79} \leqslant \alpha \leqslant 1$. Then we have and where $L(s, \chi)$ is the Dirichlet L-function.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • remark
  • Corollary 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 5 more