Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Moment of Kummer sums weighted by $L$-functions

Nilanjan Bag

Abstract

The main purpose of this article is to study higher order moments of Kummer sums weighted by $L$-functions using estimates for character sums and analytic methods. The results of this article complement a conjecture of Zhang Wenpeng (2002). Also the results in this article give analogous results of Kummer's conjecture (1846).

Moment of Kummer sums weighted by $L$-functions

Abstract

The main purpose of this article is to study higher order moments of Kummer sums weighted by -functions using estimates for character sums and analytic methods. The results of this article complement a conjecture of Zhang Wenpeng (2002). Also the results in this article give analogous results of Kummer's conjecture (1846).
Paper Structure (7 sections, 9 theorems, 60 equations)

This paper contains 7 sections, 9 theorems, 60 equations.

Table of Contents

  1. Introduction and statements of the results
  2. notations and preliminaries
  3. Some lemmas
  4. Proof of Theorem \ref{['mt1']} and \ref{['mt2']}
  5. Proof of Theorem \ref{['mt3']} and \ref{['mt4']}
  6. moment of quadratic Gauss sums
  7. Acknowledgement

Key Result

Theorem 1.2

Let $p$ an odd prime such that $p\equiv 1\bmod 3$ and $n$ be an integer. Then for any Dirichlet character $\chi \bmod p$, we have an asymptotic formula

Theorems & Definitions (15)

  • Conjecture 1.1: Patterson, 1978
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Remark 1.6
  • Lemma 3.1
  • proof
  • Lemma 3.2: Lemma 1, zhang
  • Lemma 3.3: Lemma 2, zhang
  • ...and 5 more