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A lower bound for classical Kloosterman sums and an application

Stephan Baier, Jishu Das, Jewel Mahajan

TL;DR

The paper establishes explicit lower bounds for the classical Kloosterman sums $S(a,b;c)$ under $(ab,c)=1$ with odd $c$, by treating squarefree moduli, odd prime powers, and the general case via multiplicativity. It then translates these bounds into an explicit lower bound within Petersson's trace formula, enabling a variant of Jung–Sardari where the weight $k$ and level $N$ vary independently and yielding a concrete lower bound for the weighted trace of the Hecke operator $T_n$ on $S_k(N)$ for $(n,N)=1$. The core result is Theorem Main1, which gives a sharp lower bound depending on the factorization $c=du$ with $d$ powerful and $u$ squarefree, together with residue conditions on $ab$ modulo primes dividing $d$. The work provides explicit constants and corollaries that bound $|\Delta_{k,N}(m,n)-\delta(m,n)|$ in several regimes, offering practical implications for discrepancy-type questions and for understanding eigenvalue distributions in the presence of varying weight and level.

Abstract

We present a lower bound for the classical Kloosterman sum $S(a,b;c)$ where $(ab,c)=1$ and $c$ is an odd integer. We apply this lower bound for Kloosterman sums to derive an explicit lower bound in Petersson's trace formula, subject to a given condition. Consequently, we achieve a modified version of a theorem by Jung and Sardari, where weight $k$ and level $N$ are permitted to vary independently. Using this modified version, we get a lower bound for a weighted trace of the Hecke operator $T_n$ acting on the space $S_k(N)$, of cusp forms of weight $k$ and level $N$ with $(n,N)=1$.

A lower bound for classical Kloosterman sums and an application

TL;DR

The paper establishes explicit lower bounds for the classical Kloosterman sums under with odd , by treating squarefree moduli, odd prime powers, and the general case via multiplicativity. It then translates these bounds into an explicit lower bound within Petersson's trace formula, enabling a variant of Jung–Sardari where the weight and level vary independently and yielding a concrete lower bound for the weighted trace of the Hecke operator on for . The core result is Theorem Main1, which gives a sharp lower bound depending on the factorization with powerful and squarefree, together with residue conditions on modulo primes dividing . The work provides explicit constants and corollaries that bound in several regimes, offering practical implications for discrepancy-type questions and for understanding eigenvalue distributions in the presence of varying weight and level.

Abstract

We present a lower bound for the classical Kloosterman sum where and is an odd integer. We apply this lower bound for Kloosterman sums to derive an explicit lower bound in Petersson's trace formula, subject to a given condition. Consequently, we achieve a modified version of a theorem by Jung and Sardari, where weight and level are permitted to vary independently. Using this modified version, we get a lower bound for a weighted trace of the Hecke operator acting on the space , of cusp forms of weight and level with .
Paper Structure (6 sections, 13 theorems, 67 equations)

This paper contains 6 sections, 13 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Lower bound for Kloosterman sums
  3. Squarefree case
  4. Odd prime power case
  5. General Case
  6. An application

Key Result

Theorem 1

Let $c$ be an odd natural number such that $c=du$, where $u$ is squarefree and $d$ is a powerful number with $(d,u)=1$. Let $a,b$ be integers such that $(ab,c)=1$ and $ab$ is a quadratic residue modulo each prime dividing $d$ (and hence, $ab$ is a square modulo $d$ by Hensel's lemma and Chinese rema where $\omega(d)$ is the number of distinct prime divisors of $d$, $\tau(u)$ is the number of posit

Theorems & Definitions (29)

  • Theorem 1
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • Remark 1
  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • proof
  • ...and 19 more