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On a Möbius double sum

Olivier Ramaré, Sebastian Zuniga Alterman

Abstract

We study the double sum $S_\varepsilon(X)$$=$$\sum_{\substack{d,e\le X}}\frac{μ(d)μ(e)}{[d,e]^{1+\varepsilon}}$, which converges even in the case $\varepsilon=0$, where $μ$ denotes the Möbius function and $[d,e]$ is the least common multiple of $d$ and $e$. Such expressions arise naturally in analytic number theory, notably as the diagonal contribution in certain squared mean values, and they play a significant role in zero-density estimates for the Riemann zeta function and related $L$-functions. We establish uniform upper bounds for $S_\varepsilon(X)$ across various ranges of $X$, with particular emphasis on the case $\varepsilon$ close to $0^+$.

On a Möbius double sum

Abstract

We study the double sum , which converges even in the case , where denotes the Möbius function and is the least common multiple of and . Such expressions arise naturally in analytic number theory, notably as the diagonal contribution in certain squared mean values, and they play a significant role in zero-density estimates for the Riemann zeta function and related -functions. We establish uniform upper bounds for across various ranges of , with particular emphasis on the case close to .

Paper Structure

This paper contains 7 sections, 28 theorems, 206 equations.

Table of Contents

  1. Introduction and results
  2. On the summatory function $m_q(X;1+\varepsilon)$ at $\varepsilon=0$
  3. On the summatory function of the Möbius function at $s=1+\varepsilon$
  4. Mean values of some multiplicative functions
  5. Proof of Theorem \ref{['Main']}
  6. Proof of Theorem \ref{['Mainbis']}
  7. Proofs of the auxiliary results

Key Result

Theorem 1.1

Let $X>0$. Then

Theorems & Definitions (32)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 3.1
  • Theorem 4.1
  • ...and 22 more