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Möbius function and primes: an identity factory with applications

Olivier Ramaré, Sebastian Zuniga Alterman

TL;DR

The paper develops an Identity Factory that systematizes real-analytic identities for sums of the Möbius function under coprimality constraints, enabling explicit asymptotics for $m_q(X;s)$ and the logarithmically smoothed variant $\check{m}_q(X;s)$ across complex $s$ with $\Re s>0$. By pairing $g(n)=(-1)^{n+1}$ with appropriate $H$ and $h$, main terms involving $1/\zeta(s)$ and its derivatives are extracted, together with computable error terms that depend on $\int_1^X |m_q(t)|dt$ and $q$-dependent totients; secondary order terms are explicitly revealed. The authors provide explicit bounds near the line $\Re s=1$, including numeric bounds for $q=1$, and an appendix establishing the elementary inequality $\sum_{n\le X} \frac{\Lambda(n)}{n} \le \log X$. The Identity Factory unifies and extends previous identity-based approaches, offering finer control over error terms than summation-by-parts, with potential implications for primes distribution and Möbius-sum estimates in analytic number theory.

Abstract

We investigate the sums $\sum_{n\le X, (n,q)=1}\frac{μ(n)}{n^s}\log^k\left(\frac{X}{n}\right)$, where $k\in\{0,1\}$, $s\in\mathbb{C}$, $\Re s>0$. Our goal is to obtain explicit asymptotic estimations for these quantities. To achieve this, we develop a broad framework of identities that we use to derive several applications. Building on similar principles, we also provide an appendix establishing the inequality $\sum_{n\le X}Λ(n)/n\le \log X$, valid for any $X\geq 1$.

Möbius function and primes: an identity factory with applications

TL;DR

The paper develops an Identity Factory that systematizes real-analytic identities for sums of the Möbius function under coprimality constraints, enabling explicit asymptotics for and the logarithmically smoothed variant across complex with . By pairing with appropriate and , main terms involving and its derivatives are extracted, together with computable error terms that depend on and -dependent totients; secondary order terms are explicitly revealed. The authors provide explicit bounds near the line , including numeric bounds for , and an appendix establishing the elementary inequality . The Identity Factory unifies and extends previous identity-based approaches, offering finer control over error terms than summation-by-parts, with potential implications for primes distribution and Möbius-sum estimates in analytic number theory.

Abstract

We investigate the sums , where , , . Our goal is to obtain explicit asymptotic estimations for these quantities. To achieve this, we develop a broad framework of identities that we use to derive several applications. Building on similar principles, we also provide an appendix establishing the inequality , valid for any .
Paper Structure (9 sections, 31 theorems, 165 equations)

This paper contains 9 sections, 31 theorems, 165 equations.

Table of Contents

  1. Introduction and results
  2. The Identity Factory, proof of Theorem \ref{['OFD']}
  3. Recovering earlier results through the Identity Factory
  4. Proof of Theorem \ref{['easy']}
  5. The function $m_q$
  6. Evaluating $m_q(X;s)$, proof of Theorem \ref{['mqdex']}
  7. Evaluating $\check{m}_q(X;s)$, proof of Theorem \ref{['mcheckqdex']}
  8. Estimates for $\Re s =\sigma\ge1$
  9. Bounding $\Delta_q(X,\varepsilon)$ from above

Key Result

Theorem 1.1

Let $f$ and $g$ be two arithmetic functions. We define $S_f(t)=\sum_{n\le t}f(n)$ and $S_{f\star g}(t)$ similarly. Let $h:(0,1]\rightarrow \mathbb{C}$ be Lebesgue-integrable over every segment $\subset (0,1]$ and let $H$ be a function over $[1,\infty)$ that is absolutely continuous on every finite i

Theorems & Definitions (49)

  • Theorem 1.1: The Identity Factory
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Corollary 3.1
  • Corollary 3.2
  • ...and 39 more