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A round of Pintz to celebrate oscillations in sums

Daniel R. Johnston, Tim Trudgian

TL;DR

This work revisits the Landau–Pintz method for linking oscillations in sums of arithmetic functions to zero-free regions of $L$-functions, using $\zeta(s)$ as a prototype and extending the framework toward Dirichlet $L$-functions. It presents an explicit version of Pintz’s result for simple zeros, tying a lower bound on the mean of an arithmetic transform $A(x)$ to a zero $\rho_0=β_0+iγ_0$ and analytic bounds on associated transforms $F(s)$ and $G(s)$, with concrete constants and error terms. An explicit example with $A(x)=M(x)$ demonstrates how information about zeros translates into zero-free regions and, conversely, how partial arithmetic data yield explicit mean-value bounds for $M(x)$; the paper also provides explicit bounds for $|F(s)|$ and $|G(s)|$ and discusses the limitations and potential extensions. The authors advocate a blueprint for future work: using arithmetic information to infer zero-free regions and, in turn, leveraging those regions to obtain sharper arithmetic bounds, with potential applicability to broader classes of $L$-functions and arithmetic sums.

Abstract

We outline a method, going back to Landau and developed by Pintz, for connecting sums of arithmetic functions with zero-free regions for $L$-functions. We use the Riemann zeta-function as a prototype, but outline the utility of this method in general.

A round of Pintz to celebrate oscillations in sums

TL;DR

This work revisits the Landau–Pintz method for linking oscillations in sums of arithmetic functions to zero-free regions of -functions, using as a prototype and extending the framework toward Dirichlet -functions. It presents an explicit version of Pintz’s result for simple zeros, tying a lower bound on the mean of an arithmetic transform to a zero and analytic bounds on associated transforms and , with concrete constants and error terms. An explicit example with demonstrates how information about zeros translates into zero-free regions and, conversely, how partial arithmetic data yield explicit mean-value bounds for ; the paper also provides explicit bounds for and and discusses the limitations and potential extensions. The authors advocate a blueprint for future work: using arithmetic information to infer zero-free regions and, in turn, leveraging those regions to obtain sharper arithmetic bounds, with potential applicability to broader classes of -functions and arithmetic sums.

Abstract

We outline a method, going back to Landau and developed by Pintz, for connecting sums of arithmetic functions with zero-free regions for -functions. We use the Riemann zeta-function as a prototype, but outline the utility of this method in general.

Paper Structure

This paper contains 6 sections, 4 theorems, 63 equations.

Table of Contents

  1. Introduction and background
  2. Main theorem
  3. An explicit example using $\zeta(s)$
  4. Bounds for $F(s)$ and $G(s)$
  5. Example applications with $M(x)$
  6. Conclusion and extensions

Key Result

Theorem 1

Suppose that $A(x):\mathbb{R}\to\mathbb{C}$ is a complex valued function with $|A(x)|\leq c_Ax^C$ for some $c_A>0$ and $C> 0$, with where for some $\beta_0\in(0,1]$ and $0<c_0<\beta_0$, we have that $F(s)$ is analytic for $\sigma\geq\beta_0-c_0$, and $G(s)$ for $\sigma\geq -1$. Next, suppose that in these halfplanes, one has for constants $c_F,B_F,c_G,B_G>0$. Then, if $G(s)$ has a simple zero $\

Theorems & Definitions (8)

  • Theorem 1: Explicit version of the Landau--Pintz result
  • proof
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof