$L^1$ means of exponential sums with multiplicative coefficients. I

TL;DR

This paper establishes a new lower bound for the $L^1$ norm of exponential sums with Liouville or Möbius coefficients, improving previous bounds by leveraging zeros of Dirichlet $L$-functions and unifying the treatment of these functions.

Contribution

It introduces a novel method that relates the $L^1$ norms to zeros of Dirichlet $L$-functions, providing improved lower bounds and a unified approach for Liouville and Möbius functions.

Findings

Lower bound $\, ext{gg}_{ ext{ε}} X^{1/4 - ext{ε}}$ for $L^1$ norm with Liouville or Möbius coefficients.

Improves previous bounds: $ ext{gg} X^{c/ ext{log} ext{log} X}$ for Liouville and $ ext{gg} X^{1/6}$ for Möbius.

Method exploits zeros of Dirichlet $L$-functions, unifying the treatment of these coefficients.

Abstract

We show that the $L^1$ norm of an exponential sum of length $X$ and with coefficients equal to the Liouville or M\"{o}bius function is at least $\gg_{\varepsilon} X^{1/4 - \varepsilon}$ for any given $\varepsilon$. For the Liouville function this improves on the lower bound $\gg X^{c/\log\log X}$ due to Balog and Perelli (1998). For the M\"{o}bius function this improves the lower bound $\gg X^{1/6}$ due to Balog and Ruzsa (2001). The large discrepancy between these lower bounds is due to the method employed by Balog and Ruzsa, as it crucially relies on the vanishing of $\mu(n)$. Instead our proof puts the two cases on an equal footing by exploiting the connection of these coefficients with zeros of Dirichlet $L$-functions. In the second paper in this series we will obtain a lower bound $\gg X^{\delta}$ for some small $\delta$ but for general (non-pretentious) multiplicative functions.