L ¹ mean of exponential sums with multiplicative coefficients. I.

Abstract

We show that the

norm of an exponential sum of length X , and with coefficients equal to the Liouville or Möbius function, is at least

much greater than Subscript epsilon Baseline upper X Superscript 1 divided by 4 minus epsilon ≫ ε X 1 / 4 − ε $\gg_{\varepsilon} X^{1/4 - \varepsilon}$

for any given

epsilon ε $\varepsilon$

. For the Liouville function, this improves on the lower bound

much greater than upper X Superscript 1 divided by log log upper X ≫ X 1 / log ⁡ log ⁡ X $\gg X^{1/\log\log X}$

due to Balog and Perelli. For the Möbius function, this improves the lower bound

much greater than upper X Superscript 1 divided by 6 ≫ X 1 / 6 $\gg X^{1/6}$

due to Balog and Ruzsa. 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 left parenthesis n right parenthesis μ ( n ) $\mu(n)$

. 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

much greater than upper X Superscript delta ≫ X δ $\gg X^{\delta}$

for some small

delta δ $\delta$

but for general (non-pretentious) multiplicative functions.