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- W4385018204 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 $gg X^{c/loglog X}$ 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." @default.
- W4385018204 created "2023-07-22" @default.
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- W4385018204 date "2023-07-19" @default.
- W4385018204 modified "2023-10-17" @default.
- W4385018204 title "$L^1$ means of exponential sums with multiplicative coefficients. I" @default.
- W4385018204 doi "https://doi.org/10.48550/arxiv.2307.10329" @default.
- W4385018204 hasPublicationYear "2023" @default.
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