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arXiv · 2607.15574

Improved bounds for multiplicative functions in almost all short intervals

Siddarth Menon
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Abstract

We refine the Matom\"aki-Radziwi{\l}{\l} method for short averages of multiplicative functions. For the Liouville function and multiplicative functions supported on smooth numbers, we prove decay bounds that are essentially optimal with the Matom\"aki-Radziwi{\l}{\l} method. The key new ingredient is a sharper treatment of the sieve error, achieved by introducing a more widely separated final prime range when restricting to integers with typical factorizations. We additionally give a weaker but still improved bound for arbitrary 1-bounded multiplicative functions and discuss some limitations of the method. As an application, we give an improved bound for the averaged Chowla conjecture of Matom\"aki-Radziwi{\l}{\l}-Tao that seems essentially best possible via their method.

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Siddarth Menon. 2026-07-17. Improved bounds for multiplicative functions in almost all short intervals. https://arxiv.org/abs/2607.15574

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