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Published August 26, 2021 | Submitted
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Level of distribution of unbalanced convolutions


We show that if an essentially arbitrary sequence supported on an interval containing x integers, is convolved with a tiny Siegel-Walfisz-type sequence supported on an interval containing exp((log x)^ε) integers then the resulting multiplicative convolution has (in a weak sense) level of distribution x^(1/2 + 1/66 − ε) as x goes to infinity. This dispersion estimate has a number of consequences for: the distribution of the kth divisor function to moduli x^(1/2 + 1/66 − ε) for any integer k ≥ 1, the distribution of products of exactly two primes in arithmetic progressions to large moduli, the distribution of sieve weights of level x^(1/2 + 1/66 − ε) to moduli as large as x^(1−ε) and for the Brun-Titchmarsh theorem for almost all moduli q of size x^(1−ε), lowering the long-standing constant 4 in that range. Our result improves and is inspired by earlier work of Green (and subsequent work of Granville-Shao) which is concerned with the distribution of 1-bounded multiplicative functions in arithmetic progressions to large moduli. As in these previous works the main technical ingredient are the recent estimates of Bettin-Chandee for trilinear forms in Kloosterman fractions and the estimates of Duke-Friedlander-Iwaniec for bilinear forms in Kloosterman fractions.

Additional Information

The second author would like to thank the Laboratoire de Mathématiques d'Orsay for its invitation and for its hospitality. The second author also acknowledges support of an NSERC DG grant, the CRC program and a Sloan Fellowship. We would like to thank Sandro Bettin an James Maynard for their comments on the paper.

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