A1 Refereed original research article in a scientific journal
The Chowla conjecture and Landau–Siegel zeroes
Authors: Jaskari, Mikko; Sachpazis, Stelios
Publisher: Cambridge University Press (CUP)
Publication year: 2025
Journal: Mathematical Proceedings of the Cambridge Philosophical Society
Journal name in source: Mathematical Proceedings of the Cambridge Philosophical Society
Volume: 179
Issue: 1
First page : 167
Last page: 187
ISSN: 0305-0041
eISSN: 1469-8064
DOI: https://doi.org/10.1017/S0305004125000271
Web address : https://doi.org/10.1017/s0305004125000271
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/498491837
Let k⩾2k⩾2 be an integer and let λλ be the Liouville function. Given k non-negative distinct integers h1,…,hkh1,…,hk, the Chowla conjecture claims that ∑n⩽xλ(n+h1)⋯λ(n+hk)=o(x)∑n⩽xλ(n+h1)⋯λ(n+hk)=o(x). An unconditional answer to this conjecture is yet to be found, and in this paper, we take a conditional approach. More precisely, we establish a non-trivial bound for the sums ∑n⩽xλ(n+h1)⋯λ(n+hk)∑n⩽xλ(n+h1)⋯λ(n+hk) under the existence of a Landau–Siegel zero for x in an interval that depends on the modulus of the character whose Dirichlet series corresponds to the Landau–Siegel zero. Our work constitutes an improvement over the previous related results of Germán and Kátai, Chinis and Tao and Teräväinen.
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Funding information in the publication:
During the making of this work, M. Jaskari was being supported by the Academy of Finland grant no. 346307 and the University of Turku Graduate School UTUGS. S. Sachpazis acknowledges support from the Academy of Finland grant no. 333707.
