Shubin, A. (2023). Variance Estimates in Linnik’s Problem. International Mathematics Research Notices, 2023(18), 15425–15474. https://doi.org/10.1093/imrn/rnac225
E104 - Institut für Diskrete Mathematik und Geometrie E104-05 - Forschungsbereich Kombinatorik und Algorithmen
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Journal:
International Mathematics Research Notices
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ISSN:
1073-7928
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Date (published):
Sep-2023
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Number of Pages:
50
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Publisher:
OXFORD UNIV PRESS
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Peer reviewed:
Yes
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Keywords:
Lattice points; Linnik problem; moments of L-functions
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Abstract:
We evaluate the variance of the number of lattice points in a small randomly rotated spherical ball on a surface of 3-dimensional sphere centered at the origin. Previously, Bourgain, Rudnick, and Sarnak showed conditionally on the Generalized Lindelöf Hypothesis that the variance is bounded from above by,σ (Ω)Nₙ1+ϵ, where σ (Ω) is the area of the ball Ω on the unit sphere and is Nn is the total number of solutions of Diophantine equation x²+y²+z²=n. Assuming the Grand Riemann Hypothesis and using the moments method of Soundararajan and Harper, we establish the upper bound of the form,cσ(Ω)Nₙ where c is an absolute constant. This bound is of the conjectured order of magnitude.
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Project title:
Arithmetische Zufälligkeit: I 4945-N (FWF - Österr. Wissenschaftsfonds)