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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 16 Apr 2024 13:33:51 +0000Bias in cubic Gauss sums: Patterson's conjecture
https://resolver.caltech.edu/CaltechAUTHORS:20230227-232631259
Authors: {'items': [{'id': 'Dunn-Alexander-J', 'name': {'family': 'Dunn', 'given': 'Alexander'}, 'orcid': '0000-0003-1665-7114'}, {'id': 'Radziwiłł-M', 'name': {'family': 'Radziwiłł', 'given': 'Maksym'}}]}
Year: 2023
DOI: 10.48550/arXiv.2109.07463
Let W be a smooth test function with compact support in (0,∞). Conditional on the Generalized Riemann Hypothesis for Hecke L-functions over ℚ(ω), we prove that
_p≡1∑_((mod3)) 1/2‾√p⋅ₓ(∑_((modp)) e^((2πix³/p))W(p/X)∼(2π)^(2/3)/3Γ(2/3) ∫^∞_0 W(x)x^(−1/6) dx ⋅ X^(5/6)/logX,
as X→∞ and p runs over primes. This explains a well-known numerical bias in the distribution of cubic Gauss sums first observed by Kummer in 1846 and confirms (conditionally on the Generalized Riemann Hypothesis) a conjecture of Patterson from 1978.
There are two important byproducts of our proof. The first is an explicit level aspect Voronoi summation formula for cubic Gauss sums, extending computations of Patterson and Yoshimoto. Secondly, we show that Heath-Brown's cubic large sieve is sharp up to factors of X^(o(1)) under the Generalized Riemann Hypothesis. This disproves the popular belief that the cubic large sieve can be improved.
An important ingredient in our proof is a dispersion estimate for cubic Gauss sums. It can be interpreted as a cubic large sieve with correction by a non-trivial asymptotic main term. This estimate relies on the Generalized Riemann Hypothesis, and is one of the fundamental reasons why our result is conditional.https://authors.library.caltech.edu/records/gj1ev-0dk12