Year: 2024
Type: article
Abstract: We prove a sharp upper bound for the fourth moment of the Hurwitz zeta function ζ ( s , α ) \zeta(s,\alpha) on the critical line when the shift parameter 𝛼 is irrational and of irrationality exponent strictly less than 3. As a consequence, we determine the order of magnitude of the 2 k 2k th moment for all 0 ⩽ k ⩽ 2 0\leqslant k\leqslant 2 in this case. In contrast to the Riemann zeta function and other 𝐿-functions from arithmetic, these grow like T ( log T ) k T(\log T)^{k} . This suggests, and we conjecture, that the value distribution of ζ ( s , α ) \zeta(s,\alpha) on the critical line is Gaussian.
Authors: Winston Heap, Anurag Sahay
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: Purdue University
OpenAlex ID: W4404629185