Trace Formula for Integral Points on the Three-Dimensional Sphere

Corresponding Member of the RAS V. A. Bykovskii^a^,* and M. D. Monina^a^,**

^aKhabarovsk Branch of Institute of Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences, Khabarovsk, 680000 Russia

Correspondence to: *e-mail: vab@iam.khv.ru
Correspondence to: **e-mail: monina_dvggu@mail.ru

Received 24 October, 2019

Abstract—Average values over integral points on a three-dimensional sphere with an arbitrary smooth weight function are studied. For them, an expansion of the mean product of two L-series associated with the Hecke basis in spaces of holomorphic parabolic forms of integer even weight with respect to the congruence subgroup Γ₀(4) is obtained.

Keywords: integral points on a sphere, modular functions, L-series of parabolic forms

DOI: 10.1134/S1064562420010044