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On a general divisor problem associated to general product $L$-functions over arithmetic progressions on certain integral binary quadratic forms and its application

Guodong Hua
Vol. 15 (2026), No. 2, 83–120
DOI: 10.2140/cnt.2026.15.83
Abstract

Let f be a normalized holomorphic cuspidal Hecke eigenform of even integral weight κ for the full modular group Γ = SL ⁡ (2, ℤ). We investigate the asymptotic behaviour of the higher power moments of a general divisor problem of the coefficients of ℓ-fold product L-functions associated to f on arithmetic progressions. As an application, we also provide quantitative results for the sign changes of the Dirichlet coefficients of the ℓ-fold product L-functions over arithmetic progressions. By analogy, we also consider a similar problem which is supported at certain integral binary quadratic forms.

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Keywords
general product $L$-function, divisor problem, asymptotic behaviour, binary quadratic forms, sign changes
Mathematical Subject Classification
Primary: 11F11, 11F30
Secondary: 11F66
Milestones
Received: 6 June 2025
Revised: 27 February 2026
Accepted: 30 March 2026
Published: 10 May 2026
Authors
Guodong Hua
Research Institute of Qindong Mathematics
School of Mathematics and Statistics
Weinan Normal University
Weinan
China
© 2026 MSP (Mathematical Sciences Publishers).