A family of L-functions related to the Riemann zeta function.

Professor Kane, Benjamin Robert   (Principal Investigator (PI))

36

2018-09-01

2021-08-31

456452

A family of L-functions related to the Riemann zeta function.

Mellin transforms, meromorphic modular forms, regularized integrals, Riemann zeta function, sum-of-divisors function

Pure Mathematics

Physical Sciences (P)

17303618

General Research Fund (GRF)

2018

Completed

1 Construct a family of L-functions which have an additional parameter (a certain complex number) and yield the Riemann zeta function as a limit. 2 Construct a regularized Mellin transform that allows as input functions which have poles in the upper half plane. 3 Compute the regularized Mellin transform on a family of weight 2 polar harmonic Maass forms; construct a family of L-functions related to the sum-of-divisors function. 4 Construct a family of weight 1/2 meromorphic modular modular forms closely associated to the weight 2 polar harmonic Maass forms. 5 Compute the regularized Mellin transform on the family of weight 1/2 forms; construct a family of L-functions related to the Riemann zeta function. 6 Consider applications of the properties of the L-functions constructed in previous steps on the L-series obtained in the limit. Investigate any possible implications about the zeros of L-series. 7 Consider applications to the class numbers of imaginary quadratic fields via twists of the above constructions and the class number formula.