On the value-distribution of the logarithms of symmetric power L-functions in the level aspect

TL;DR

This paper studies the distribution of logarithmic values of symmetric power L-functions associated with modular forms, establishing integral formulas involving a density function linked to the Sato-Tate measure, under certain conditions.

Contribution

It proves that averages of these L-function values can be expressed as integrals with a specific density function, extending to general symmetric powers under certain test functions.

Findings

Established integral formulas involving the M-function density

Demonstrated the parity phenomenon in the density function

Extended results to general symmetric power L-functions

Abstract

We consider the value distribution of logarithms of symmetric power L-functions associated with newforms of even weight and prime power level. In the symmetric square case, under certain plausible analytical conditions, we prove that certain averages of those values in the level aspect, involving continuous bounded or Riemann integrable test functions, can be written as integrals involving a density function (the "M-function") which is related with the Sato-Tate measure. Moreover, even in the case of general symmetric power L-functions, we show the same type of formula when for some special type of test functions. We see that a kind of parity phenomenon of the density function exists.