On sign changes of q-exponents of generalized modular functions

TL;DR

This paper investigates the sign distribution of q-exponents in generalized modular functions, demonstrating their equidistribution for prime indices using advanced equidistribution theorems.

Contribution

It establishes the equidistribution of signs of q-exponents in generalized modular functions of weight zero, linking to eigenforms of integral weight.

Findings

Signs of q-exponents at prime indices are equidistributed.

The result applies to functions with zero divisor and real q-exponents.

Uses equidistribution theorems for normalized cuspidal eigenforms.

Abstract

Let f be a generalized modular function of weight 0 of level N such that its q-exponents c(n)(n>0) are all real, and div(f) is zero. In this note, we show the equidistribution of signs for c(p)(p prime) by using equidistribution theorems for normalized cuspidal eigenforms of integral weight.