On sums involving products and quotients of $L$-functions over function fields

Abstract

We estimate the sum of products or quotients of $L$-functions, where the sum is taken over all quadratic extensions of given genus over a fixed global function field. Our estimate for the sum of the quotient of two $L$-functions is analogous to a result of Schmidt where he estimates the sum of the quotient of two $L$-series, where the sum is over quadratic extensions of $\mathbb{Q}$ with absolute value of the discriminant less than a given bound.