Mean value of the class number in function fields revisited

Abstract

In this paper an asymptotic formula for the sum L(1, χ) is established for the family of quadratic Dirichlet L-functions over the rational function field over a finite field F q with q fixed. Using the recent techniques developed by Florea we obtain an extra lower order terms that was never been predicted in number fields and function fields. As a corollary, we obtain a formula for the average of the class number over function fields which also contains strenuous lower order terms and so improving on previous results of Hoffstein and Rosen.

The first moment of $L\bigl(\frac{1}{2},\chi\bigr)$ for real quadratic function fields

“…A e g−1,1,1 = − 2] du. (5.15) We see that (5.14) and (5.15) are precisely the terms S 1 (V = ) and S 2 (V = ) in the statement of Lemma 5.6.…”

“…Using the techniques presented by Florea, Andrade and Jung [2] improved the asymptotic formula (1.6) by obtaining a secondary main term of size q g 3 and bounding the error term by q g for any > 0. In particular, they proved that (1.7)…”