Sums of Quadratic Functions with Two Discriminants and Farkas' Identities with Quartic Characters.

Abstract

Zagier in [21] discusses a construction of a function Fk;D(x) de ned for an even integer k 2, and a positive discriminant D. This construction is intimately related to half-integral weight modular forms. In particular, the average value of this function is a constant multiple of the D-th Fourier coe cient of weight k + 1=2 Eisenstein series constructed by H. Cohen in [2]. In this dissertation, we consider a construction which works both for even and odd positive integers k. Our function Fk;D;d(x) depends on two discriminants d and D with signs sign(d) = sign(D) = (􀀀1)k, degenerates to Zagier's function when d = 1, namely, Fk;D;1(x) = Fk;D(x); and has very similar properties. In particular, we prove that the average value of Fk;D;d(x) is again a Fourier coe cient of H. Cohen's Eisenstein series of weight k + 1=2, while now the integer k 2 is allowed to be both even and odd. In [6] Farkas introduces a new arithmetic function and proves an identity involving this function. Guerzhoy and Raji [8] generalize this function for primes that are congruent to 3 modulo 4 by introducing a quadratic Dirichlet character and nd another identity of the same type. We look at the case when p 5 (mod 8) by introducing quartic Dirichlet characters and prove an analogue of their generalization.