Sums of quadratic functions with two discriminants

Abstract In a 1999 paper, Zagier discusses a construction of a function F k , D ( x ) defined 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 coefficient of weight k + 1 / 2 Eisenstein series constructed by H. Cohen. In this note we consider a construction which works both for even and odd positive integers k . Our function F k , D , d ( x ) depends on two discriminants d and D with signs sgn ( d ) = sgn ( D ) = ( − 1 ) k , degenerates to Zagier's function when d = 1 , namely, F k , D , 1 ( x ) = F k , D ( x ) , and has very similar properties. In particular, we prove that the average value of F k , D , d ( x ) is again a Fourier coefficient 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.