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

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Title:Sums of Quadratic Functions with Two Discriminants and Farkas' Identities with Quartic Characters.
Authors:Wong, Ka Lun
Contributors:Mathematics (department)
number theory
modular forms
Date Issued:Aug 2017
Publisher:University of Hawaiʻi at Mānoa
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.
Description:Ph.D. Thesis. University of Hawaiʻi at Mānoa 2017.
Rights:All UHM dissertations and theses are protected by copyright. They may be viewed from this source for any purpose, but reproduction or distribution in any format is prohibited without written permission from the copyright owner.
Appears in Collections: Ph.D. - Mathematics

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