PhD Defense – Ka Lun Wong @ Keller 401

July 11, 2017 @ 3:00 pm – 4:00 pm
Keller Hall
Honolulu, HI 96822

Title: Sums of quadratic functions with two discriminants and Farkas’ identities with quartic characters

Dissertation draft

In a 1999 paper, Zagier discusses a construction of a function $F_{k,D}(x)$ defined for an even integer $k ge 2$, 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 cite{Cohen}.

In this dissertation, 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 $sign(d)=sign(D)=(-1)^k$, degenerates to Zagier’s function when $d=1$, namely,
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 $k ge 2$ is allowed to be both even and odd.

In a 2004 paper, Farkas introduces a new arithmetic function and proves an identity involving this function. Guerzhoy and Raji cite{Guerzhoy} generalize this function for primes that are congruent to 3 modulo 4 by introducing a quadratic Dirichlet character and find another identity of the same type. We look at the case when $p equiv 5 Mod 8$ by introducing quartic Dirichlet characters and prove an analogy of their generalization.
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