Half-integral weight Kloosterman sums and integer partitions

Sun, Qihang

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https://hdl.handle.net/2142/124560 Copy

Description

Title

Half-integral weight Kloosterman sums and integer partitions

Author(s)

Sun, Qihang

Issue Date

2024-04-24

Director of Research (if dissertation) or Advisor (if thesis)

Ahlgren, Scott

Doctoral Committee Chair(s)

Ford, Kevin

Committee Member(s)

• Berndt, Bruce C.
• Thorner, Jesse

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(s)

• Kloosterman sum
• integer partition
• modular form
• Maass form

Abstract

Kloosterman sums are special exponential sums which appear in many problems in number theory. Kloosterman first introduced these sums in \cite{Kloosterman1926firstdef} to investigate whether the quadratic form $a_1n_1^2+a_2n_2^2+a_3n_3^2+a_4n_4^2$ with fixed $a_i\in \mathbb N$ represents all sufficiently large natural numbers. Another application is to estimate the shifted sum of divisor functions. Let $\tau(n)$ be the number of divisors of the positive integer $n$ and \[D(N,f)\defeq \sum_{n=1}^N \tau(n)\tau(n+h),\quad \text{for some fixed integer }h\geq 1. \] Heath-Brown \cite{HeathBrown1979ShiftedDivisor} applied the Weil bound \eqref{Weilbound standard Kl sum} of Kloosterman sums to prove that \[D(N,f)=\text{explicit main terms}+O(N^{\frac 56+\ep}),\quad \text{uniformly for }1\leq h\leq N^{\frac 56}. \] Using Kuznetsov's trace formula, Deshouillers and Iwaniec \cite{DeshouillersIwaniec1982ShiftDivisor} obtained a much better error bound $O(N^{\frac 23+\ep})$ for all $h\geq 1$. The integer partition function $p(n)$, which is the number of ways to write $n$ as a sum of positive integers, has been researched for remarkable properties by Euler, Hardy and Ramanujan \cite{HardyRamanujan1918Asymp}. Rademacher's exact formula \cite{Rademacher1937pn} states that $p(n)$ can be written as a sum of exponential sums. The generating function of $p(n)$ is $q^{\frac1{24}}/\eta(z)$, where $\eta(z)$ is Dedekind's eta function with $q=e^{2\pi i z}$ and $\im z>0$. Since $\eta(z)$ is a weight $\frac12$ modular form, using the definition of multiplier systems, we are able to rewrite the exponential sums in Rademacher's exact formula as generalized Kloosterman sums. The bounds on Kloosterman sums give the growth rate of errors for such approximations. There are very famous congruence properties of the partition function $p(n)$ by Ramanujan: \[p(5n+4)\equiv 0\Mod 5,\quad p(7n+5)\equiv 0\Mod7, \quad p(11n+6)\equiv 0\Mod {11}. \] In 1944, Dyson \cite{Dyson} defined the rank of a partition of $n$. If we let $N(a,b;n)$ denote the number of partitions of $n$ with rank congruent to $a\Mod b$, then Dyson conjectured that $5N(j,5;5n+4)=p(5n+4)$ and $7N(j,7;7n+5)=p(7n+5)$ for all $j$. By the work of Bringmann and Ono \cite{BrmOno2006ivt,BrmOno2010}, the generating functions for the ranks of partitions have similar properties as $q^{\frac1{24}}/\eta(z)$. The work of Bringmann and Ono in the theory of harmonic Maass forms discovers beautiful properties about the rank of partitions. For example, in \cite{BrmOno2006ivt} they proved the exact formula for the modulo $2$ case, which perfected the asymptotics by Ramanujan, Dragonette \cite{Dragonette1952} and Andrews \cite{Andrews1966}. If we have better estimates for the sums of half-integral weight Kloosterman sums, we are able to obtain better tail bounds for the Rademacher-type exact formulas, which control the efficiency of their convergence. The recent work by Ahlgren and Andersen \cite{AAimrn}, Ahlgren and Dunn \cite{ahlgrendunn}, and Andersen and Wu \cite{AndersenWu2022bound36_publishedver} provide improved error bounds based on their improvement on the estimates for Kloosterman sums. The author \cite{QihangFirstAsympt,QihangSecondAsympt} generalized their work to the Kloosterman sums with a wider class of multiplier systems, which are half-integral weight and include the commonly used theta- and eta-multipliers twisted by quadratic characters. The resulting estimates give a uniform version of the general result by Goldfeld and Sarnak \cite{gs} for sums of such Kloosterman sums with a power-saving bound in the parameters $m$ and $n$. Following the method in \cite{BrmOno2006ivt}, the author provided a detailed proof of the exact formula for the rank modulo 3 case in \cite{QihangFirstAsympt}. Then what about the exact formulae in the rank modulo 5 and 7 cases, where Ramanujan's congruences appear? Bringmann \cite{BringmannTAMS} proved the general asymptotics for all odd moduli, while the Kloosterman type sums are hard to interpret as Kloosterman sums. Thanks to the theory of vector-valued Maass forms from \cite{BrmOno2010} and the explicit transformation laws by Garvan \cite{GarvanTransformationDyson2017}, the author finds the interpretation as vector-valued Kloosterman sums. Combining with some generalization of \cite{gs}, the author finally provides the proof for the exact formula of rank modulo primes $p\geq 5$. The author also has a striking observation between the interesting cases $p=5,7$, where the Kloosterman sums become identically zero (or become equal for those defined on different cusp pairs). After a long study of the cases depending on congruence properties of the Dedekind sums, the author proves this cancellation property and provides a new proof for the Dyson's conjecture $5N(a,5;5n+4)=p(5n+4)$ and $7N(a,7;7n+5)=p(7n+5)$ which implies Ramanujan's congruences.

Graduation Semester

2024-05

Type of Resource

Thesis

Copyright and License Information

Copyright 2024 Qihang Sun

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