Topics in the theory of automorphic L-functions

Automorphic $L$-functions play a fundamental role in number theory, linking the analytic and algebraic properties of automorphic forms to the distribution of prime numbers. Their Fourier coefficients encode significant arithmetic data, while the distribution of their zeros provides valuable insight into their analytic behavior. These themes are central to major conjectures in number theory such as the Generalized Riemann Hypothesis (GRH) and the Generalized Ramanujan Conjecture (GRC). In this thesis, we investigate various questions related to the properties of Fourier coefficients and zeros of automorphic $L$-functions, and explore the deep interactions between these two topics. First, we address the question concerning the size of the least quadratic non-residue modulo a prime $p$. Assuming GRH, it is known that the smallest quadratic non-residue modulo $p$ is less than or equal to \log² p. In intervals slightly larger, of size (\log p)^A with $A > 2$, although GRH implies the existence of quadratic non-residues modulo each prime $p$, this does not give any clue about how these quadratic non-residues are distributed. Our work on this topic achieves three main objectives: establish statistical distributions of quadratic non-residues in such small ranges for almost all primes; going beyond GRH by proving results also when $A \in (1,2]$; and doing so unconditionally. Second, we study the classical problem of Landau--Siegel zeros. These are possible zeros of Dirichlet \(L\)-functions $L(s,\chi)$ associated to quadratic number fields that lie close to \(s=1\). They serve as potential counterexamples to GRH and have significant implications for various topics, such as primes in arithmetic progressions, class number formula and character sums. In this work, we prove that for certain families of $L$-functions, if each $L$-function in the family has only real zeros in a fixed yet arbitrarily small neighborhood of $s=1$, then one may considerably improve existing results on Landau--Siegel zeros. This recovers earlier results of Sarnak and Zaharescu under a significantly weaker hypothesis. Third, we investigate the non-vanishing problem for primitive Dirichlet \(L\)-functions at the central point. The non-vanishing of \(L\)-functions at the central point is a fundamental problem in number theory with far-reaching arithmetic implications. For e.g., by the Birch and Swinnerton-Dyer Conjecture, the central value of the $L$-function of an elliptic curve over a number field vanishes precisely when the curve has positive rank. Assuming GRH, it is known that at least half of the central values $L(\frac{1}{2},\chi)$ are non-vanishing as $\chi$ ranges over primitive characters modulo $q$. Unconditionally, this is known on average over both $\chi$ modulo $q$ and $Q/2 \leq q \leq 2Q$. We prove that if one considers shorter averages (a power savings in the number of $q$'s used around $Q$), one can still produce non-vanishing proportions arbitrarily close to $\tfrac{1}{2}$. Finally, we explore the topic of correlations of zeros of automorphic $L$-functions. Assuming the Riemann Hypothesis, Montgomery established results concerning the pair correlation of zeros of the Riemann zeta function. Rudnick and Sarnak extended these results to automorphic \( L \)-functions and all level correlations. In our work, we discover additional geometric structures associated to the zeros of automorphic \( L \)-functions. In the case of pair correlation, these structures form certain surfaces which display Gaussian behavior. For triple correlation, these structures exhibit characteristics of the Laplace and Chi-squared distributions, revealing an unexpected phase transition.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois Urbana-Champaign

Year dc:date

2025

Author dc:creator

• Basak, Debmalya

Contributors dc:contributor

• Zaharescu, Alexandru
• Thorner, Jesse
• Ford, Kevin
• Berndt, Bruce

• Gaussian Distribution
• Quadratic Residues And Non-residues
• Primes In Short Intervals
• Square-free Numbers
• Poisson Distribution
• Dirichlet $l$-functions
• Landau--siegel Zeros
• Quadratic Characters
• Non-vanishing At Central Point
• Kloosterman Sums
• Short Intervals
• Arithmetic Progressions
• Automorphic $l$-functions
• Non-trivial Zeros, Correlation Surfaces
• Laplace And Chi-squared Distributions.

Statement dc:rights

• Copyright 2025 Debmalya Basak

Language dc:language

en, eng

Handle dc:identifier

https://hdl.handle.net/2142/130092

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F130092