L-Functions in Analytic Number Theory

I will start with a survey on sums of random multiplicative functions, focusing on distributional questions and almost sure upper bounds and $\Omega$-results. In this context, I will describe previous work with Jake Chinis on a central limit theorem for correlations of Rademacher multiplicative functions, as well as ongoing work on establishing almost sure sharp bounds for them.

I am going to discuss various results on moments of symmetric square L-functions and some of their applications. I will mainly focus on a recent result of R. Khan and M. Young and our improvement of it. Khan and Young proved a mean Lindelöf estimate for the second moment of Maass form symmetric-square L-functions $L(\mathop{sym}^2 u_j, 1/2 + it)$ on the short interval of length $G >> |t_j|^{(1 + \epsilon)/t^{(2/3)}}$, where $t_j$ is a spectral parameter of the corresponding Maass form. Their estimate yields a subconvexity estimate for $L(\mathop{sym}^2 u_j, 1/2 + it)$ as long as $|t_j|^{(6/7 + \delta)} << t < (2 - \delta)|t_j|$. We obtain a mean Lindelöf estimate for the same moment in shorter intervals, namely for $G >> |t_j|^{(1 + \epsilon)/t}$. As a corollary, we prove a subconvexity estimate for $L(\mathop{sym}^2 u_j, 1/2 + it)$ on the interval $|t_j|^{(2/3 + \delta)} << t << |t_j|^{(6/7 - \delta)}$. This is joint work with Olga Balkanova.

I will discuss recent work with Chantal David, Alexander Dunn, and Joshua Stucky, in which we prove that a positive proportion of Hecke L-functions associated to the cubic residue symbol modulo square-free Eisenstein integers do not vanish at the central point. Our principal new contribution is the asymptotic evaluation of the mollified second moment. No such asymptotic formula was previously known for a cubic family (even over function fields).

Our new approach makes crucial use of Patterson's evaluation of the Fourier coefficients of the cubic metaplectic theta function, Heath-Brown's cubic large sieve, and a Lindelöf-on-average upper bound for the second moment of cubic Dirichlet series that we establish. The significance of our result is that the family considered does not satisfy a perfectly orthogonal large sieve bound. This is quite unlike other families of Dirichlet L-functions for which unconditional results are known (namely the family of quadratic characters and the family of all Dirichlet characters modulo q). Consequently, our proof has fundamentally different features from the corresponding works of Soundararajan and of Iwaniec and Sarnak.

Bounds on the logarithmic derivative and the reciprocal of the Riemann zeta function are studied as they have a wide range of applications, such as computing bounds for Mertens function. In this talk, we are mainly concerned with explicit bounds. Obtaining decent bounds are tricky, as they are only valid in a zero-free region, and the constants involved tend to blow up as one approaches the edge of the region, and a potential zero. We will discuss such bounds, their uses, and the computational and analytic techniques involved. Finally, we also show how to obtain a power savings in the case of the reciprocal of zeta.

The least quadratic non-residue has been a central problem in number theory for centuries. The average least quadratic non-residue was explored by Erdős in the 1960s, and many extensions of this problem such as to the average least character non-residue (Martin, Pollack) have been explored. In this talk, we look in to the average first sign change of Fourier coefficients of newforms (equivalently Hecke eigenvalues). We discuss the distribution of Hecke eigenvalues through the so-called 'horizontal' and 'vertical' Sato-Tate distributions, and we also discuss large sieve inequalities for cusp forms that are uniform in both the weight and the level.