# Number Theory

## Moments in the Chebotarev density theorem

In joint work with Régis de la Bretéche and Daniel Fiorilli, we consider weighted

moments for the distribution of Frobenius substitutions in conjugacy classes of

Galois groups of normal number field extensions. The question is inspired by work

of Hooley and recent progress by de la Bretéche–Fiorilli in the case of moments for

primes in arithmetic progressions. As in their work, the results I will discuss are

conditional on the Riemann Hypothesis and confirm that the moments considered

should be Gaussian. Time permitting, I will address a different notion of moments

that can be considered in the same context and that leads to non-Gaussian families

for particular Galois group structures.

## The Riemann hypothesis via the generalized von Mangoldt function

Based on work previously done by Gonek, Graham, and Lee, we show that the Riemann Hypothesis (RH) can be reformulated in terms of certain asymptotic estimates for twisted sums with k-fold convolution of von Mangoldt function and the generalized von Mangoldt function. For each $k \in\mathbb{N}$, we study two types of twisted sums:

1. $\sum_{n\leq x} \Lambda^k(n)n^{-iy}$, where $\Lambda^k(n) = \underbrace{\Lambda\star\cdots\Lambda}_\text{k copies}$

2. $\sum_{n\leq x} \Lambda_k(n)n^{-iy}$, where $\Lambda_k(n) :=\sum_{d|n}\mu(d)\left(\log{\frac{n}{d}}\right)^k$.

Where $\Lambda$ is the von Mangoldt function and $\mu$ is the Möbius function, and establish similar connections with RH.

## A simple proof of the Wiener–Ikehara Tauberian theorem

The Wiener–Ikehara Tauberian theorem is an important theorem giving an asymptotic formula for the sum of coefficients of a Dirichlet series. In this talk, we present a simple and elegant proof of the Wiener–Ikehara Tauberian theorem which relies only on basic Fourier analysis and known estimates for the given Dirichlet series. This method allows us to derive a version of the WienerIkehara theorem with an error term. This is joint work with Prof. M. Ram Murty and Prof. Akshaa Vatwani.

## Shifting the ordinates of zeros of the Riemann zeta function

Let $y\neq 0$ and $C>0$. Under the Riemann Hypothesis, there is a number $T_* > 0$ (depending on $y$ and $C$) such that for every $T>T_*$, both

$$

\zeta(\frac{1}{2}+i\gamma) = 0 \qquad \mbox{and} \qquad

\zeta(\frac{1}{2} + i(\gamma + y))\neq 0

$$

hold for at least one $\gamma$ in the interval $[T, T(1+\epsilon]$, where $\epsilon := T^{-C/\log\log T}$.

## Remarks on Landau–Siegel zeros

One of the central problems in comparative prime number theory involves understanding primes in

arithmetic progressions. The distribution of primes in arithmetic progressions are sensitive to real zeros near $s = 1$ of L-functions associated to primitive real Dirichlet characters. The Generalized Riemann Hypothesis implies that such L-functions have no zeros near $s = 1$. In 1935, Siegel proved the strongest known upper bound for the largest such real zero, but his result is vastly inferior to what is known unconditionally for other L-functions. We exponentially improve Siegel’s bound under a mild hypothesis that permits real zeros to lie close to $s = 1$. Our hypothesis can be verified for almost all primitive real characters. Our work extends to other families of L-functions. This is joint work with Jesse Thorner and Alexandru Zaharescu.

## Distribution of Gaussian primes and zeros of L-functions

In this talk, we are interested in the following question: among primes that can be written as a sum of two squares $p = a^2 + 4b^2$ with $a > 0$, how is the congruence class of a distributed? This will lead us to study the distribution of values of Hecke characters from the point of view of Chebyshev’s bias, as well as the distribution of zeros of the associated L-functions and in particular their vanishing at $1/2$.

## Quantitative upper bounds related to an isogeny criterion for elliptic curves

Let $E_1$ and $E_2$ be two non-CM elliptic curves defined over a number field $K$. By an isogeny theorem due to Kulkarni, Patankar, and Rajan, the two curves are geometrically isogenous if and only if the density of primes for which their Frobenius field coincide is positive. In this talk, we present a quantitative upper bounds of this criterion that improves the result of Baier–Patankar and Wong. The strategy relies on effective versions of the Chebotarev Density Theorem. This is joint work with Alina Cojocaru and Auden Hinz.

## Joint Distribution of primes in multiple short intervals

Assuming the Riemann hypothesis (RH) and the linear independence conjecture (LI), we show that the weighted count of primes in multiple disjoint short intervals has a multivariate Gaussian logarithmic limiting distribution with weak negative correlation. As a consequence, we derive short-interval counterparts for many important works in the literature of the Shanks–Rényi prime number race, including a sharp phase transition from all races being asymptotically unbiased to the existence of biased races. Our result remains novel, even for primes in a single moving interval, especially under a quantitative formulation of the linear independence conjecture (QLI).

## Explicit estimates for the Mertens function

We prove explicit estimates of $1/\zeta(s)$ of various orders, and use an improved version of the Perron formula to get explicit estimates for the Mertens function $M(x)$ of order $O(x)$, $O(x/\log^k x)$, and $O(x\log x exp(−\sqrt{\log{x}})$. These estimates are good for small, medium, and large ranges of $x$, respectively.

## Counting “supersingularity” in arithmetic statistics

Supersingularity is a notion to describe certain elliptic curves defined over a field with positive characteristic $p > 0$. Supersingular elliptic curves possess many special properties, such as larger endomorphism rings, extremal point counts, and special p-torsion group scheme structures. This notion was then generalized to higherdimensional abelian varieties. A global function field is associated with an algebraic curve defined over a finite field; the supersingularity of the Jacobian would affect the prime distribution of this function field. In this talk, I want to discuss the effect of supersingularity on prime distribution for function fields and introduce some perspectives to study this phenomenon.