Number Theory

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 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.

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.

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.

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.