Scientific

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.

Let f be a self-dual Maass form for $SL(n, Z)$. We write $L_f (s)$ for the Godement–Jacquet L-function associated to $f$ and $L_{f\times f} (s)$ for the Rankin–Selberg L-function of $f$ with itself. The inverse of $L_{f\times f} (s)$ is defined by
$$
\frac{1}{L_{f\times f}(s)} := \sum_{m=1}^\infty \frac{c(m)}{m^s}, \mathfrak{R}(s) > 1.
$$
It is well known that the classical Mertens function $M(x) := \sum_{m\leq x} \mu(m)$ is related to
$$
\frac{1}{\zeta(s)} = \sum_{m=1}^\infty \frac{\mu(m)}{m^s}, \mathfrak{R}(s) > 1.
$$
We define the analogue of the Mertens function for $L_{f\times f} (s)$ as $\widetilde{M}(x) := \sum_{m\leq x} c(m)$ and obtain an upper bound for this analogue $\widetilde{M}(x)$, similar to what is known for the Mertens function $M(x)$. In particular, we prove that $\widetilde{M}(x) \ll_f x \exp(−A\sqrt{\log{x}}$ for sufficiently large $x$ and for some positive constant $A$. This is a joint work with my Ph.D. supervisor Prof. A. Sankaranarayanan.

In 2020, Fiorilli and Jouve proved an unconditional Chebyshev bias result for a Galois extension of number fields under a group theoretic condition on its Galois group. We extend their result to a larger family of groups. This leads us to characterize abelian groups enabling extreme biases. In the case of prime power degree extensions, we give a simple criterion implying extreme biases and we also investigate the corresponding Linnik-type question.

As a refinement of Goldfeld’s conjecture, there is a conjecture of Keating–Snaith asserting that $\log L(1/2,E_d)$ for certain quadratic twists $E_d$ of an elliptic curve $E$ behaves like a normal random variable. In light of this, Radziwill and Soundararajan conjectured that the distribution of $\log(|Sha(E_d)|/\sqrt{|d|}$ is approximately Gaussian for these $E_d$, and proved that the conjectures of Keating–Snaith and theirs are both valid “from above”. More recently, under GRH, they further established a lower bound for the involving distribution towards Keating–Snaith’s conjecture. In this talk, we shall discuss the joint distribution of central values and orders of Sha groups of $E_d$ and how to adapt Radziwill–Soundararajan’s methods to study upper bound and lower bounds for such a joint distribution if time allows.

Let $\pi(x; q, a)$ be the number of primes $p\leq x$ such that $p \equiv a (\mod q)$. The classical Shanks–Rényi prime number race problem asks, given positive integers $q \geq 3$ and $2 \leq r \leq \phi(q)$ and distinct reduced residue classes $a_1, a_2, . . . , a_r$ modulo $q$, whether there are infinitely many integers $n$ such that $\pi (n; q, a1) > \pi(n; q, a2) > \cdots > \pi(n; q, ar)$. In this talk, I will describe what is known on this problem when the number of competitors $r \geq 3$, and how this compares to the Chebyshev’s bias case which corresponds to $r = 2$.