## Number Theory Seminar

### Wednesday, November 29th, 2017

**Time:** 1:30 p.m. **Place:** Jeffery Hall 319

**1 ^{st} Speaker:** M. Ram Murty

**Title:** Hilbert’s tenth problem over number fields

**Abstract: ** Hilbert's tenth problem for rings of integers of number fields remains open in general,

although a conditional negative solution was obtained by Mazur and Rubin assuming some unproved conjectures about the Shafarevich-Tate groups of elliptic curves. In this talk, we highlight how the non-vanishing of certain L-functions is related to this problem. In particular, we show that Hilbert's tenth problem for rings of integers of number fields is unsolvable assuming the automorphy of L-functions attached to elliptic curves and the rank part of the Birch and Swinnerton-Dyer conjecture. This is joint work with Hector Pasten.

**2 ^{nd} Speaker:** François Séguin

**Title:** A lower bound for the two-variable Artin conjecture

**Abstract: **In 1927, Artin conjectured that any integer other than -1 or a perfect square generates the multiplicative group Z/pZ× for infinitely many p. In a 2000 article, Moree and Stevenhagen considered a two-variable version of this problem, and proved a positive density result conditionally to the generalized Riemann Hypothesis by adapting a 1967 proof by Hooley for the original conjecture. During this talk, we will prove an unconditional lower bound for this two-variable problem. This is joint work with Ram Murty and Cameron Stewart.

**3 ^{rd} Speaker:** Arpita Kar

**Title:** On a Conjecture of Bateman about $r_5(n)$

**Abstract: **Let $r_5(n)$ be the number of ways of writing $n$ as a sum of five integer squares. In his study of this function, Bateman was led to formulate a conjecture regarding the sum $$\sum_{|j| \leq \sqrt{n}}\sigma(n-j^2)$$ where $\sigma(n)$ is the sum of positive divisors of $n$. We give a proof of Bateman's conjecture in the case $n$ is square-free and congruent to $1$ (mod $4$). This is joint work with Prof. Ram Murty.

**4 ^{th} Speaker:** Siddhi Pathak

**Title:** Derivatives of L-series and generalized Stieltjes constants

**Abstract: **Generalized Stieltjes constants occur as coefficients of (s-1)^k in the Laurent series expansion of certain Dirichlet series around s=1. The connection between these generalized Stieltjes constants and derivatives of L(s,f) for periodic arithmetical functions f, at s=1 is known. We utilize this link to throw light on the arithmetic nature of L'(1,f) and certain Stieltjes constants. In particular, if p is an odd prime greater than 7, then we deduce the transcendence of at least (p-7)/2 of the generalized Stieltjes constants, {gamma_1(a,p) : 1 \leq a < p }, conditional on a conjecture of S. Gun, M. Ram Murty and P. Rath.