An explicit upper bound on the least primitive root

Speaker: 

Kevin McGown

Institution: 

CSU Chico

Time: 

Thursday, October 10, 2019 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306
Let $p$ be an odd prime. A classical problem in analytic number theory is to give an upper bound on the least primitive root modulo $p$, denoted by $g(p)$. In the 1960s Burgess proved that for any $\varepsilon>0$ one has $g(p)\ll p^{1/4+\varepsilon}$ for sufficiently large $p$. This was a consequence of his landmark character sum inequality, and this result remains the state of the art. However, in applications, explicit estimates are often required, and one needs more than an implicit constant that depends on $\varepsilon$. Recently, Trudgian and the speaker have given an explicit upper bound on $g(p)$ that improves (by a small power of log factor) on what one can obtain using any existing version of the Burgess inequality. In particular, we show that $g(p)<2r\,2^{r\omega(p-1)}p^{\frac{1}{4}+\frac{1}{4r}}$ for $p>10^{56}$, where $r\geq 2$ is an integer parameter. \[ \ \] In 1952 Grosswald showed that if $g(p)<\sqrt{p}-2$, then the principal congruence subgroup $\Gamma(p)$ for can be generated by the matrix $[1,p;0,1]$ and $p(p-1)(p+1)/12$ other hyperbolic matrices. He conjectured that $g(p)<\sqrt{p}-2$ for $p>409$. Our method allows us to show that Grosswald's conjecture holds unconditionally for $p> 10^{56}$, improving on previous results.

The Weyl law for algebraic tori

Speaker: 

Ian Petrow

Institution: 

ETH Zurich

Time: 

Wednesday, May 1, 2019 - 3:00pm to 4:00pm

Host: 

Location: 

RH 440R

A basic but difficult question in the analytic theory of automorphic forms is: given a reductive group G and a representation r of its L-group, how many automorphic representations of bounded analytic conductor are there? In this talk I will present an answer to this question in the case that G is a torus over a number field.

Factorization of Hasse-Weil zeta functions of Dwork surfaces

Speaker: 

Lian Duan

Institution: 

Univ. of Mass, Amherst

Time: 

Thursday, April 11, 2019 - 3:00pm to 4:00pm

Host: 

Location: 

RH 306

As a classical example of K3 surfaces, the Dwork surface family is of interest in algebraic geometry and number theory. A lot of work has been done to understand the Hasse-Weil zeta functions of these surfaces. Recent works show that people can totally determine the algebraic part of the zeta function for a general Dwork surface. In this talk, we discuss how to use geometric method to find the explicit factorization of the algebraic part.

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