Prime Number Error Terms
Mon, Jun 17, 2024, Location: PIMS, University of British Columbia Conference: Comparative Prime Number Theory CRG: L-Functions in Analytic Number Theory
Abstract:
In 1980 Montgomery made a conjecture about the true order of the error term in the prime number theorem. In the early 1990s Gonek made an analogous conjecture for the sum of the Mobius function. In 2012 I further revised Gonek’s conjecture by providing a precise limiting constant. This was based on work on large deviations of sums of independent random variables. Similar ideas can be applied to any prime number error term. In this talk I will speculate about the true order of prime number error terms.
The sum of the Möbius function
Date: Mon, Feb 27, 2023. Conference: PIMS Network Wide Courses: Analytic Number Theory II
CRG: L-Functions in Analytic Number Theory
Moments of L-functions
(joint lecture with Alia Hamieh (UNBC))
21:09-27:43
Date: Nov. 19 ,2022. Conference: L-functions in Analytic Number Theory Launch, BIRS
CRG: L-Functions in Analytic Number Theory
Moments of the Riemann zeta function
Date: Nov. 7, 2021. Conference: Alberta Number Theory Days XIII, BIRS.
Abstract:
For over 100 years, I_k(T), the 2k-th moments of the Riemann zeta function on the critical line, have been extensively studied. In 1918 Hardy-Littlewood established an asymptotic formula for the second moment (k=1) and in 1926 Ingham established an asymptotic formula for the fourth moment (k=2). Since then, no other moments have been asymptotically evaluated. In the late 1990's Keating and Snaith gave a conjecture for the size of I_k(T) based on a random matrix model. In this talk I will give a historical overview of the advances on I_k(T) and the techniques used to study them since the beginning of the twentieth century.
Inclusive Prime Number Races
Abstract:
Let $\pi(x;q,a)$ denote the number of prime $x$ that are congruent to a (mod q). A "prime number race", for fixed modulus q and residue classes $a_1, \ldots, a_r$ investigates the system of inequalities $\pi(x;q,a_1) > \cdots $\pi(x;q,a_r)$. We expect that this system should have arbitrarily large solutions x, and moreover we expect the same to be true no matter how we permute the residue classes a_j; if this is the case, the prime number race is called "inclusive". Rubinstein and Sarnak proved conditionally that every prime number race is inclusive; they assumed not only the generalized Riemann hypothesis but also a strong statement about the linear independence of the zeros of Dirichlet L-functions. We show that the same conclusion can be reached with a substantially weaker linear independence hypothesis. This is joint work with Greg Martin.
The linear independence conjecture for zeros of L-functions
Riemann Zeta Function Workshop:
November 3, 2012
Dr. Nathan Ng,
University of Lethbridge
The distribution of the Mobius function in short intervals
Date: Friday, May 16th, 2008, Location: IRMACS, SFU, Conference: The mathematical interests of Peter Borwein.
Abstract:
In this talk we study the distribution of the Mobius function in a short interval [x, x + h] with x less than N. We give an argument which suggests that the distribution is approximately normal with mean 0 and variance 6h/π2 . This argument is based on Montgomery and Soundararajan’s work on primes in short intervals.