We study sums of the form ∑n≤xf(n)n−iy, where f is an arithmetic function, and we establish an equivalence between the Riemann Hypothesis and estimates on these sums. In this talk, we will explore examples of such sums with specific arithmetic functions, as well as discuss potential implications and future research directions.
We will present a matching upper and lower bound for the right tail
probability of the maximum of a random model of the Riemann zeta function over
short intervals. In particular, we show that the right tail interpolates
between that of log-correlated and IID random variables as the interval varies
in length. We will also discuss a new normalization for the moments over short
intervals. This result follows the recent work of Arguin-Dubach-Hartung and is inspired by a conjecture by Fyodorov-Hiary-Keating on the local maximum over
short intervals.
Lethbridge Number Theory and Combinatorics Seminar
Abstract:
In 1973, assuming the Riemann hypothesis (RH), Montgomery studied the vertical distribution of zeta zeros, and conjectured that they behave like the eigenvalues of some random matrices. We will discuss some models for zeta zeros starting from the random matrix model but going beyond it and related questions, conjectures and results on statistical information on the zeros. In particular, assuming RH and a conjecture of Chan for how often gaps between zeros can be close to a fixed non-zero value, we will discuss our proof of a conjecture of Berry (1988) for the number variance of zeta zeros, in a regime where random matrix models alone do not accurately predict the actual behavior (based on joint work with Meghann Moriah Lugar and Micah B. Milinovich).
Lethbridge Number Theory and Combinatorics Seminar
Abstract:
Let G be a graph with adjacency matrix A. A continuous quantum walk on G is determined by the complex unitary matrix U(t)=exp(itA), where i2=−1andtisarealnumber.Here,Grepresentsaquantumspinnetwork,anditsverticesandedgesrepresenttheparticlesandtheirinteractionsinthenetwork.ThepropagationofquantumstatesinthequantumsystemdeterminedbyGisthengovernedbythematrixU(t).Inparticular,|U(t)_{u,v}|^2maybeinterpretedastheprobabilitythatthequantumstateassignedatvertexuistransmittedtovertexvattimet$. Quantum walks are of great interest in quantum computing because not only do they produce algorithms that outperform classical counterparts, but they are also promising tools in the construction of operational quantum computers. In this talk, we give an overview of continuous quantum walks, and discuss old and new results in this area with emphasis on the concepts and techniques that fall under the umbrella of discrete mathematics.
In this talk, we will discuss a well-known formula of Ramanujan and its relationship with the partial sums of the Möbius function. Under some conjectures, we analyze a finer structure of the involved terms. It is a joint work with Steven M. Gonek (University of Rochester).
A zero-free region of the Riemann zeta-function is a subset of the
complex plane where the zeta-function is known to not vanish. In this talk we
will discuss various computational and analytic techniques used to enlarge the
zero-free region for the Riemann zeta-function, when the imaginary part of a
complex zero is large. We will also explore the limitations of currently known
approaches. This talk will reference a number of works from the literature,
including a joint work with M. Mossinghoff and T. Trudgian.
If we assume the relevant Riemann hypotheses, after a suitable rescaling many functions counting certain primes become almost periodic. There are different notion of almost periodicity in use; here we consider the notion induced by the norm ||f||=sup. We show that if a function f can be approximated by linear combinations of periodic functions with respect to this norm, then the level sets \left\{x: f(x) \geq t\right\} are almost periodic for all real t with at most countably many exceptions. We also compare this notion to other notions of almost periodicity in use.
Please note, the wrong video feed was captured for this lecture so the writing on the blackboard is not legible.
Given an elliptic curve E/\mathbb{Q}, we can consider its trace of Frobenius, denoted as a_p(E), where p is a prime. We will discuss the race problem arising from these ap values and the general strategy in attacking these problems.
In 1999, Gadiyar and Padma discovered a simple heuristic to derive the generalized twin prime conjecture using an orthogonality principle for Ramanujan sums originally discovered by Carmichael. We derive a limit formula for higher convolutions of Ramanujan sums, generalizing an old result of Carmichael. We then apply this in conjunction with the general theory of arithmetical functions of several variables to give a heuristic derivation of the Hardy–Littlewood formula for the number of prime k-tuples less than x.
It has been known since the 80s, thanks to Conrey and Ghosh, that the average of the square of the Riemann zeta function, summed over the extreme points of zeta up to a height T, is \frac{1}{2}(e^2 −5)\log T as T\rightarrow \infty. This problem and its generalisations are closely linked to evaluating asymptotics of joint moments of the zeta function and its derivatives, and for a time was one of the few cases in which Number Theory could do what Random Matrix Theory could not. RMT then managed to retake the lead in calculating these sorts of problems, but we may now tell the story of how Number Theory is fighting back, and in doing so, describe how to find a full asymptotic expansion for this problem, the first of its kind for any nontrivial joint moment of the Riemann zeta function. This is joint work with Chris Hughes and Solomon Lugmayer