Number theory seminar

St. Petersburg State University and Euler International Mathematical Institute are happy to announce a new Number Theory seminar, headed by Eric Mortenson. To participate in the seminar, join zoom channel 310-172-1994, password: the number of quadratic non-residues modulo 23.

Forcoming talks

October 1, 18:00-19:00

Frank Garvan

A new approach to Dyson’s rank conjectures

In 1944 Dyson defined the rank of a partition as the largest part minus the number of parts, and conjectured that the residue of the rank mod 5 divides the partitions of 5n+4 into five equal classes. This gave a combinatorial explanation of Ramanujan’s famous partition congruence mod 5. He made an analogous conjecture for the rank mod 7 and the partitions of 7n+5. In 1954 Atkin and Swinnerton-Dyer proved Dyson’s rank conjectures by constructing several Lambert-series identities basically using the theory of elliptic functions. In 2016 the author gave another proof using the theory of weak harmonic Maass forms. In this talk we describe a new and more elementary approach using Hecke-Rogers series.

October 5, 17:00-18:00

Damaris Schindler

On the distribution of Campana points on toric varieties

In this talk we discuss joint work with Marta Pieropan on the distribution of Campana points on toric varieties. We discuss how this problem leads us to studying a generalised version of the hyperbola method, which had first been developed by Blomer and Bruedern. We show how duality in linear programming is used to interpret the counting result in the context of a general conjecture of Pieropan-Smeets-Tanimoto-Varilly-Alvarado.

Past talks

September 24, 18:00-19:00

Matthias Beck

Partitions with fixed differences between largest and smallest parts

Enumeration results on integer partitions form a classic body of mathematics going back to at least Euler, including numerous applications throughout mathematics and some areas of physics. We study the number p(n,t) of partitions of n with difference t between largest and smallest parts. For example, p(n,0) equals the number of divisors of n, the function p(n,1) counts the nondivisors of n, and p(n,2) = \binom{ \left\lfloor \frac n 2 \right\rfloor }{ 2 }. Beyond these three cases, the existing literature contains few results about p(n,t), even though concrete evaluations of this partition function are featured in several entries of Sloane’s Online Encyclopedia of Integer Sequences.

Our main result is an explicit formula for the generating function P_t(q) := \sum_{ n \ge 1 } p(n,t) \, q^n. Somewhat surprisingly, P_t(q) is a rational function for t>1; equivalently, p(n,t) is a quasipolynomial in n for fixed t>1 (e.g., the above formula for p(n,2) is an example of a quasipolynomial with period 2). Our result generalizes to partitions with an arbitrary number of specified distances.

This is joint work with George Andrews (Penn State) and Neville Robbins (SF State).

Presentation slides

 

September 16, 19:00-20:00

Bruce Berndt

The Circle Problem of Gauss, the Divisor Problem of Dirichlet, and Ramanujan’s Interest in Them

Let r_2(n) denote the number of representations of the positive integer n as a sum of two squares, and let d(n) denote the number of positive divisors of n. Gauss and Dirichlet were evidently the first mathematicians to derive asymptotic formulas for \sum_{n\leq x}r_2(n) and \sum_{n\leq x}d(n), respectively, as x tends to infinity. The magnitudes of the error terms for the two asymptotic expansions are unknown. Determining the exact orders of the error terms are the Gauss Circle Problem and Dirichlet’s Divisor Problem, respectively, and they represent two of the most famous and difficult unsolved problems in number theory.

Beginning with his first letter to Hardy, it is evident that Ramanujan had a keen interest in the Divisor Problem, and from a paper written by Hardy and published in 1915, shortly after Ramanujan arrived in England, we learn that Ramanujan was also greatly interested in the Circle Problem. In a fragment published with his Lost Notebook, Ramanujan stated two doubly infinite series identities involving Bessel functions that we think Ramanujan derived to attack these two famous unsolved problems. The identities are difficult to prove. Unfortunately, we cannot figure out how Ramanujan might have intended to use them. We survey what is known about these two unsolved problems, with a concentration on Ramanujan’s two marvelous and mysterious identities. Joint work with Sun Kim, Junxian Li, and Alexandru Zaharescu is discussed.

Presentation slides

 

September 7, 17:00-18:00

Ken Ono

Variations of Lehmer’s Conjecture on the Nonvanishing of the Ramanujan tau-function

In the spirit of Lehmer’s unresolved speculation on the nonvanishing of Ramanujan’s tau-function, it is natural to ask whether a fixed integer is a value of \tau(n), or is a Fourier coefficient of any given newform. In joint work with J. Balakrishnan, W. Craig, and W.-L. Tsai, the speaker has obtained some results that will be described here. For example, infinitely many spaces are presented for which the primes \ell\leqslant 37 are not absolute values of coefficients of any new forms with integer coefficients. For Ramanujan’s tau-function, such results imply, for n>1, that
\tau(n)\notin \{\pm \ell\,:\, \ell<100\,\text{is odd prime}\}.

Presentation slides