# Alexander Berkovich

## $$(1-q)(1-q^2)\dots(1-q^m)$$ Revisited

#### When/Where:

October 31, 2017, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

Let
$$\displaystyle (1-q)(1-q^2) … (1-q^m) = \sum_{n>= 0} c_m (n) q^n.$$

In this talk I discuss how to use $$q$$- binomial theorem  together with   the Euler pentagonal number theorem to show that  $$\max( |c_m|) = 1$$  iff $$m = 0,\ 1,\ 2,\ 3,\ 5$$.   There are many other similar results.  For example, it can be  proven  that no positive integer m exists such that $$\max( |c_m|) = 9$$.

This talk is based on my recent joint work with  Ali Uncu  (RISC, Linz).

# Larry Rolen

## Jensen-Pólya Criterion for the Riemann Hypothesis and Related Problems

#### When/Where:

October 24, 2017, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

In this talk, I will summarize forthcoming work with Griffin, Ono, and Zagier. In 1927 Pólya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann’s Xi-function. This hyperbolicity has been proved for degrees $$d\leq3$$. We obtain an arbitrary precision asymptotic formula for the derivatives $$\Xi^{(2n)}(0)$$, which allows us to prove the hyperbolicity of $$100\%$$ of the Jensen polynomials of each degree. We obtain a general theorem which models such polynomials by Hermite polynomials. This general condition also confirms a conjecture of Chen, Jia, and Wang.

# Peter Elliott

## Group of Rationals, then ad now

#### When/Where:

March 21, 2017, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

An overview of application of the Fourier analysis applied to the study of multiplicative groups of rationals.

# George E. Andrews

## The combinatorics of the mock theta functions $$\nu(q)$$

#### When/Where:

March 14, 2017, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

The third order mock theta function $$\nu(q)$$ has a variety of interesting combinatorial aspects.  We shall explore several and shall introduce a new class of partitions complete with a new crank and new congruence properties.

# Alexander Berkovich

## Non-negative Thoughts

#### When/Where:

February 28, 2017, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

I discuss some new  inequalities involving certain  partitions with simple restrictions on largest and smallest parts. I will employ  these inequalities to show that

$$\displaystyle \sum_{n>0} \frac{q^{n(n+1)/2}}{(-q;q)_n(1-q^{L+n})(q;q)_L}$$

has non-negative  q-series coefficients for all positive integers $$L$$.

This talk is based on my current joint work with Ali K.  Uncu.

# Colin Defant

## Unitary Cayley Graphs

#### When/Where:

February 14, 2017, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

If $$R$$ is a commutative ring with unity, then the unitary Cayley graph of $$R$$, denoted $$G_R$$ , is the graph with vertex set $$V(G_R)=R$$ and edge set $$E(G_R)=\{\{x,y\} : x-y\text{ is a unit in }R\}$$. We will focus specifically on the unitary Cayley graph of $$\mathbb{Z}/n\mathbb{Z}$$, which we may view as the graph with vertices $$0,1,…,n-1$$ in which two vertices are adjacent if and only if their difference is relatively prime to $$n$$. We provide the values of many graph parameters of these unitary Cayley graphs and find that they are intimately related to some interesting arithmetic functions. We also discuss an open problem concerning the domination numbers of these graphs.

# Ali K. Uncu

## Some new observations on partitions and divisors

#### When/Where:

January 31, 2017, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

We will look through the history and many proofs from many academics of the result best known due to Fokkink, Fokkink and Wang. We will add more proofs to the list of proofs of this identity. Later we will move onto refinements of these results.

This discussion is about my recent joint work with Alexander Berkovich.

# George E. Andrews

## Topics in Partitions

#### When/Where:

January 24, 2017, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

I will begin with the Bhargava-Adiga summation and how it relates Gauss’s Eureka Theorem to partitions.  Assuming that time permits, I will conclude by discussing joint work with David Newman on bizarre factorizations of the classical theta functions.

# Atul Dixit

## Transformations involving $$r_k(n)$$ and Bessel functions

#### When/Where:

January 10, 2017, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

Let $$r_k(n)$$ denote the number of representations of the positive integer $$n$$ as the sum of $$k$$ squares, where $$k\geq 2$$. In 1934, the Russian mathematician Alexander Ivanovich Popov, who is more popularly known as one of the world experts in Finno-Ugric Linguistics, obtained a beautiful transformation between two series involving $$r_k(n)$$ and Bessel functions. Unfortunately, Popov’s proof appears to be defective since there are subtleties involved in extending the available results on $$r_2(n)$$ to those involving $$r_k(n)$$, $$k>2$$, and since the usual techniques do not carry over. In this work, we give a rigorous proof of Popov’s result by observing that N. S. Koshliakov’s ingenious proof of the Voronoi summation formula for coefficients of a Dirichlet series satisfying a functional equation with one gamma factor circumvents these difficulties. We then obtain an analogue of a double Bessel series identity on page 335 of Ramanujan’s Lost Notebook in the spirit of Popov’s identity.
In the second part of our talk, we will obtain a proof of a more general summation formula for $$r_k(n)$$ due to A. P. Guinand, which is claimed in his work without proof and without any conditions, and apply it to obtain a new transformation of a series involving $$r_k(n)$$ and a product of two Bessel functions. This transformation can be considered as a massive generalization of many well-known results in the literature, for example, those of A.L. Dixon and W.L. Ferrar, of G.H. Hardy, and of a classical result of Popov. This is joint work with Bruce C. Berndt, Sun Kim and Alexandru Zaharescu.

# Frank Garvan

## New Mock Theta Function Identities II

#### When/Where:

December 1, 2016, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

This is a continuation of my previous talk.
In his last letter to Hardy, Ramanujan defined there mock theta functions of order 7 and stated that these three functions are not related. We find that there are actually some surprising relationships between these functions.