# Robert C. Vaughan

## Zeros of Dirichlet series

#### When/Where:

April 7, 2015, 1:55 — 2:45pm at LIT 368.

#### Abstract:

We are concerned here with Dirichlet series

$$f(s) = 1 +sum_{n=2}^{infty} frac{c(n)}{n^s}$$

which satisfy a function equation similar to that of the Riemann zeta function, typically of the form

$$f(s) = 2^s q^{1/2-s} pi^{s-1} Gamma(1-s) big(sintextstylefrac{pi}{2}(s+kappa)big) f(1-s),$$

but for which the Riemann hypothesis is false. We establish a plethora of such functions and show that the zeros of such functions are ubiquitous in the complex plane.

# Robert C. Vaughan

## Goldbach: A modern panorama

#### When/Where:

April 7, 2015, 10:40 — 11:30am at LIT 339 (Atrium).

#### Abstract:

Here I will build on my first talk by showing how many of the ideas stimulated by interest in the Goldbach conjectures and related questions have had widespread applications.  We illustrate this by considering the Montgomery-Hooley theorem on the distribution of primes in arithmetic progression, variants of the Goldbach questions in which the primes are restricted in peculiar ways, the question of prime partitions, a proof of the Bombieri–Vinogradov theorem, and the recent work of Zhang, Maynard and Tao on primes in short intervals which brings many of these ideas together.

# George E. Andrews

## Partitions With Early Conditions and Mock Theta Functions

#### When/Where:

March 31 2015, 1:55 — 2:45pm at LIT 368.

#### Abstract:

This talk is devoted to joint work with Stephen Hill (a Penn State undergraduate). In 1961, Basil Gordon proved a sweeping generalization of the Rogers-Ramanujan identities. His theorem may be broadly characterized as identifying the generating function for partitions having specified difference conditions on the parts with the quotient of two theta functions. We shall provide a new class of partitions (similar to those studied by Gordon) where the generating function is identified with the quotient of a Hecke-type theta series divided by the Dedekind eta function. The simplest case is related to one of the fifth order mock theta functions of Ramanujan. The partitions in question are similar in kind to those described in two earlier papers, Partitions with initial repetitions, Acta Math. Sinica, English Series, 25(2009), 1437-1442, and, Partitions with early conditions, Advances in Combinatorics Waterloo Workshop in Computer Algebra, W80 May. 26-29, 2011.

# Ali Kemal Uncu

## A new companion to Capparelli’s identities and the allied inequalities

#### When/Where:

March 17 2015, 1:55 — 2:45pm at LIT 368.

#### Abstract:

We will start by the discussion of the new companion results to Capparelli’s identities found in a joint work with Prof Berkovich late 2014. Later, I will state and prove a new partition identity which is a refinement of this companion result. Then we will start the comparison of our refinement and the refinement that Alladi and Andrews proved in 1994. Lastly, I will discuss some new developments in these refinements and some related combinatorial inequalities that are still open questions to date.

# Alexander Berkovich

## On a new companion to Capparelli partition theorem

#### When/Where:

December 2, 1:55 — 2:45pm at LIT 368.

#### Abstract:

I will report on my recent joint work with Ali Uncu.

In particular, I will prove the following Theorem:

Let $$A(n)$$ be the number of partitions of n of the form $$p_1+ p_2+ p_3 +…$$, where $$p_{2i-r}-p_{2i-r+1} >r$$ for $$r=0$$ or $$1$$. Moreover, $$p_{2i} \not= 2$$ mod $$3$$ and $$p_{2i+1} \not= 1$$ mod $$3$$ .

Let $$C(n)$$ be the number of partitions of $$n$$ into distinct parts $$\not= 1$$ or $$5$$ mod $$6$$.

Then $$A(n) = C(n)$$.

If time permits, I will discuss 3 parameter refinement of this result

# Olivia Beckwith

## Multiplicative properties of the number of k-regular partitions

#### When/Where:

October 21, 1:55 — 2:45pm at LIT 368.

#### Abstract:

Earlier this year, Bessenrodt and Ono proved surprising multiplicative properties of the partition function. In this project, we deal with $$k$$-regular partitions. Extending the generating function for $$k$$-regular partitions multiplicatively to a function on $$k$$-regular partitions, we show that it takes its maximum at an explicitly described small number of partitions, and thus can be easily computed. The basis for this is an extension of a classical result of Lehmer, from which we prove an inequality for the number of $$k$$-regular partitions which seems not to have been noticed before.

# Michael Griffin

## Theorems at the interface of number theory and representation theory

#### When/Where:

October 21, 11:45 — 12:35pm at LIT 305.

#### Abstract:

We will discuss recent work on Moonshine and the Rogers-Ramanujan identities. The Rogers-Ramanujan identities express two infinite product modular forms as number theoretic q-series. These identities give rise to the Rogers-Ramanujan continued fraction, whose values at CM points are algebraic integral units. In recent joint work with Ono, and Warnaar, we obtain a comprehensive framework of identities for infinite product modular forms in terms of Hall-Littlewood q-series. This work characterizes those integral units that arise from this theory.

In a related direction, we revisit the theory of Monstrous Moonshine which asserts that the coefficients of the modular j-function are “dimensions” of virtual characters for the Monster, the largest of the simple sporadic groups. There are 194 irreducible representations of the Monster, and it has been a longstanding open problem to determine the distribution of these representations in Moonshine. In joint work with Ono and Duncan, we obtain exact formulas for these distributions. Moonshine have phenomena have also been observed connecting other sporadic groups such as the Mathieu group M24 to certain mock modular forms. We will also discuss recent developments in this theory of ”Umbral Moonshine.”

# Frank Patane

## An identity connecting theta series associated with binary quadratic forms of discriminant Delta and Delta p^2

#### When/Where:

October 7, 2014, at 1:55pm, in LIT 368

#### Abstract:

I will state and prove a new identity which connects theta series associated with binary quadratic forms of idoneal discriminants $$\Delta$$ and $$\Delta p^2$$, for $$p$$ a prime. I will then illustrate how to use this identity to derive Lambert series identities and hence product representation formulas for certain forms. Last but not least, I discuss generalizations to non-idoneal discriminants and the resulting theta series identities one may derive.

# Alexander Berkovich

## Some implications of the 1907 Hurwitz formula II

#### When/Where:

September 16, 2014, at 1:55pm, in LIT 368

#### Abstract:

Let $$s(n)$$ denote $$\#$$ of representations of integer $$n$$ by $$x^2+y^2+z^2$$. Let $$n$$ be odd positive integer. In my first talk I showed that $$s(n^2)\geq 6n$$ with equality if and only if all prime divisors of $$n$$ are congruent to $$1 \pmod{4}$$. I employ the above inequality and a special case of the Jacobi triple product identity to prove certain conjectures of Kaplansky. In particular, I will show that $$4x^2 + 9y^2 + 32z^2 +4xy$$ represents, exclusively, all positive integers not of the form

• $$4^a(4m+2), a=0,1$$
• $$4^a(8m+3), a=0,1$$
• $$4^a(8m+5), a=0,1$$
• $$4^a(8m+7), a\geq 0,$$
• or $$M^2$$

where $$a,m,M$$ are non-negative integers and $$M$$ is generated by $$1$$ and primes congruent to $$1 \pmod{4}$$.

# Alexander Berkovich

## Some implications of the 1907 Hurwitz formula

#### When/Where:

September 9, 2014, at 1:55pm, in LIT 368

#### Abstract:

I start with a brief review of Siegel’s formula for the number of representations of a positive integer by a genera of a quadratic form. I show how this formula implies the 1907 observation by Hurwitz, who proposed that $$\left|\{ (x,y,z) \in \mathbb{Z}^3 \mid n^2 = x^2+y^2+z^2 \}\right|$$ may be expressed as a simple finite function of the divisors of $$n \in \mathbb{N}$$. I use this formula together with the Jacobi triple product identity to prove (among other things) that $$9x^2+ 16y^2 +36z^2 + 16yz+ 4xz + 8xy$$ represents exclusively all positive integers not of the form

• $$4^a(8m+7)$$;
• $$4^a(8m+3)$$, $$a=0,1,2$$;
• $$4^a(4m+2)$$, $$a=0,1,2$$;
• $$4^a(8m+5)$$, $$a=0,1$$;
• $$M^2$$;
• or $$4M^2$$;

where $$a,m,M$$ are non-negative integers and $$M$$ is generated by 1 and primes congruent to $$1\pmod{4}$$.