# Alexander Berkovich

## New Weighted Partition Identities, the Smallest Part of Partition and all that 3

#### When/Where:

October 6, 2016, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

I explain how to  use the Jackson transformation  to prove  new partition theorem. This theorem  involves over-partitions counted with the weight  $$(-1)^{1+ \#(\pi) + s(\pi)}$$ and  $$\#$$  representation of $$|\pi|$$  as a sum of two squares. Here $$s(\pi) :=$$ the smallest part of partition $$\pi$$.
If time permits, I will discuss some new results for partitions with distinct even parts.

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

# Alexander Berkovich

## New Weighted Partition Identities, the Smallest Part of Partition and all that 2

#### When/Where:

September 29, 2016, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

I explain how to  use the Jackson transformation  to establish  two new weighted partition theorems. These identities involve unrestricted partitions and over-partitions.  Smallest part of the partition plays an important role in this analysis.

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

# Alexander Berkovich

## New Weighted Partition Identities, the Smallest Part of Partition and all that

#### When/Where:

September 22, 2016, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

I explain how to  use the q-binomial theorem, the q-Gauss sum, and the   transformation of Jackson to discover and prove many new weighted partition identities.These identities involve unrestricted partitions, over-partitions and partitions with distinct even parts. Smallest part of the partitions plays an important role in this analysis.

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

# Michael D. Hirschhorn

## Partitions in three colors

#### When/Where:

April 12, 2016, 3:00 — 3:50pm at LIT 368.

#### Abstract:

We study $$p_3(n)$$, the number of partitions of $$n$$ in three colors, and show that certain sub-sequences are divisible by surprisingly high powers of $$3$$. These results are analogs of results of Ramanujan and Watson for $$p(n)$$ and powers of $$5$$, $$7$$ and $$11$$.

# Colin Defant

## Revisiting the Riemann Zeta Function at Positive Even Integers

#### When/Where:

April 5, 2016, 3:00 — 3:50pm at LIT 368.

#### Abstract:

In his number theory seminar course, Professor Alladi provided a simple inductive proof that $$\zeta(2k)$$ is a rational multiple of $$\pi^{2k}$$ for each positive integer $$k$$. The argument relies on little more than Parseval’s Identity and basic calculus. In this talk, we will begin by proving a new identity involving Bernoulli numbers. We then describe how to use this identity in order to extend Professor Alladi’s argument and obtain a new proof of Euler’s explicit formula for $$\zeta(2k)$$.

# Alexander Berkovich

## New weighted partition theorems inspired by the work of  Alladi

#### When/Where:

March 15, 2016, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

In my first lecture I discussed certain restricted over-partitions related to the Rogers  false theta function   $$\Psi (q^2, q)$$.   In this  lecture I will focus on  partitions with distinct odd parts subject to some additional conditions. This talk is based on my recent joint work with Ali Uncu.

# Alexander Berkovich

## New weighted partition theorems inspired by the work of  Alladi

#### When/Where:

March 8, 2016, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

I will show how to use the Rogers-Fine identities for false theta functions to gain new insights into properties of certain over-partitions and   partitions with distinct odd parts subject to some additional conditions. This talk is based on my recent joint work with Ali Uncu.

# George E. Andrews

## 4-Shadows in q-Series, the Kimberling Index, and Garden of Eden Partitions

#### When/Where:

February 23, 2016, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

This talk is devoted to discussing the implications of a very elementary technique for proving mod 4 congruences
in the theory of partitions.  It leads in unexpected ways to partitions investigated by Clark Kimberling and
to Garden of Eden partitions

## On the local distribution of the number of small prime factors

#### When/Where:

February 16, 2016, 3:00 — 3:50 pm at LIT 368.

#### Abstract:

In the second lecture, I discussed the my work on the distribution of the number of prime factors among integers all of whose prime factors are large, or all are small. With that as a background, I will discuss in this third lecture the local distribution of $$\nu_y(n)$$, the number of prime factors of $$n$$ which are less than $$y$$. This leads to Todd Molnar’s thesis. More precisely, I will consider asymptotic estimates for $$N_k(x,y)$$, the number of integers $$\le x$$ for which $$\nu_y(n)=k$$, as $$y$$ varies with $$x$$ and $$k$$ varies as well. When $$\frac{\log x}{\log y} >1$$, the behavior of $$N_k(x,y)$$ is strikingly different from the classical case, but as k tends to $$\log\log y$$, the mean, the behavior is as in the classical case. To study this problem, we investigate sums involving $$z^{\nu_y(n)}$$, where $$z$$ is a complex number. Previously,I had investigated such sums when $$0<z<1$$ using sieve methods. The investigation for complex $$z$$ can be done using the analytic method of Selberg when $$y$$ is small, and by the use of difference-differential equations when $$y$$ is large. The interplay of a variety of techniques is fascinating.