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

Krishnaswami Alladi

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.

Krishnaswami Alladi

The number of prime factors function and integers with restricted prime factors , part 2

When/Where:

February 9, 2016, 3:00 — 3:50 pm at LIT 368.

Abstract:

The study of the distribution of of values of the number of prime factors of the integers goes back to Hardy-Ramanujan. Then with the work of Turan and Erdos-Kac, Probabilistic Number Theory was born. Selberg showed how analytic methods could be used to study the “local” distribution of the number of prime factors.

The distribution of integers all of whose prime facts are small, or all are large, is of fundamental importance. N. G. deBruijn obtained strong uniform estimates for these two problems. My work thirty years ago involved a study of the distribution of the number of prime factors among integers all of whose prime factors are large, or all are small. In this first of two talks, I shall review all of the above mentioned results as a background for recent work on the number of restricted prime factors that reveals some surprising variations on the classical theme. These recent observations that lead to the thesis of Todd Molnar, will be the theme of the second lecture.