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.

The number of prime factors function and integers with restricted prime factors

When/Where:

February 2, 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.

Legendre Theorems, Mock theta functions and Overpartitions

When/Where:

January 19, 2016, 3:00 — 3:50 pm at LIT 368.

Abstract:

This is a report on two papers which are joint work with Atul Dixit and Ae Ja Yee.  We begin by studying the generating function for partitions with repeated (resp. distinct) parts such that each odd part is less than twice the smallest part. Surprisingly, the generating function turns out to be \(\omega(q)q\) (resp. \(\nu(-q)\), where \(\omega(q)\) and \(\nu(q)\) are two of the third order mock theta functions of Ramanujan.  This work is related to recent work of Garvan and Jennings-Shaffer.  There are neat Legendre theorems for both \(\omega(q)\) and \(\nu(q)\), and this leads to comparable Legendre theorems for subclasses of overpartitions that also naturally arise from the Garvan/Jennings-Shaffer work.

Extending Ramanujan’s Dyson rank function identity to all primes greater than 3

When/Where:

December 1, 2015, 3:00 — 3:50 pm at LIT 368.

Abstract:

Let \(R(z,q)\) be the two-variable generating function for Dyson’s rank function. In his lost notebook Ramanujan gives the 5-dissection of \(R(\zeta_p,q)\)
where \(\zeta_p\)  is a primitive \(p\)-th root of unity and \(p=5\). This result is related to Dyson’s famous rank conjecture which was proved by Atkin
and Swinnerton-Dyer.
We show that there is an analogous result for the \(p\)-dissection of \(R(\zeta_p,q)\) when \(p\) is any prime greater than 3.
This extends previous work of Bringmann and Ono, and Ahlgren and Treneer.

Ranges of Divisor Functions

When/Where:

November 10, 2015, 3:00 — 3:50pm at LIT 368.

Abstract:

For any complex number \(c\), let \(\sigma_c : N \rightarrow C\) be the divisor function given by \(\sigma_c(n) =\displaystyle\sum_{d|n} d^c\).

In this talk, I will discuss some of the history of divisor functions. I will then discuss some of my work concerning the ranges \(\sigma_c(N)\). In particular, I will focus on some of the basic topological properties of \(\sigma_{-r}(N)\) for \(r > 1\).

Combinatorial interpretation of the Berkovich-Warnaar identity
for Rogers-Szego polynomials

When/Where:

November 3, 2015, 3:00 — 3:50pm at LIT 368.

Abstract:

I will show that (2004)  Berkovich-Warnaar identity for Rogers-Szego polynomials  can be proven by means of the Sylvester map:  partitions into odd parts maps to  partitions into distinct parts.
This talk is basted on  joint work with Ali Uncu.

On Ramanujan’s identities involving the Eisenstein series and hyper-geometric series

When/Where:

October 20, 2015, 3:00 — 3:50pm at LIT 368.

Abstract:

We shall study the properties of elliptic functions based on the differential equations
\(y’^2 = T_n(y) – (1 – 2\mu^2); \)
where \(\mu^2\) is a constant, \(T_n(x)\) are the Chebyshev polynomials with \(n = 3; 4; 6\) and the initial condition for each case is to be given later. The solutions of the differential equations will be expressed explicitly in terms of the Weierstrass elliptic function and can be used to construct theories of elliptic functions based on \({}_2F_1(1/4; 3.4; 1; z)\) and \({}_2F_1(1/3; 2/3; 1; z)\) and \({}_2F_1(1/6; 5/6; 1; z)\). From the perspective of differential equations, they are natural analogues of the classical elliptic functions of Jacobi and Weierstrass derived, respectively, from the solutions of the differential equations

\(y’2 = (1- y^2)(1 – k^2y^2);\;\; y(0) = 0\)

and
\(y’^2 = 4y^3 – g_3y – g_3; y(0) = \infty.\)

On some partitions with conditions on even/odd indexed odd parts II

When/Where:

September 1, 2015, 3:00 — 3:50pm at LIT 368.

Abstract:

I discuss my recent joint work with Ali Uncu.

In particular, I will show that generating function for

partitions into parts \(\leq 2L\) with   BG-rank \(=k\) is given by

 

\(\displaystyle\frac{1}{ (q^2;q^2)_{L-k}(q^2;q^2)_{L+k}}\)

 

and  discuss combinatorial  explanation  of this result.

On some partitions with conditions on even/odd indexed odd parts

When/Where:

August 25, 2015, 3:00 — 3:50pm at LIT 368.

Abstract:

I discuss my recent joint work with Ali Uncu.

In particular, I will show that generating function for

partitions into parts \(\leq 2L\) with BG-rank \(=k\) is given by

 

\(\displaystyle\frac{1}{ (q^2;q^2)_{L-k}(q^2;q^2)_{L+k}}\)

The Combinatorics of the leading root of Ramanujan’s function

When/Where:

August 21, 2015, 1:55 — 2:45pm at LIT 368.

Abstract:

I consider the leading root \(x_0(q)\) of Ramanujan’s function
\(\sum\limits_{n=0}^\infty\frac{(-x)^nq^{n^2}}{(1-q)(1-q^2)\ldots(1-q^n)}\).
I prove that its formal power series expansion
\(qx_0(-q)=1+q+q^2+2q^3+4q^4+8q^5+\ldots\)

has positive integer-valued coefficients, by giving an explicit combinatorial interpretation
of these numbers in terms of trees whose vertices are decorated with polyominos.

Similar results are obtained for the leading roots of the partial Theta function and the
Painleve Airy function.