Schmidt type partitions and Partition Analysis

When/Where:

January 18, 2022, 1:55 — 2:45 pm at LIT 368.

Abstract:

In 1999, Frank Schmidt noted that the number of partitions of integers with distinct parts in which the first, third, fifth, etc., summands add to n is equal to p(n), the number of partitions of n. In this talk, we shall provide a context for this result via MacMahon’s Partition Analysis which leads directly to many other theorems of this nature, and which can be viewed as a continuation of our work on elongated partition diamonds. We find that generating functions which are infinite products built by the Dedekind eta function lead to interesting arithmetic theorems and conjectures for the related partition functions.

This is joint work with my PhD student Peter Paule.

Parity questions concerning the generalized divisor function with restrictions on the prime factors

When/Where:

April 9, 2019, 3:00 — 3:50 pm at LIT 368.

Abstract:

Let \(\nu_y(n)\) denote the number of distinct prime factors of n which are \(<y\). We discuss the asymptotic behavior of the sum

\( S_{-k}(x,y)=\sum_{n\le x}(-k)^{\nu_y(n)}.\)

The emphasis is on uniform estimates as y varies in the interval \([2,x]\). We study this sum using a combination of analytic techniques, sieve methods, and difference- differential equations. It turns out that there is a difference in behavior when \(k=p-1\), and \(k\ne p-1\), where p is a prime, which is clearly understood by considering certain Dirichlet series and representing them in terms of the Riemann zeta function.

This is joint work with my PhD student Ankush Goswami.

The parity of the number of prime factors among integers with restrictions

When/Where:

April 2, 2019, 3:00 — 3:50 pm at LIT 368.

Abstract:

This talk will provide the background for my second talk next week on the parity of the generalized divisor function, which is recent joint work with my PhD student Ankush Goswami.

The celebrated Prime Number Theorem is equivalent to the statement that the number of prime factors function takes odd and even values with asymptotically equal frequency. More refined versions of this statement are equivalent to the Riemann Hypothesis. In this talk, as a background, I will present my results dating back to 1977-87 on the parity of the number of prime factors in which we consider
(i) only integers all whose prime factors are >y, or
(ii) integers for which all prime factors are <y, or
(iii) all integers, but their prime factors are <y.

Here the integers are from an interval [1,x], and the emphasis is on uniform estimates as y varies with x. These questions provide striking variations of the classical theme. The problems are attacked by using a combination of classic analytic techniques, sieve methods, and difference-differential equations

The Ramanujan-Dyson identities and George Beck’s Congruence Conjectures

When/Where:

March 19, 2019, 3:00 — 3:50 pm at LIT 368.

Abstract:

Dyson’s famous conjectures (proved by Atkin and Swinnerton-Dyer) give a combinatorial interpretation of Ramanujan’s congruences for the partition function. The proofs of these conjectures center on the universal mock theta function associated with the rank of a partition. George Beck has generalized the study of partition function congruences related to the rank by considering the total number of parts in the partitions of n. The related generating functions are no longer part of the world of mock theta functions.

However, George Beck has conjectured that certain linear combinations of the related enumerating functions do satisfy congruences modulo 5 and 7.

We shall describe the proofs of these conjectures.

Ramanujan Graphs and Error Correcting Codes

When/Where:

March 12, 2019, 3:00– 3:50 pm at the 3rd floor Atrium.

Abstract:

While many of the classical codes are cyclic, a long standing conjecture asserts that there are no `good’ cyclic codes. In recent years, interest in symmetric codes has been stimulated by Kaufman, Sudan, Wigderson and others

From Expander Graphs to High Dimensional Expanders

When/Where:

March 12, 2019, 1:55 — 2:45 pm at the 3rd floor Atrium.

Abstract:

How Ramanujan May Have Thought of the Mock Theta Functions

When/Where:

February 19, 2019, 3:00 — 3:50 pm at LIT 368.

Abstract:

Polynomial identities implying two Capparelli’s partition theorems

When/Where:

February 12, 2019, 3:00 — 3:50 pm at LIT 368.

Abstract:

I continue to review my recent joint work with Ali Uncu entitled ” Polynomial polynomial identities implying Capparreli’s parttion theorems”. I will show how to use the duality transformation (\(q->1/q\)) to derive two new partition identities.

Elementary polynomial identities implying two Capparelli’s partition theorems

When/Where:

February 5, 2019, 3:00 — 3:50 pm at LIT 368.

Abstract:

I review my recent joint work with Ali Uncu, entitled ” Elementary polynomial identities involving q-Trinomial coefficients” .

Remarks on Bressoud’s positivity conjecture

When/Where:

January 29, 2019, 3:00 — 3:50 pm at LIT 368.

Abstract:

I will use q-trinomial coefficients to prove Bressoud’s conjecture for certain infinite class of polynomials.