Progressing with the series of Departmental Seminars, Department of Mathematics, SRM University-AP, Andhra Pradesh invites Dr Gaurav Bhatnagar, Visiting Associate Professor, Department of Mathematics, Ashoka University to deliver a lecture titled “The Partition-Frequency Enumeration Matrix” on March 10, 2021, at 3 PM.
The Partition-Frequency Enumeration (PFE) matrix is an infinite upper-triangular number-theoretic matrix that is used to enumerate partition-like objects as an elementary approach. This matrix unifies voluminous results connecting number-theoretic functions to partition-type functions. The calculus is extended to arbitrary generating functions and functions with Weierstrass products. As a by-product, some well-known recurrence relations for many number-theoretic functions are recovered. These include the sum of divisors function, Ramanujan’s $\tau$ function, sums of squares and triangular numbers, and $\zeta(2n)$, where $n$ is a positive integer. As an application, Ramanujan’s famous congruences $p(5n+4)\equiv 0$ (mod $5)$ and $\tau(5n+5)\equiv 0$ (mod $5)$ are embedded into an infinite family of such congruences. During the lecture, Dr Gaurav Bhatnagar will highlight two other congruence results concerning the sum of the divisor function.