Alexandru Pascadi (Mathematical Institute, Oxford) -- Thursday, Apr 4, 2024 - 4:00 PM (IST)

Dear all,

The next talk is by Alexandru Pascadi of the University of Oxford. The announcement is as follows. 

Talk Announcement: 

Title:  The dispersion method and beyond: from primes to exceptional Maass forms

Speaker:  Alexandru Pascadi (Mathematical Institute, University of Oxford)
When: April 4, 2024, 4:00 PM- 5:00 PM IST (11:30 AM BST)

Where: Zoom: Write to the organisers for the link

Live LInk: https://youtube.com/live/h9Nil7L8tOI?feature=share

Abstract:

The dispersion method has found an impressive number of applications in analytic number theory, from bounded gaps between primes to the greatest prime factors of quadratic polynomials. The method requires bounding certain exponential sums, using deep inputs from algebraic geometry, the spectral theory of GL2 automorphic forms, and GLn automorphic L-functions. We'll give a broad outline of this process, which combines various types of number theory; time permitting, we'll also discuss the key ideas behind some new results.

Gaurav Bhatnagar (Ashoka University) -- Thursday, Mar 21, 2024 - 4:00 PM (IST)

 Dear all,

The next talk is by Gaurav Bhatnagar of Ashoka University. The announcement is as follows. 

Talk Announcement: 

Title:  Elliptic enumeration and identities

Speaker:  Gaurav Bhatnagar (Ashoka University)
When: Mar 21, 2024, 4:00 PM- 5:00 PM IST 

Where: Zoom: Please write to the organisers for the link.

Live LInk: https://youtube.com/live/cpqHK-R2oXg?feature=share

Abstract

Many of the ideas of $q$-counting and $q$-hypergeometrics are now being extended to the elliptic case. The approach is not very far from the $q$-case. In this talk, we show several examples to illustrate this idea. First we extend some Fibonacci identities using combinatorial methods. Many such identities can be found by telescoping, so we next use telescoping to find elliptic extensions of elementary identities such as the sum of the first  odd or even numbers, the geometric sum and the sum of the first  cubes. In the course of our study, we obtained an identity with many parameters, which appears to be new even in the $q$-case. Finally, we introduce elliptic hypergeometric series and give an extension of some important identities of Liu. As applications, we find 5 double summations and 4 new elliptic transformation formulas. Again, these are new in the $q$-hypergeometric case, where the nome $p$ is 0. 

This is a report of joint work with Archna Kumari and Michael Schlosser. 

Shivani Goel (IIIT, Delhi) - Thursday, Mar 7, 2024 - 4:00 PM (IST)

 Dear all,

The next talk is by Shivani Goel, of the Indraprastha Institute of Information Technology (IIIT), Delhi. The announcement is as follows. 

Talk Announcement: 

Title:  Distribution and applications of Ramanujan sums

Speaker:  Shivani Goel (IIIT, Delhi)
When: Mar 7, 2024, 4:00 PM- 5:00 PM IST 

Where: Zoom: Ask the organisers for the link.

Live LInk: https://youtube.com/live/mQ9EiVeqimI?feature=share

Abstract

While studying the trigonometric series expansion of certain arithmetic functions, Ramanujan, in 1918, defined a sum of the $n^{th}$ power of the primitive $q^{th}$ roots of unity and denoted it as $c_q(n)$. These sums are now known as Ramanujan sums.

Our focus lies in the distribution of Ramanujan sums. One way to study distribution is via moments of averages. Chan and Kumchev initially considered this problem. They estimated the first and second moments of Ramanujan sums.  Building upon their work, we extend the estimation of the moments of Ramanujan sums for cases where $k\ge 3$.  Apart from this, We derive a limit formula for higher convolutions of Ramanujan sums to give a heuristic derivation of the Hardy-Littlewood formula for the number of prime $k$-tuplets less than $x$.