Speaker :- Dr. Neelam Kandhil

Title of the Talk :- Exploring the interplay of Euler-Kronecker constants, prime numbers, and $L$-value

Date, Time and venue :- 16 June, at 11:30 AM to 12:30 PM at Seminar Hall, Dept of Mathematics.

Abstract :- The Euler-Kronecker constant of a number field $K$ is defined as the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function associated with $K$ at $s=1$. Investigating the distribution of the normalized difference of the Euler-Kronecker constants of the prime cyclotomic field $\mathbb{Q}(\zeta_q)$ and its maximal real subfield, we connect it with Kummer's conjecture. This conjecture predicts the asymptotic growth of the relative class number of prime cyclotomic fields. The ratio of this class number and the simple asymptotic that Kummer predicted for it is now called the Kummer ratio. Furthermore, we discuss recent advancements in understanding the irregularities in the behavior of the Kummer ratio shedding light on deep questions related to prime number distribution. In addition, assuming the Generalized Riemann Hypothesis and a pair correlation conjecture for the zeros of Dirichlet $L$-functions, we establish the truth of a conjecture of Montgomery on the magnitude of the error term in the prime number theorem in arithmetic progressions.

About the speaker :- Dr. Neelam Kandhil, currently a Postdoctoral Researcher at the University of Hong Kong, Hong Kong SAR, and her research interests lie in Number Theory. Prior to  this, she has done  a postdoc at the Max-Planck-Institut für Mathematik, Germany.  She obtained, Ph.D. in Mathematics from the Institute of Mathematical Sciences, Chennai, in 2022.