## Number Theory Seminar

### Wednesday, June 27th, 2018

**Time:** 2:30-3:20p.m. **Place:** Jeffery Hall 422

**Speaker:** Atul Dixit (IIT Gandhinagar, India)

**Title:** RAMANUJAN'S FORMULA FOR ODD ZETA VALUES AND SUBSEQUENT DEVELOPMENTS.

**Abstract:** While the Riemann zeta function $\zeta(s)$ at even positive integers is known to be always transcendental, the arithmetic nature of $\zeta(s)$ at odd positive integers remains mysterious with only $\zeta(3)$ known to be irrational, thanks to Ap\'{e}ry. In 1901, Lerch obtained a beautiful result involving $\zeta(2m+1), m$ odd, and an Eisenstein series, which implies that either $\zeta(2m+1)$ or the Eisenstein series is transcendental. In his famous notebooks, Srinivasa Ramanujan obtained a beautiful result generalizing that of Lerch. This result has had a tremendous impact on Mathematics with its applications in modular forms, computer science, to name a few. A brief historical survey on Ramanujan's formula will be given. We will then concentrate on the generalized Lambert series $\sum_{n=1}^{\infty}\frac{n^{N-2h}}{e^{n^{N}x}-1}$ studied by Kanemitsu, Tanigawa and Yoshimoto, which we recently found to be located on page $332$ of Ramanujan's Lost Notebook in a slightly more general form. In a joint work with Bibekananda Maji, we have extended a transformation of this series given by the above authors. The novel feature of this extension is that it not only gives Ramanujan's formula for $\zeta(2m+1)$ but also its beautiful new generalization linking $\zeta(2m+1)$ and $\zeta(2Nm+1), N\in\mathbb{N}$. We then discuss some results associated with a more general Lambert series $\displaystyle\sum_{n=1}^{\infty}\frac{n^{N-2h}\exp{(-an^{N}x)}}{1- \exp{(-n^{N}x)}}$ that we have obtained in a joint work with Bibekananda Maji, Rahul Kumar and Rajat Gupta. Applications of many of these formulas towards new transcendence results of Zudilin- and Rivoal-type are given.