#### IMSc Webinar

#### Notes on primes in some special subsets modulo q

#### Olivier Ramare

##### Institut de Mathematiques de Luminy

*We prove some closed formulas for Euler products of the form*

\prod_{p+qZ \in A}R(1/p^s) for a proper rational fraction R, where A is

among a collection of subsets of Z/qZ that we describe. These formulas

extend two formulas of D. Shanks in the sixties and lead to very fast

numerical evaluation in some cases. We shall show the link between our

formulas and abelian field theory. We will continue our journey with

some problems on the existence of primes in the coset of some subgroup

of (Z/qZ)^x of finite index. This is based on joined works with R.

Balasubramanian, S. Ettahri, S. Laishram, P. Srivastav and L. Surel.

Google meet link for this talk is

meet.google.com/rzt-jmep-hmt

Done