L-functions in Analytic Number Theory: Jyothsnaa Sivaraman

In 1944, Linnik showed that the least prime in an arithmetic progression given by a mod q for (a,q)=1 is at most cq^L for some absolutely computable constants c and L. A lot of work has gone in computing explicit bounds for c and L. The best known bound is due to Xylouris (2011) who showed that c can be taken to be 1 and L to be 5 for q sufficiently large. In 2018, Ramaré and Walker gave a completely explicit result if one prime is replaced by a product of primes. They showed that each co-prime class modulo q contains a product of three primes each less than q^(16/3). This was improved by Ramaré, Srivastava and Serra to 650q^3 in 2020. In this talk we will introduce analogous results in the set up of narrow ray class fields of number fields. This is joint work with Deshouillers, Gun and Ramaré.