Description
CACAAG seminar.
Speaker: Priyamvad Srivastav.
Affiliation: IMSc, Chennai.
Date and Time: Thursday 18 April, 5.15 pm - 6.15 pm.
Venue: Ramanujan Hall, Department of Mathematics.
Title: Product of primes in arithmetic progression.
Abstract: Let $q$ be a positive integer and let $(a,q)=1$ be a given
residue class. Let $p(a,q)$ denote the least prime congruent to $a
\mod{q}$. Linnik's theorem tells us that there is a constant $L>0$, such
that the $p(a,q) \ll q^L$. The best known value today is $L = 5.18$.
A conjecture of Erdos asks if there exist primes $p_1$ and $p_2$, both
less than $q$, such that $p_1 p_2 \equiv a \mod{q}$. Recently, Ramar\'{e}
and Walker proved that for all $q \geq 2$, there are primes $p_1, p_2,
p_3$, each less than $q^{16/3}$, such that $p_1 p_2 p_3 \equiv a \mod{q}$.
Their proof combines additive combinatorics with sieve theoretic
techniques. We sketch the ideas involved in their proof and talk about a
joint work with Olivier Ramar\'{e}, where we refine this method and obtain
an improved exponent of $q$.