Lethbridge Number Theory and Combinatorics Seminar: Peng-Jie Wong

The study of the asymptotic behaviour of the summatory function of the number of divisors of shifted primes was initiated by Titchmarsh, who used the generalised Riemann hypothesis to establish a main term and remainder for the sum of tau(p-a) for p < x, where tau denotes the divisor function. This formula was first proved unconditionally by Linnik via the dispersion method. Moreover, applying the celebrated Bombieri-Vinogradov theorem, Halberstam and Rodriguez independently gave another proof.

In this talk, we shall study the Titchmarsh divisor problem in arithmetic progressions by considering the same sum, but restricting to primes that are congruent to b (modulo r).
Also, we will try to explain how to obtain an asymptotic formula for the same, uniform in a certain range of the modulus r. If time allows, we will discuss a number field analogue of this problem by considering the above sum over primes satisfying Chebotarev conditions.

(This is joint work with Akshaa Vatwani.)