A binary quadratic Titchmarsh divisor problem

TL;DR

This paper investigates a binary quadratic variant of the Titchmarsh divisor problem, providing an asymptotic formula for sums involving the divisor function over primes satisfying a quadratic relation.

Contribution

It introduces a new binary quadratic variant of the Titchmarsh divisor problem and derives an asymptotic formula for the sum involving primes and the divisor function.

Findings

Derived an asymptotic formula for the sum over primes satisfying a quadratic relation.

Extended the classical Titchmarsh divisor problem to a binary quadratic setting.

Provided new insights into the distribution of primes in quadratic forms.

Abstract

We consider a binary quadratic variant of the Titchmarsh divisor problem and give an asymptotic formula for $\sum_{p^2+q^2\leq N} \tau(p^2+q^2+1)$, where $p,q$ are primes.