On the second moment for primes in an arithmetic progression

Abstract

Assuming the Generalized Riemann Hypothesis, we obtain a lower bound within a constant factor of the conjectured asymptotic result for the second moment for primes in an individual arithmetic progression in short intervals. Previous results were averaged over all progression of a given modulus. The method uses a short divisor sum approximation for the von Mangoldt function, together with some new results for binary correlations of this divisor sum approximation in arithmetic progressions.

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• The Path to Recent Progress on Small Gaps Between Primes
• Higher Correlations of Divisor Sums Related to Primes II: Variations of the error term in the prime number theorem