Pair Correlation of zeros of Dirichlet $L$-Functions: A possible path towards the conjectures of Chowla, Elliott-Halberstam and Montgomery

Assuming the Generalized Riemann Hypothesis and a pair correlation conjecture for the zeros of Dirichlet $L$-functions, we establish the truth of a conjecture of Montgomery (in its corrected form stated by Friedlander and Granville) on the magnitude of the error term in the prime number theorem in arithmetic progressions. As a consequence, we obtain that, under the same assumptions, the Elliott-Halberstam conjecture holds true. As another consequence, under the same assumptions, we will show that the number of Dirichlet characters $χ\pmod{q}$ for which $L(\frac{1}{2},χ)=0$ is of order less than $q^{1/2+ε}$.

💡 Research Summary

The paper investigates the pair correlation of zeros of Dirichlet L‑functions and shows how, under the Generalized Riemann Hypothesis (GRH) together with a suitable pair‑correlation conjecture, one can settle several deep conjectures concerning primes in arithmetic progressions, the Elliott–Halberstam conjecture, and the non‑vanishing of L‑functions at the central point.

The authors begin by recalling Montgomery’s original pair‑correlation conjecture for the Riemann zeta‑function and its extensions to Dirichlet L‑functions studied by Özlük, Yıldırım and others. They point out that for Dirichlet L‑functions the imaginary parts of zeros are not symmetric about the real axis and that the possibility of a zero at s = ½ must be taken into account. To overcome these difficulties they introduce a new correlation sum

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