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I am writing my senior thesis on Montgomery's pair correlation conjecture, and in his first lemma, he uses the following explicit formula: $$\sum_{n \leq x} \Lambda(n) n^{-s} = -\frac{\zeta'}{\zeta}(s)...

This question will be based roughly on the Bourgade–Keating review on Zeta function and eigenvalue asymptotics (BK): https://link.springer.com/chapter/10.1007/978-3-0348-0697-8_4 To set up the ...

Montgomery's pair correlation conjecture states that the distribution of the pair correlations of the zeroes of the Riemann zeta function (normalized to have average spacing 1) is given by the ...