Under GRH, the 2k-th moment of Im log L(1/2+it, χ) for fixed odd squarefree conductor q is bounded by (C_K k log log(qT))^k for k up to order log log(qT).
Notes on Pair Correlation of Zeros and Prime Numbers
1 Pith paper cite this work. Polarity classification is still indexing.
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abstract
These notes are based on my four lectures given at the Newton Institute in April 2004 during the Recent Perspectives in Random Matrix Theory and Number Theory Workshop. Their purpose is to introduce the reader to the analytic number theory necessary to understand Montgomery's work on the pair correlation of the zeros of the Riemann zeta-function and subsequent work on how this relates to prime numbers. A very brief introduction to Selberg's work on the moments of $S(T)$ is also given.
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A Tsang-range high-moment bound for $\operatorname{Im}\log L(\tfrac12+it,\chi)$ under GRH
Under GRH, the 2k-th moment of Im log L(1/2+it, χ) for fixed odd squarefree conductor q is bounded by (C_K k log log(qT))^k for k up to order log log(qT).