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- W32649130 abstract "This chapter discusses the application of mean values of the Riemann zeta-function to the distribution of zeros. The precise results about the horizontal distribution of the zeros of the Riemann zeta-function are deduced from mean value theorems that involve the zeta-function multiplied by a Dirichlet polynomial. The first result of this sort that required an arithmetic argument involving the coefficients of the Dirichlet polynomial to estimate the mean in question is because of Selberg in his proof that a positive proportion of the zeros are on the critical line. Improvements in the lower bound for the proportion of zeros on the critical line depended in part on better choices for the mollifier, which have been found through the use of the calculus of variations. Further developments in this method of mollifying have yielded lower bounds for the proportion of zeros on the critical line. On generalized Riemann hypothesis (GRH), a positive proportion of the zeros of the zeta-function of a quadratic number field is simple and the gaps between consecutive zeros of the zeta-function are infinitely larger than 2.68 times the average spacing." @default.
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- W32649130 date "1989-01-01" @default.
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- W32649130 title "Mean Values of the Riemann Zeta-Function with Application to the Distribution of Zeros" @default.
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- W32649130 doi "https://doi.org/10.1016/b978-0-12-067570-8.50017-2" @default.
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