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AbstractFor the completed Riemann zeta function ξ(s){\xi(s)}, it is known that the Riemann hypothesis for ξ(s){\xi(s)}implies the Riemann hypothesis for ξ(m)(s){\xi^{(m)}(s)}, where m is any positive integer.In this paper, we investigate the distribution of the fractional parts of the sequence (αγm){(\alpha\gamma_{m})}, where α is any fixed non-zero real number and γm{\gamma_{m}}runs over the imaginary parts of the zeros of ξ(m)(s){\xi^{(m)}(s)}.We also obtain a zero density estimate and an explicit formula for the zeros of ξ(m)(s){\xi^{(m)}(s)}.In particular, all our results hold uniformly for 0≤m≤g(T){0\leq m\leq g(T)}, where the function g(T){g(T)}tends to infinity with T and g(T)=o(loglogT){g(T)=o(\log\log T)}.
Forum Mathematicum – de Gruyter
Published: Jan 1, 2020
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