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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On the zeros of the error term for the mean square of $ \vert \zeta(\frac {1}{2}+it)\vert $

Authors: A. Ivić and H. J. J. te Riele
Journal: Math. Comp. 56 (1991), 303-328
MSC: Primary 11M06; Secondary 11Y35
MathSciNet review: 1052096
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Abstract: Let $ E(T)$ denote the error term in the asymptotic formula for

$\displaystyle \int_0^T {{{\left\vert {\zeta \left( {\frac{1}{2} + it} \right)} \right\vert}^2}dt.} $

The function $ E(T)$ has mean value $ \pi $. By $ {t_n}$ we denote the nth zero of $ E(T) - \pi $. Several results concerning $ {t_n}$ are obtained, including $ {t_{n + 1}} - {t_n} \ll t_n^{1/2}$. An algorithm is presented to compute the zeros of $ E(T) - \pi $ below a given bound. For $ T \leq 500000$, 42010 zeros of $ E(T) - \pi $ were found. Various tables and figures are given, which present a selection of the computational results.

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Keywords: Riemann zeta function, mean square, zeros, gaps between zeros
Article copyright: © Copyright 1991 American Mathematical Society