On the zeros of the error term for the mean square of |𝜁(\frac{1}2+𝑖𝑡)|

Ivić, A.; te Riele, H. J. J.

doi:10.1090/S0025-5718-1991-1052096-7

On the zeros of the error term for the mean square of $\vert \zeta (\frac {1}{2}+it)\vert$
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Math. Comp. 56 (1991), 303-328 Request permission

Abstract:

Let $E(T)$ denote the error term in the asymptotic formula for \[ \int _0^T {{{\left | {\zeta \left ( {\frac {1}{2} + it} \right )} \right |}^2}dt.} \] 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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