On a conjecture of Shanks (Q5906619)

The author gives some refinements of his previous work on Shanks' conjecture that \(\zeta' ({1\over 2}+ i\gamma)\) is real positive on average over the imaginary part \(\gamma\) of the non-trivial zeros of the Riemann zeta function \(\zeta (s)\). The paper contains four mean value theorems of the above type, whose statements are too long to be reported here. The proofs are based on the approximate functional equation and on some arguments in \textit{S. Gonek} [Invent. Math. 75, 123-141 (1984; Zbl 0531.10040)].