Notes on \(\log(\zeta(s))''\) (Q2374295)

In this paper, the zeros of the second logarithmic derivative of the Riemann zeta-function \(\zeta(s)\), \(s=\sigma+it\), are studied. Since \[ \log (\zeta(s))''=\bigg(\frac{\zeta'(s)}{\zeta(s)}\bigg)'=\frac{\zeta(s)\zeta''(s)-\zeta'(s)^2}{\zeta(s)^2}, \] let \(\nu(s):=\zeta(s)\zeta''(s)-\zeta'(s)^2\) or, in terms the von Mangoldt function \(\Lambda(n)\) and the divisor function \(\tau(n)\), \(\nu(s)=\sum_{n}\big(\sum_{d |n}\Lambda(d)\log (d)\tau\big(\frac{n}{d}\big)\big)n^{-s}\). Note that the zeros of \(\log (\zeta(s))''\) are the zeros of \(\nu(s)\). The one of results obtained in the paper gives a zeros-free region, i.e., it is shown that, for \(\mathrm{Re}(s)>4.25\), \(\nu(s) \not =0\). The second result deals with an asymptotic estimate for the number of nontrivial zeros \(\rho\) to height \(T\), i.e., it is proved that, for \(0<\Im (\rho)<T\) and \(-4<\mathrm{Re}(\rho)\), this number is \( 2\big(\frac{T}{2\pi}\log\big(\frac{T}{2\pi}\big)-\frac{T}{2\pi}\big)-\frac{\log(2)}{\pi}T+O(\log(T)). \) Also, the zero density estimate is given, i.e., for positive \(\delta\), the number of zeros of the function \(\nu(s)\) in the region \(|\Im(s)|\leq T\), \(\frac{5}{6}+\delta\leq \mathrm{Re}(s)\) is \(\ll_{\delta}T\).