The pair correlation of homothetic images of zeros of the Riemann zeta-function (Q441979)

The authors prove that NEWLINE\[NEWLINE\begin{multlined} \mathop{{\sum}^*}\frac{x^{\rho+\rho'}}{\rho+\rho'} = \frac{2x^{(k+1)/(2k)}T}{\pi(k+1)}\left(\log x - \frac{2k}{k+1}\right) + \\ O_k\left(x^{(k+1)/(2k)}\log^4T\left(\frac{x^{(2k-1)/k}}{T}+ \exp\left(-c(\log x)^{3/5}(\log\log x)^{-1/5}\right) \right)\right).\end{multlined}\tag{1} NEWLINE\]NEWLINE NEWLINEHere \(k\in \mathbb N\) is fixed, \(2\leq x \leq T^{k/(2k-1)}\), \(c>0\), \({\sum}^{*}\) denotes summation over complex zeros of \(\zeta(s)\) and \(\zeta(k(s-1/2)+1/2)\), respectively, such that \(-T \leq \text{Im}\,\rho, \text{Im}\,\rho' \leq T\), the last error term coming from the currently strongest known error term in the prime number theorem. To obtain (1), the authors first consider the sum NEWLINE\[NEWLINE\sum_{\substack{ \zeta(\rho)=0,\zeta(k\rho-(k-1)/2)=0,\\ -T \leq \text{Im}\, \rho,\text{Im}\,\rho' \leq T}} x^{\rho+\rho'}=\sum_{\substack{ \zeta(\rho)=0,\\ -T \leq\text{Im}\, \rho \leq T}} x^\rho\sum_{\substack{\zeta(k\rho-(k-1)/2)=0,\\ -T \leq \text{Im}\, \rho' \leq T}} x^{\rho'},\tag{2}NEWLINE\]NEWLINEand for the inner sum they use a version of the Landau-Gonek formula (see e.g., \textit{S.M. Gonek} [Contemp. Math. 143, 395--413 (1993; Zbl 0791.11043)]) which has a good error term. This entails expressions involving the function \(n_x\), the closest prime power to \(x\), which they successfully evaluate. Then by integration of (2) over \(x\) they obtain (1). Two results related to (1) are also obtained.