Remainder estimation in an approximate functional equation (Q2857900)

In the paper, the remainder estimation in the Hardy-Littlewood type approximate functional equation for the Riemann zeta-function \(\zeta(s)\), \(s=\sigma+it\), is obtained. It is proved, that, for \(\sigma>0\), \(t>0\), \(x \geq \frac{t}{\pi}\) and semi-integer \(x\), the following equation NEWLINE\[NEWLINE \zeta(s)=\sum_{n \leq x} \frac{1}{n^s}-\frac{x^{1-s}}{1-s}+\frac{s\zeta(2)}{4\pi^2 x^{s+1}}+\frac{s(s+1)(s+2)}{x^{s+3}2^3 \pi^4}\zeta(4)\bigg(1-\frac{1}{2^3}\bigg)+R_4 NEWLINE\]NEWLINE holds with NEWLINE\[NEWLINE R_4=s(s+1)(s+2)(s+3)\int_{x}^{\infty}\frac{1}{y^{s+4}}\sum_{k=1}^{\infty}\frac{\cos 2 \pi ky}{8\pi^4 k^4}\,dy. NEWLINE\]