A note on the approximate functional equation for \(\zeta^ 2(s)\). I,II
Let \(\zeta(s)^ 2=\sum_{n\leq x}d(n)n^{-s}+\chi(s)^ 2\sum_{n\leq y}d(n)n^{s-1}+D(s,X)\), where \(s=\sigma +it\), \(0<\sigma<1\), \(t\geq 2\) and \(XY=(t/2\pi)^ 2\). These two very interesting papers consider D(s,t/2\(\pi)\) \((=D(s)\), say). It was shown by Titchmarsh that \(D(s)\ll t^{1/2-\sigma}\log t.\) The first paper improves this to \(t^{1/3- \sigma}\log t\). The second gives an asymptotic expansion for D(s), with error term O(1/t), in which the first term is \(-2(\pi /t)^{1/2}\chi(s)\Delta(t/2\pi);\) here \(\Delta\) (x) is the error term in the Dirichlet divisor problem. These results go a long way towards explaining the apparent relationship between \(\Delta\) (x) and the remainder term in the mean-value formula for \(| \zeta(1/2+it)|^ 2\). The proofs, which are rather condensed, start from the approximate functional equation for \(\zeta\) (s). This is squared up, and then information from the Riemann-Siegel formula is used. Related results have been given by \textit{M. Jutila} (to appear) who remarks that for \(t\ll X\ll t\) one has \(D(s,X)=\Omega(t^{1/2-\sigma}\log t)\). Thus D(s,X) is untypically small when \(X=t/2\pi\).
scientific article; zbMATH DE number 3859234