An explicit formula for the square of the Riemann zeta-function on the critical line
The author proves the explicit formula \[ \begin{aligned} |\zeta(\tfrac 12+ iT)|^2 & = \sqrt{2}\sum_{2T/\pi+1<n\leq T_C(\alpha)}{(-1)^nd(n)\over \sqrt{n}({1\over 4} - T(2\pi n))^{1/4}}\cos(f_C(T,n))\\ & + 2\sum_{n\leq T\alpha/(2\pi)}d(n)n^{-1/2}\cos(T\log(T/(2\pi n e)) - \pi/4) + O(T^{\varepsilon}).\end{aligned} \tag{1} \] Here \(d(n)\) is the number of divisors of \(n\), \(\theta \leq \alpha < 1\) is a constant for a given \(0 < \theta < 1\), \(T_C(\alpha) = (1+\alpha)^2T/(2\pi)\), and \[ f_C(T,n) = 2T\operatorname {arcosh}\left(\sqrt{\pi n\over 2T} \right) - 2\pi n\sqrt{{1\over 4} - {T\over 2\pi n}} + {\pi\over 4}, \] \[ \text{arcosh}(z) = \log(z + \sqrt{z^2-1})\qquad(|z|> 1). \] As an application, by exponential averaging with the Gaussian weight, one obtains from (1) the classical bound \(\zeta({1\over 2} + it) \ll_\varepsilon |t|^{1/6+\varepsilon}\). The proof is similar to the approach used by \textit{M. Jutila} [J. Number Theory 18, 135-156 (1984; Zbl 0533.10034)], who transformed Dirichlet polynomials of the type \(\sum_{N<n\leq N'\leq 2N}d(n)n^{-it}\) with the aid of Voronoi's formula, inserting the (trivial) factor \(1 = \exp(2\pi in)\) in the sum to regulate the location of the saddle points of the ensuing exponential integrals. The present author does not work with Dirichlet polynomials, but works directly with the classical approximate functional equation \[ |\zeta(\tfrac 12+iT)|^2 = 2\sum_{n\leq T/(2\pi)}d(n)n^{-1/2}\cos(T\log(T/(2\pi ne)) - \pi/4) + O(\log T).\tag{2} \] The portion of the sum in (2) for which \(n < \alpha T/(2\pi)\) is kept intact, and the remaining portion is transformed by the Voronoi formula, after the factor \(1 = \exp(-2\pi in)\) is inserted in the sum. This produces the first sum on the right-hand side of (1), plus a small error term. The analysis required for the treatment of various sums and integrals that arise in the course of the proof is delicate, and the author displays considerable skill in dealing with them.
scientific article; zbMATH DE number 1822084