Transformations of zeta-sums
The author provides an interesting transformation formula for the zeta-sum \[ S := \sum_{N_1\leq n \leq N_2}n^{-1/2-it}\qquad(N_1 \asymp N_2), \] which, in the context of the estimation of \(\zeta(1/2+it)\), is especially interesting when \(N_i \asymp \sqrt{t/2\pi}\). The initial step is to write \[ S = \sum_{N_1\leq n \leq N_2}n^{-1/2-it}e({\textstyle{1\over2}}rn^2) e(-{\textstyle{1\over2}}rn^2) \] and to use a Poisson-type summation formula for sums of \(f(n)e(\frac{h}{k}n^2)\), after applying a smoothing process to \(S\). Here \(e(x) = \exp(2\pi ix)\) and \(r = h/k\) is a suitably chosen rational number. Then the saddle point method for the estimation of exponential sums is applied. The end result is a (complicated) expression for \(S\), involving the Gauss sums \[ G(a,b;k) = \sum_{x\pmod k}e\left(\frac{ax^2+bx}{k}\right). \] The author says: ``This argument gives an approach to the estimation of the (Riemann) zeta-function, and it remains to study its scope.'' Indeed, it would be very interesting to analyze whether new bounds for \(\zeta(1/2+it)\) can be obtained from the author's transformation formula.
scientific article; zbMATH DE number 6345310