Transformation formulae for Dirichlet polynomials (Q789433)

In this note the author considers Dirichlet polynomials related to the series \(\zeta^ 2(s)=\sum^{\infty}_{n=1}d(n)n^{-s}\) and to \(\phi(s)=\sum^{\infty}_{n=1}a(n)n^{-s},\) where the a(n) are the Fourier coefficients of a cusp form f of weight k for the full modular group, esp. \(a(n)=\tau(n)\), the Ramanujan arithmetical function. Let \(S(M_ 1,M_ 2;t)=\sum_{M_ 1\leq m\leq M_ 2}d(m)m^{-1/2-it}.\) By the Voronoi summation formula the asymptotic behaviour of \(S(M_ 1,M_ 2;t)\) for special values of \(M_ 1\) and \(M_ 2\) is estimated (too complicated to be given here explicitly). As an application a formula for the mean square of \(| \zeta(1/2+it)|\) is obtained, which allows to relax the conditions in a result of Atkinson concerning \(\int^{T}_{0}| \zeta(1/2+it)|^ 2dt.\) Now, let \(S_{\phi}(M_ 1,M_ 2;t)=\sum_{M_ 1\leq m\leq M_ 2}\quad a(m)m^{-k/2-it}.\) With the same method analogous results are proved, where the estimate \(a(n)\ll n^{(k-1)/2+\epsilon},\) a deep result of Deligne, is used. An application to the zeros of \(\phi\) (s) on the critical line is given: Suppose that a(n) is real for all n. Then the function \(\phi\) (s) has a zero \(k/2+it\) with \(| T-t| \leq T^{1/3+\epsilon},T>T_ 0(\epsilon).\) Finally it is sketched how to avoid the result of Deligne but to use a result of Rankin (concerning \(\sum_{n\leq x}| a(n)|^ 2)\) and a suitable averaging device.