Hardy's theorem for zeta-function of quadratic forms
Let \(Q(u_1,u_2, \dots, u_\ell)\) be a positive definite quadratic form in \(\ell\) \((\geq 3)\) variables, \(Z_Q(s)\) the zeta-function associated with \(Q\), and \(N(\alpha,T)\) the number of zeros \(\rho= \beta+ i\gamma\) of \(Z_Q(s)\) for which \(\beta\geq\alpha\), \(T\leq\gamma\leq 2T\). The main result of the present paper are the bounds \[ N(\alpha,T) \gg T\log T, \quad N(\beta,T)\ll T, \] which are valid for \(\alpha= {\ell-1 \over 2}- \delta\) \((\delta>0\) an arbitrary constant) and \(\beta ={\ell+1 \over 2} +\delta\). The proofs are based on several mean value results which are due to R. Balasubramanian and K. Ramachandra, and in particular on the bounds \[ \int_T^{2T} |Z_Q (\sigma+it) |\;dt \gg T^{1+\delta} \quad \left(\sigma= {\ell-1\over 2} -\delta \right), \] \[ \int^{2T}_T \left|Z_Q \left({\ell-1 \over 2}- \delta+ it\right) \right |^2dt \ll T^{1+2 \delta} \left({1\over 2}+\varepsilon \leq {\ell-1\over 2}- \delta\leq {\ell-1\over 2}- \varepsilon \right), \] where \(\varepsilon >0\) is a small constant.
positive definite quadratic form in many variables
scientific article; zbMATH DE number 955206
scientific article