Some generalized Pólya-Turán inequalities
Let \[ F(z):=2\int^{\infty}_{0}\Phi (t)\cos (zt)dt, \] where \[ \Phi (t):=\sum^{\infty}_{n=1}a_ n(t), \] and \[ a_ n(t):=4(2\pi^ 2n^ 4e^{9t}-3n^ 2\pi e^{5t})\exp (-n^ 2\pi e^{4t}). \] Then the Riemann hypothesis is equivalent to the statement that F(z) has only real zeros. A necessary condition for the validity of the Riemann hypothesis is that the Turán inequalities hold; that is, \[ (*)\quad (\hat b)^ 2_ m-\frac{2m-1}{2m+1}\hat b_{m-1}\hat b_{m+1}\geq 0\quad (m=1,2,3,...), \] where \[ \hat b_ m:=\int^{\infty}_{0}t^{2m}\Phi (t)dt\quad (m=0,1,2,...). \] In 1927, Pólya conjectured the validity of the inequalities (*) and recently (*) has been proved [cf. \textit{G. Csordas}, \textit{T. S. Norfolk} and \textit{R. S. Varga}, Trans. Am. Math. Soc. 296, 521-541 (1986; Zbl 0602.30030)], \textit{G. Csordas} and \textit{R. S. Varga} [Constructive Approximation 4, No.2, 175-198 (1988; Zbl 0683.30032)] and \textit{Yu. V. Matiyasevich} [Cybernetics 18, 705-707 (1983; Zbl 0516.10032)]. The author uses a classical result of Hilbert to generate inequalities which are more general than the Turán inequalities (*).
scientific article; zbMATH DE number 4181711
scientific article