The integral of the Riemann \(\xi\)-function
The Riemann \(\xi\)-function is the entire function defined by the formula NEWLINE\[NEWLINE\xi(s) =\frac{1}{2}s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s).NEWLINE\]NEWLINE The \(\xi\)-function satisfies the functional equation \(\xi(s)=\xi(1-s)\), and its zeros are exactly the non-trivial zeros of the Riemann zeta function \(\zeta (s)\). The rescaled function \(\Xi(z) :=\xi(1/2+iz)\) has the Fourier integral representation NEWLINE\[NEWLINE \Xi(z):=2\int_{0}^{\infty}\Phi(u)\cos(zu)\,du,NEWLINE\]NEWLINE where NEWLINE\[NEWLINE \Phi (u):=\sum_{n=1}^{\infty} ( 4\pi^2 n^{4} e^{9u/2} - 6\pi n^{2} e^{5u/2})\exp \left(-\pi n^{2} e^{2u}\right), \quad 0\leq u< \infty. NEWLINE\]NEWLINE In this interesting paper under review, the authors study the integral of the Riemann \(\xi\)-function defined by \(\xi^{(-1)} (s)=\int_{1/2}^s\xi(w)\,dw\), \(s=1/2+iz\). They investigate a one-parameter family of functions given by Fourier integrals which satisfy a functional equation. Members of this family are shown to have only finitely many zeros on the critical line, with \(\xi^{(-1)} (s)\) having exactly one zero on the critical line, \(s = 1/2\). The authors prove that the zeros of \(\xi^{(-1)} (s)\) lie arbitrarily far away from the critical line. The rescaled function \(\Xi^{-1}(z)=-i\xi^{(-1)} (1/2+iz)\) has the Fourier integral representation NEWLINE\[NEWLINE \Xi^{-1}(z):=2\int_{0}^{\infty}\Phi(u) \left(\frac{\sin zu}{u}\right)\, du. NEWLINE\]NEWLINE The authors study the locations of zeros of this function and other entire functions related to \(\xi^{(-1)} (s)\). In addition, they introduce an analogue of the de Bruijn-Newman constant for this family, and prove that it is infinite.NEWLINENEWLINEIt is worth quoting here the authors's perspicuous statement about the purpose of their research. ``To add perspective to the results above, we study the effect of the inverse operation of integration applied to the Riemann \(\xi\)-function on the zeros of the resulting function. Since differentiation seems to smooth the distribution of zero spacings, we may anticipate that integration will 'roughen' their distribution, and even force zeros off the critical line. Our object is to obtain quantitative information in this direction''.
scientific article; zbMATH DE number 6079048