Convolution operators and zeros of entire functions (Q2781343)

The author studies convolution operators with distribution functions as the measure and real entire functions \(G\) of order less than 2 and generalizes Pólya's theorem by proving that these convolutions have only real zeros. Pólya's application of the theorem was to study the Riemann zeta function. Some examples of distributions and entire functions \(G\) being represented as a Fourier transform are considered. The author links these results with the famous Riemann hypothesis about the zeros of the zeta function on the critical line \({\mathfrak R}s= 1/2\).