Wigner Distribution Analysis Applied to Lehmer’s Conjecture on the Ramanujan tau Function

Wigner distribution is a tool for signal processing to obtain instantaneous spectrum of a signal. By using Wigner distribution analysis, another representation of the Euler product can be obtained for Dirichlet series of the Ramanujan tau function. From which, it can be proved that the Ramanujan tau function never become zero for all numbers.

NONVANISHING OF THE RAMANUJAN TAU FUNCTION IN SHORT INTERVALS

We provide new estimates for the gap function of the Delta function and for the number of nonzero values of the Ramanujan tau function in short intervals.