Zeros of general L-functions on the critical line[HBNI Th52]

Abstract

The author studies the gaps between consecutive zeros on the critical line for the Riemann zeta function, and some of its generalisations, namely, the Epstein zeta function and the Selberg class of functions. First a simplified exposition of a result of Ivic and Jutila on the large gaps between consecutive zeros of Riemann zeta function on the critical line is given. Then presented a generalisation of this result to the case of the Epstein zeta function associated to a certain binary, positive definite, integral quadratic form Q(x, y). Then established the analogue of Hardy's theorem, namely, - that there are infinitely many zeros on the critical line, for degree 2 elements of the Selberg class of L-functions whose Dirichlet coefficients satisfy certain mild growth conditions. The study concludes with a conditional version of Hardy's theorem for the degree d > 2 elements of the Selberg class.