On the Riemann zeta-function, Parts IV-V

Details On the Riemann zeta-function, Parts IV-V On the Riemann zeta-function, Parts IV-V

2007-05-31

arxiv.org/abs/0705.4593v2

Mathematics

General Mathematics

12 pages Added Part V to a contracted version of Part IV

Scientific paper

In Part I an odd meromorphic function f(s) has been constructed from the Riemann zeta-function evaluated at one-half plus s. The conjunction of the Riemann hypothesis and hypotheses advanced by the author in Part I is assumed. In Part IV we derive the two-sided Laplace transform representation of f(s) on the open vertical strip V of all s with real part between zero and four. An additional hypothesis is used to prove that the Laplace density of f(s) on the strip V is positive. Let z(n) be the nth critical zero of the Riemann zeta-function of positive imaginary part in order of magnitude thereof. In Part V an expression is derived for z(1). A relation is obtained of the pair z(n) and the first derivative thereat of the zeta-function to the preceding such pairs.

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