## An Expression For The Argument of ζ at Zeros on the Critical Line

**Authors:** Stephen Crowley

Abstract. It is conjectured that when t=t_n is the imaginary part of the n-th zero of ζ on the critical line, the normalised argument S(t_)_=π^(-1)argζ(1/2+i t__) is equal to S(t)=S_n(t_n)=_n-3/2-(ϑ(t_n_))/π where ϑ(t) is the Riemann-Siegel ϑ function. If S(t_n)=S_n(t_n)∀n∈ℤ^+ then the exact transcendental equation for the Riemann zeros has a solution for each positive integer n which proves that Riemann's hypothesis is true since the counting function for zeros on the critical line is equal to the counting function for zeros on the critical strip in that case.

**Comments:** 6 Pages.

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### Submission history

[v1] 2017-03-05 22:06:49

[v2] 2017-03-07 22:44:56

[v3] 2017-03-09 16:06:14

[v4] 2017-03-14 18:11:01

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