All Zeros of the Riemann Zeta Function in the Critical Strip are Located on the Critical Line and are Simple
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arXiv
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| Format: | Preprint |
| Publié: |
2017
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| _version_ | 1866913455423881216 |
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| author | Stenger, Frank |
| author_facet | Stenger, Frank |
| contents | In this paper we study the function G(z) :=
int{0,infinity} y^{z-1}(1 + \exp(y))^{-1} dy,
for z in C. We derive a functional equation
that relates G(z) and G(1 - z) for all z in C,
and we prove:
-- That G and the Riemann Zeta function Zeta have
exactly the same zeros in the critical region
D := z in C: Re z in (0,1);
-- All the zeros of the Riemann Zeta function
located on the critical line are simple; and
-- The Riemann hypothesis, i.e., that all of the zeros
of G in D are located on the critical line L :=
{z in D : Re z = 1/2}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1708_01209 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | All Zeros of the Riemann Zeta Function in the Critical Strip are Located on the Critical Line and are Simple Stenger, Frank General Mathematics 11M06, 42A38 In this paper we study the function G(z) := int{0,infinity} y^{z-1}(1 + \exp(y))^{-1} dy, for z in C. We derive a functional equation that relates G(z) and G(1 - z) for all z in C, and we prove: -- That G and the Riemann Zeta function Zeta have exactly the same zeros in the critical region D := z in C: Re z in (0,1); -- All the zeros of the Riemann Zeta function located on the critical line are simple; and -- The Riemann hypothesis, i.e., that all of the zeros of G in D are located on the critical line L := {z in D : Re z = 1/2}. |
| title | All Zeros of the Riemann Zeta Function in the Critical Strip are Located on the Critical Line and are Simple |
| topic | General Mathematics 11M06, 42A38 |
| url | https://arxiv.org/abs/1708.01209 |