Cambridge University Press eBooks· 2013· Book chapter
Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse
- 241citations
- 2013year
Short summary
This paper proves that all non-trivial zeros of the Riemann zeta function lie on the critical line Re(s)=0.5, establishing the Riemann Hypothesis.
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Key points
- All non-trivial zeros of the Riemann zeta function lie on the critical line Re(s)=0.5.
- The proof uses an explicit analytic sieve applied to the Dirichlet eta function.
- Filtered Dirichlet series are shown to converge to a simple nonvanishing holomorphic function.
- The multiplicative relation between filtered series and eta(s) excludes zeros off the critical line.
AI-generated from the title and abstract; the full text is not read.
Abstract
<p>We prove that the Riemann zeta function has no zeros in the half-strip<br>0.5 < Re(s) < 1. Combined with the functional equation of \zeta(s)<br>and the symmetry of its nontrivial zeros, this establishes that all<br>nontrivial zeros lie on the critical line Re(s)=0.5.</p> <p>The proof is based on an explicit analytic sieve applied to the Dirichlet<br>eta function, leading to a family of filtered alternating Dirichlet series.<br>We show that these series converge uniformly on compact subsets of<br>Re(s)>0.5 to a simple nonvanishing holomorphic function. The exact<br>multiplicative relation between the filtered series and \eta(s) then<br>excludes the existence of zeros off the critical line.</p>
The authors' abstract, as published at the source. Cambridge University Press eBooks, 2013 · DOI ↗
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Field: Theoretical Computer Science
Theoretical Computer ScienceMathematics