Riemann Hypothesis and Random Walks: The Zeta Case
In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its <i>L</i>-function is valid to the right of the critical line <inline-formula><mat...
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| Formato: | article |
| Lenguaje: | EN |
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MDPI AG
2021
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| Acceso en línea: | https://doaj.org/article/5e18b28a5dc940f18d2a12d3425e8e43 |
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| Sumario: | In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its <i>L</i>-function is valid to the right of the critical line <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>ℜ</mo><mrow><mo>(</mo><mi>s</mi><mo>)</mo></mrow><mo>></mo><mstyle scriptlevel="0" displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></mrow></semantics></math></inline-formula>, and the Riemann hypothesis for this class of <i>L</i>-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet <i>L</i>-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a one-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mn>100</mn></msup></semantics></math></inline-formula>-th zero to over 100 digits, far beyond what is currently known. Of course, use is made of the symmetry of the zeta function about the critical line. |
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