Timeline for Collection of equivalent forms of Riemann Hypothesis
Current License: CC BY-SA 4.0
5 events
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| Sep 6, 2021 at 20:46 | comment | added | GH from MO | @JoséHdz.Stgo. The RH implies $|\pi(x)-\mathrm{li}(x)|<\frac{1}{8\pi}\sqrt{x}\log x$ for all $x\geq 2657$. This is Corollary 1 in Schoenfeld: Sharper bounds for the Chebyshev functions $\theta(x)$ and $\psi(x)$. II, Math. Comp. 134 (1976), 337-360. Checking the finite range $2657>x\geq 2$ by a computer program, we find that the weaker inequality $|\pi(x)-\mathrm{li}(x)|<\sqrt{x}\log x$ holds for all $x\geq 2$. Conversely, if we assume this inequality for all $x\geq 2$, then the RH is true by Theorem 15.2 in Montgomery-Vaughan: Multiplicative number theory I. | |
| Sep 6, 2021 at 16:51 | comment | added | José Hdz. Stgo. | Good morning! Do you know of a textbook wherein this equivalence is established in detail? Thanks in advance for your reply. | |
| Aug 10, 2019 at 9:31 | history | edited | GH from MO | CC BY-SA 4.0 |
added 6 characters in body
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| S Aug 10, 2019 at 9:22 | history | answered | GH from MO | CC BY-SA 4.0 | |
| S Aug 10, 2019 at 9:22 | history | made wiki | Post Made Community Wiki by GH from MO |