Timeline for Zeros of polynomial approximations of the Riemann $\zeta$ function
Current License: CC BY-SA 4.0
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| May 5, 2019 at 11:28 | history | edited | Alexandre Eremenko | CC BY-SA 4.0 |
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| May 5, 2019 at 6:49 | comment | added | KConrad | While not directly related, another example where finite truncations shed no light on zeros of the zeta-function is related to a failed attempt at RH suggested by Turan (1948): the full series $\zeta(s) = \sum_{n \geq 1} 1/n^s$ has no zeros when ${\rm Re}(s) > 1$ and this suggests that the truncations $\zeta_N(s) = \sum_{n=1}^N 1/n^s$ should have "few" zeros there for large $N$. Turan showed that if $\zeta_N(s)$ for large $N$ has no zeros when ${\rm Re}(s) > 1$ then RH is a consequence, but Monach (1980) showed $\zeta_N(s)$ does have a zero with ${\rm Re}(s) > 1$ when $N \geq 31$. | |
| May 4, 2019 at 23:46 | history | answered | Alexandre Eremenko | CC BY-SA 4.0 |