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Mathematics
General Math
Is Riemann's Zeta at 2 Related to Pi through Prime Numbers?
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[QUOTE="sysprog, post: 6616864, member: 617516"] We know that the uncountably infinite is of greater magnitude than the countably infinite, and we know that there are fewer algebraic irrationals than transcendentals, but we have not found a simple way to prove an arbitrary designatable real or imaginary number to be transcendental. Regarding the special case of the transcendental number ##\pi##, Euler showed that ##\frac π 4 = \frac 3 4 \cdot \frac 5 4 \cdot \frac 7 8 \cdot \frac {11} {12} \cdot \frac {13} {12} \cdots## with the numerators being the odd primes (and the denominators being the nearest thereto multiples of 4) ##\dots## Using e.g. the[URL='https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes'] Sieve of Eratosthenes[/URL] to find the odd primes to obtain that series to produce digits of ##\pi## is not as fast as using the more rapidly converging [URL='https://en.wikipedia.org/wiki/Gauss%E2%80%93Legendre_algorithm']Gauss-Legendre Algorithm[/URL], but that algorithm is a core-hog (i.e. requires much memory) ##-## as you know, CPU time vs memory is a frequently-occurring trade-off. :wink: [/QUOTE]
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Is Riemann's Zeta at 2 Related to Pi through Prime Numbers?
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