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-   -   Zeta(1+s) (https://www.mersenneforum.org/showthread.php?t=11442)

mart_r 2009-02-01 18:39

Zeta(1+s)
 
I can't find anything about it on the web, so I'll post a question here:

What is known about the series expansion of Zeta(1+s) when s gets near zero?
I found Zeta(1+s) = 1/s + Gamma + s/13.73327... - s²/206.39... - s³/2921.6... + ...

(Maybe I have some more questions or results of other calculations of mine I'll post on this thread later on.)

XYYXF 2009-02-02 23:26

[url]http://mathworld.wolfram.com/StieltjesConstants.html[/url]

mart_r 2009-02-03 17:28

Darn, I really should read more carefully; I've been on this site before. Sorry.

Now to something completely different - was someone ever interested in the sum of reciprocals of full reptend primes ([URL]http://mathworld.wolfram.com/FullReptendPrime.html[/URL])?
I figured the sum 1/7+1/17+1/19+... exceeds 1 at about p=10.7*10^9.
General formula: Sum[SIZE=1](f.r.p.)[/SIZE][SIZE=2](1/p) ~ (log log p - 0.4655)*Artin's constant.[/SIZE]
Any objections? Formulae for bases other than 10?

XYYXF 2009-02-04 14:16

Yes, the sum should be

Artin*(log log p) + O[1]

mart_r 2009-02-04 17:19

[quote=XYYXF;161535]Yes, the sum should be

Artin*(log log p) + O[1][/quote]


I suppose that's equivalent to Artin*(log log p + c) (± prime number irregularities) for some constant c for each base b. At least until I'm more familiar with the niceties of the big-oh notation.


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