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Old 2009-02-01, 18:39   #1
mart_r
 
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Default 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.)
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Old 2009-02-02, 23:26   #2
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http://mathworld.wolfram.com/StieltjesConstants.html
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Old 2009-02-03, 17:28   #3
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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 (http://mathworld.wolfram.com/FullReptendPrime.html)?
I figured the sum 1/7+1/17+1/19+... exceeds 1 at about p=10.7*10^9.
General formula: Sum(f.r.p.)(1/p) ~ (log log p - 0.4655)*Artin's constant.
Any objections? Formulae for bases other than 10?
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Old 2009-02-04, 14:16   #4
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Yes, the sum should be

Artin*(log log p) + O[1]
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Old 2009-02-04, 17:19   #5
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Quote:
Originally Posted by XYYXF View Post
Yes, the sum should be

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

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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