## Re: zeta limit

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• ... Speaking of converging to one, I have wondered if there is an exponent k such that (1/2)^k + (1/3)^k + (1/5)^k + (1/7)^k + ... (1/p)^k + ... has a limit
Message 1 of 5 , Apr 6, 2007
> >
> > Limit pn/Sum[Zeta(i)*(pi-p(i-1)),{i,2,n}]=1
>
> > 1) Does someone can prove the firs limit?
>
> Yes, it's trivial and uninteresting.
>

Speaking of converging to one, I have wondered if there is an exponent
k such that

(1/2)^k + (1/3)^k + (1/5)^k + (1/7)^k + ... (1/p)^k + ...

has a limit of one. I just tried k = sqrt(2), and the first million
primes yields a sum of about .97486

For comparison, I just tried the same thing on the natural numbers
starting at two.

(1/2)^k + (1/3)^k + (1/4)^k + (1/5)^k + (1/6)^k + ...

If k = sqrt(3), the first million terms sums to about .99378

Just some reversion, no need for comment. A reflective Easter, everyone.

Mark

.

> > 2) Does someone can prove that the second limit converges?
>
> Yes, it's trivial and uninteresting.
>
> Exactly the same type of relation holds true for any sequence that
converges to
> 1, not just {Zeta(i)}. There are an infinitude of these, your
relation doesn't
> connect Zeta to the primes in any way.
>
> Take the fine structure constant alpha ~1/137, and replace Zeta(i) by
> alpha/CF_i(alpha) where CF_i is the i-th continued fraction convergent.
>
> Congratulations, your expression connects prime gaps to the fine
structure
> constant.
>
> Not!
>
> Phil
>
> Phil
>
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• ... 1/2^k + 1/3^k + ... + 1/p^k + ... just equals to Sum[MoebiusMu[n]*Log[Zeta[n*k]]/n, {n, 1, +Infinity}] which converges rapidly; so it can be used to
Message 2 of 5 , Apr 6, 2007
> I have wondered if there is an exponent
> k such that
>
> (1/2)^k + (1/3)^k + (1/5)^k + (1/7)^k + ... (1/p)^k + ...
>
> has a limit of one.

1/2^k + 1/3^k + ... + 1/p^k + ...

just equals to

Sum[MoebiusMu[n]*Log[Zeta[n*k]]/n, {n, 1, +Infinity}]

which converges rapidly; so it can be used to estimate the waned value of k as
1.3994333287263303...

Best,

Andrey

[Non-text portions of this message have been removed]
• ... 1.399433328726330318202807214745644327904727429484383941274765822888062492487247800233390475384227... (if you need more digits) Best, Andrey [Non-text
Message 3 of 5 , Apr 6, 2007
> 1.3994333287263303...

1.399433328726330318202807214745644327904727429484383941274765822888062492487247800233390475384227...

(if you need more digits)

Best,

Andrey

[Non-text portions of this message have been removed]
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