> >

> > 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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