Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

It is said that the Euler product $$\prod_p \frac{1}{1-p^{-s}}$$ diverges as $s \to 1^+$ proves we can't find constants $C$,$\theta$ with $\theta < 1$ such that $\pi(x) < C x^\theta$ because that would imply that the product converges for $s > \theta$.

I don't understand this deduction at all, how is the fact about the prime counting function concluded?

share|improve this question

1 Answer 1

up vote 7 down vote accepted

Taking logs of the Euler product, expanding out the resulting series, and comparing all the higher-order terms (i.e. terms involving $p^{-ns}$ for $n > 2$) to the convergent series $\sum_{m = 1}^{\infty} m^{-2s}$, we find that the divergence as $s \to 1^{+}$ is equivalent to the divergence of the series $\sum_p p^{-s}$ as $s \to 1^{+}$. Now if $\pi(x) < C x^{\theta}$, then the $n$th prime would have size $\sim n^{1/\theta}$, and so this series would behave like $\sum_{n = 1}^{\infty} n^{- s/\theta},$ which converges as $s \to 1^+$ (because $1/\theta > 1$), a contradiction.

share|improve this answer
    
Matt, if I may ask, where'd the $m$ in the first sum come from? –  J. M. May 16 '11 at 2:33
    
@J.M. Dear J.M., This is a typo, about to be corrected. Thanks! Regards, –  Matt E May 16 '11 at 2:51
    
Matt, isn't the connection between pi(n) and the size of the n-th prime true only in average (so that the last sentence of the answer is correct, if read as an Abel summation or Tauberian argument)? –  zyx Oct 5 '11 at 19:00
    
@zyx: Dear zyx, I didnt' think carefully about the argument for the last sentence; I think you are right that some kind of Tauberian argument might work, or perhaps if one takes into account a suffciently strong error term in the PNT? Best wishes, –  Matt E Oct 6 '11 at 1:23

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.