Talk:Chebyshev function

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[edit] Riemann hypothesis

I doubt the equation

\left|\sum_{\rho} \frac{x^{\rho}}{\rho}\right| \le \left(\sum_{\gamma} \frac{1}{\gamma}\right) \sqrt x = O(\sqrt x \log^2 x)

is meaningful. If the sum \sum_\gamma\frac1\gamma is over all nontrivial roots (ordered according to the absolute value of the imaginary part, per the usual convention), then the positive and negative terms cancel, and the result is zero, which is clearly false. If the sum is intended to run only over roots with positive imaginary part, then it is divergent, as there are Θ(logn) zeros with imaginary part in [n, n + 1]. I'm afraid that the reason for the O(\sqrt x\log^2x) bound is much more subtle than what is alluded here. -- EJ 20:11, 30 August 2006 (UTC)

got a question..if Ψ(x)˜x for big x then would it hold that \frac{d\Psi (x)}{dx} \sim 1 for big x??? --85.85.100.144 13:40, 13 December 2006 (UTC)

[edit] Asymptotics

Are there any good asymptotics for the theta function? I see that A002110 has \vartheta(n)\sim(1 + o(1)) \cdot n \cdot \log(n) = \ln(n^n(1+o(1))^{n\log n}), but this seems to be worse than \vartheta(n)\sim n and the latter is alredy an overstatement for 0 < n < 2,000,000,000 by my calculation. Any thoughts? CRGreathouse (t | c) 04:32, 16 September 2006 (UTC)

I'm surprised better estimates aren't mentioned here. The relevant sections to look at would be Hardy and Wright's first chapter on the series of primes or almost any intro analytic number theory textbook. We have \vartheta(x) \sim \phi(x) \sim x (a statement which is in fact equivalent to the Prime Number Theorem. It is easy to show that \vartheta(x) \sim \phi(x) The second estimate is essentially how one normally proves the prime number theorem. Explicit results about the error bound are known - I think the best currently is due to Dusart. If you want, I can go look them up. JoshuaZ 04:59, 16 September 2006 (UTC)
Oh, I feel foolish now. I actually have Dusart's paper on my hard drive, but I forgot it had information on this. I even have his research report, not just the MComp paper... I'll look through it and update the article. Gosh do I feel stupid!
By the way—are there better results conditional on the RH? CRGreathouse (t | c) 05:07, 16 September 2006 (UTC)
Yes. In fact that is how one shows that's one standard RH gives better bounds on the PNT. The rough idea is that as zeros are pushed away from the zero line we get better estimates on \vartheta(x) and φ(x) and hence a better bound on Π(x). The relevant chapter of Hardy and Wright (or Apostol's Intro Analytic Number Theory) should give a decent explanation of the basic ideas. Edward's book on the zeta function has all the relevant details worked out. JoshuaZ 05:42, 16 September 2006 (UTC)
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