Talk:Chebyshev function

message: Does anyone know anything about the first integers where the Chebyshev functions evaluate to greater than their inputs? I think that would be a nice thing to include in this article, even if the answer is that the first such examples are unknown.  Kyle1009 (talk) 03:24, 17 August 2014 (UTC)
thread_title:  Crossover Points 

message: 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 \Theta(\log n) 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)
replies: text: I know I'm replying to a very old post, but just in case: since psi(x) has a slope of 0 at (all? almost all?) points, I think the derivative  \frac{d\Psi (x)}{dx}  would be zero as well. Pieater3.14159265 (talk) 01:02, 31 July 2015 (UTC)
replies: text: Yes, i agree a derivative wouldn't make sense since it is a step function. Approxilator (talk) 21:18, 1 January 2024 (UTC)
thread_title:  Riemann hypothesis 

message: The text of the first section states "The Chebyshev function is often used in proofs relate ...." Yet the text prior states two different Chebyshev functions. Which one or both?  — Preceding unsigned comment added by Reddwarf2956 (talk • contribs) 12:29, 18 September 2012 (UTC)
thread_title:  Chebyshev function 

message: I have two questions relating to Chebyshev's First Function.  First, is it known that  \vartheta(x) < x  for all x (or that it's not)?  Are there any conjectures that would imply this?  Are there any other helpful bounds on  \vartheta(x)  given by someone other than Dusart (I think I've found all of his papers on the subject) without assuming RH?  The only other ones I can find are slight improvements on his, which don't seem to be all that helpful.  Second, is there any good way of calculating  \vartheta(x)  other than summing the logarithms of primes?  This ends up either taking too long or losing a startling degree of accuracy.  Pieater3.14159265 (talk) 07:24, 19 June 2015 (UTC)
thread_title:  Bounds on Chebyshev's First Function 

message: Does anyone object to my replacing the upper bound on  \theta(x)  currently here (from Rosser and Schoenfeld) with Pierre Dusart's bound ( \theta(x) < 1.000028x  from http://arxiv.org/pdf/1002.0442v1.pdf, page 2)?  There it is listed as   \theta(x)-x < \frac{1}{36260}x.  Pieater3.14159265 (talk) 23:18, 10 August 2015 (UTC)
thread_title:  Upper Bound on theta(x) 

message: I think "ln" is a better notation than "log" for the following reasons:
*Consistency: At present this article uses both log and ln for the same function
*Clarity: ln is unambiguous contrary to log
*Compliance with standards: ln is probably more widely recognizable, and recommended by ISO.
The only argument in favor of "log" seems to be that "log" is more accepted in the field of number theory. I think this argument weighs less than the previous three.2A01:CB00:A34:1000:ED0E:D7C1:D5BD:64BC (talk) 14:21, 13 June 2021 (UTC)
thread_title:  Log vs Ln 

message: It appears this section is incorrect. See https://math.stackexchange.com/q/3339574.
StvC (talk) 19:56, 18 June 2022 (UTC)
replies: text: Removed. Another way this can be shown to be false is by, for fixed c>1, choosing a sequence of functions tending to 1, e.g. of form f(x)=e^{-dx} where d \to 0^+ . The first term is negative and the second term goes to infinity. Kclisp (talk) 06:50, 30 November 2023 (UTC)
thread_title:  Variational formulation