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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
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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
162:
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.
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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.
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but if the
Schmidt result is correct this is impossible. Moreover, I was trying to understand the Schmidt result and I could find the result that appears in wikipedia. So I hope somebody could explain what is correct.
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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.
1218:. 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.
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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
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I also cannot find this result in the
Schmidt paper. He states 4 theorems. Kindly indicate which one is it that you think is the one cited in Knowledge. Also, Schmidt does not seem to discuss psi(x).
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Seems to me that the article would be improved by just stating the Hardy and
Littlewood result and omitting the Schmidt result. Do you happen to know if theta(x)-x changes sign infinitely often?
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As far as I know, theta(x) < x, at least for x<8*10^11 (some Dusart paper), but I don't know whether or not it's true for all x (it never changes sign). If you find out, let me know!
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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
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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?
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a problem is that this constant has, indeed, to be smaller than 1/29, whereas we want it to be arbitrarily large. Now, the question is what is in the Hardy and
Littlewood paper.—
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for many values of x. Those values below 100 are 19, 31, 32, 43, 47, 49, 53, 61, 73, 74, 75, 79, 83, 84, 89, and this inequality seems to hold for *most* large values of x.
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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.
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397:{\displaystyle \left|\sum _{\rho }{\frac {x^{\rho }}{\rho }}\right|\leq \left(\sum _{\gamma }{\frac {1}{\gamma }}\right){\sqrt {x}}=O({\sqrt {x}}\log ^{2}x)}
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That is speaks of Π instead of ψ is no problem, partial summation applied to the above results immediately implies that there are arbitrarily large
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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
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Using the syntax of the
Wolfram Language, the asymptotic for the theta function is Sum x^(1/n), {n, 1, Floor]}], which is illustrated at
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other than summing the logarithms of primes? This ends up either taking too long or losing a startling degree of accuracy.
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for all x (or that it's not)? Are there any conjectures that would imply this? Are there any other helpful bounds on
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Erhard
Schmidt, "Über die Anzahl der Primzahlen unter gegebener Grenze", Mathematische Annalen, 57 (1903), pp.195-204.
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Third inequality has reference to x. this cannot be correct since left side does not mention x at all.
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holds, although I can't prove or find a proof or counterexample to that assertion being true for all x.
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Hardy and
Littlewood state Schmidt's result essentially as I wrote above (without specifying a value for
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From my experience, both are used in proofs related to prime numbers, but the second
Chebyshev function
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I saw some manuscripts that stablishe RH is true because the Tchebychev function has this asymptotic
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on Knowledge. If you would like to participate, please visit the project page, where you can join
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1544:{\displaystyle \Pi (x)-\operatorname {Li} (x)<-{\tfrac {1}{29}}{\frac {\sqrt {x}}{\log x}}.}
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Compliance with standards: ln is probably more widely recognizable, and recommended by ISO.
863:{\displaystyle \vartheta (n)\sim (1+o(1))\cdot n\cdot \log(n)=\ln(n^{n}(1+o(1))^{n\log n})}
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is probably used more often especially in proofs relating to the prime counting function.
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I have two questions relating to Chebyshev's First Function. First, is it known that
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Hmm. As far as I can see, Schmidt proves in his paper that there are infinitely many
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Consistency: At present this article uses both log and ln for the same function
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I can't find the exact source, but I'm pretty sure that for x below 8*10^11
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Yes, i agree a derivative wouldn't make sense since it is a step function.
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I think "ln" is a better notation than "log" for the following reasons:
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currently here (from Rosser and Schoenfeld) with Pierre Dusart's bound (
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Removed. Another way this can be shown to be false is by, for fixed
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Are there any good asymptotics for the theta function? I see that
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zeros with imaginary part in . I'm afraid that the reason for the
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1870:{\displaystyle \psi (x)-x<-K{\sqrt {x}}\log \log \log x.}
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By the way—are there better results conditional on the RH?
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bound is much more subtle than what is alluded here. --
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This does imply that Schmidt's formula holds for every
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Does anyone object to my replacing the upper bound on
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1369:{\tfrac {1}{29}}{\frac {\sqrt {x}}{\log x}},}": -->
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905:and the latter is alredy an overstatement for
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622:{\displaystyle {\frac {d\Psi (x)}{dx}}\sim 1}
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1885:, but it is no longer Schmidt's result.—
1687:is any constant smaller than 1/29. What
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526:{\displaystyle O({\sqrt {x}}\log ^{2}x)}
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678:{\displaystyle {\frac {d\Psi (x)}{dx}}}
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2285:2A01:CB00:A34:1000:ED0E:D7C1:D5BD:64BC
1790:K{\sqrt {x}}\log \log \log x,}" /: -->
1711:such that there are arbitrarily large
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2011:Bounds on Chebyshev's First Function
1721:K{\sqrt {x}}\log \log \log x,}": -->
95:This article is within the scope of
2199:, page 2)? There it is listed as
898:{\displaystyle \vartheta (n)\sim n}
577:for big x then would it hold that
38:It is of interest to the following
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2102:{\displaystyle \vartheta (x)}
2072:{\displaystyle \vartheta (x)}
1153:{\displaystyle \vartheta (x)}
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109:and see a list of open tasks.
2458:B-Class mathematics articles
2393:{\displaystyle f(x)=e^{-dx}}
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1189:and hence a better bound on
244:{\displaystyle \psi (x): -->
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2426:{\displaystyle d\to 0^{+}}
2150:{\displaystyle \theta (x)}
1031:. It is easy to show that
407:is meaningful. If the sum
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263:21:54, 20 June 2015 (UTC)
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1989:{\displaystyle \psi (x)}
1255:First Chebyshev Function
1182:{\displaystyle \phi (x)}
141:project's priority scale
2298:Variational formulation
2124:Upper Bound on theta(x)
1211:{\displaystyle \Pi (x)}
685:would be zero as well.
98:WikiProject Mathematics
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251:x}" /: -->
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