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that this short equation is an estimation of. (That saying it is not impossible to derive the error term.) The reason it is here is that it is au pair with corresponding formula for Gram points and it is examined even beyond what
Riemann zeta zeros are, and there is no similar short formula in the text. Words "approximate" and "asymptotical" used in the text are the exact description of the formula. If nothing the work deserves a link to it somewhere. Otherwise the imaginary parts look kind of magical and difficult to derive, while we do have at least some estimate. There is nowhere in the text any sort of general estimates for the values, while we have them. But, It is up to you. Sure, the authors came more from physics, but the formula is very sound and practical.
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borders of the critical stripe there are special behaviors. On the critical line the zero of the real and imaginary part coincide. For y=1 the real part does not have any zeros and the absolute function does not either. For y=0 the situation is different. It is like that the real and imaginary part do a schwebung and the absolute function is the upper limit of the schwebung without a zero possibly. This is hard to prove. The absolute function of the zeta function diverges at most.
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All important results are based on the sieves. These are hard to do and there is only computational result. So to be honest the density of the prime numbers vanishes in the limit of all positive integers. But it does this only in the limit and for any finite and so big integer there is a finite prime
965:
I suggest to improve the article with my picture of the first two nontrivial zeros. This shows exemplary and fundamental the behavior on the critical line in the critical stripe in the complex plane. This supports but does not prove the
Riemann conjecture. It suggests on the other hand the path to a
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The nature of this formula is the same as similar formula that approximates Gram points. It is as the sentence describes asymptotically correct for all the zeros on the critical line. Its derivation in the origin is correct and the error term exists not for this estimation, but the original equation
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If one proof is not enough, I guess one could add another proof – a candidate would be the (original) Fourier series proof from
Titchmarsh, though I don't know if it makes sense to include it here – on the one hand, it should not be copied from Titchmarsh word for word; on the other hand, excluding
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All the zeros look like touches to the surface 0, like that there is a pencil pointing to 0, like dip, the curves look like roots or potency functions from the zero on the critical line to the borders of the stripe. I can offer some pictures showing that exemplary and fundamental behavior. On the
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of zeros on the critical line are given by a certain equation ((13) in the paper). Then, by ignoring the limit term in (13) (although pointing it is usually not zero), they consider a simplified equation (62), whose solutions (63)
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The formula in the following referenced assertion, which has been recently inserted in the subsection « Other results » of « Zeros, the critical line, and the
Riemann hypothesis » is problematic, mainly because of the use by
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when rounding up numerical value to various decimals). Moreover the formula in the following referenced assertion is in fact not given under this form in the cited paper. What the authors state is that the ordinates
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showing these values look indeed close to each other, but they never rigorously say what they mean by « approximate ». All of this is extremely sloppy, and the shortcut adopted below makes things even sloppier.
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I suppressed the (unsourced; and by the way the only appeal to
Titchmarsh’s book is wrong) « Proof 2 » of the functional equation, which is incorrect for several reasons. In particular the series
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I find the description accurate. I don't think there's disagreement that the formula is a good approximation, but I haven't found a paper with actual error bounds, asymptotic or otherwise. I
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The answer there is still vague: "expected to have integer part correct": in what sense is "expected" used? Expected by whom? Is this proven? Conditionally proven? Conjectured? —
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Thank you very much for spotting the error and removing the content. The alleged proof is a "modification of
Titchmarsh's Fourier series proof" (see
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LeClair, André; França, Guilherme. "Transcendental equations satisfied by the individual zeros of
Riemann ζ, Dirichlet and modular L-functions".
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1427:{\displaystyle \sum _{n=2}^{\infty }{\frac {\zeta (n)-1}{n}}\operatorname {Im} {\bigl (}(1+i)^{n}-(1+i)^{n}{\bigr )}={\frac {\pi }{4}}}
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This together gives reasons for the idea that the nontrivial zeros are isolated touches on the surface 0 in the
Riemann zeta function.
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Before we can answer that question, we need to know: what is the context of the sum in published reliable sources that discuss it? —
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they state are approximate solutions for the ordinates of the
Riemann zeros. They produce tables of computed values of
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749:{\displaystyle \gamma _{n}\approx 2\pi {\frac {n-{\frac {11}{8}}}{W({\frac {n-{\frac {11}{8}}}{e}})}}\qquad }
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Picture of the absolute function of the
Riemann zeta function with the first two nontrivial zeros shown.
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You got an answer there and it can only find the integer part 14.134725, 21.022040, 25.010858 ...
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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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Okay, forget about that. I found it in the source, the formula as given in the article was wrong.
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a few years back without any result. So it's interesting but probably not ready for inclusion. -
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I'll get back to that when I have a reference for you. An unrelated question: In
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The surface of the Riemann zeta function looks like this in the critical stripe:
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If this description is accurate, it sounds too non-rigorous to include to me. —
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The discussion of the critical stripe and the critical line is incomplete!
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the details that make the proof rigorous is not a good idea anyway.
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Clarification needed on approximate values of ordinates of zeros
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