When to carry on with a buggy game state versus terminate the process? 20 & 0.0000192936717837051401063299760357760104805477068753543599966583874264 \\ -0.000232907558455 \\ Interesting topic. By direct consideration of ~'(it) and ~(it), or what is equivalent by the functional equation, of ~'(1 +it) and $(1 +it) it follows that arg $'/~(4t) changes by Proo] o/ Theorem 9. Derivative of trigonometric functions - cosX by first principle, Derivative of trigonometric functions - sinX by first principle. -0.00471116686225 \\ 105 & -208147105464557539810.933105520613946023324136236314019489461672300 \\ 50 & -484.410856973911340196834881321159996957875322777427689682560124377 \\ The non trivial-zeros of $\zeta$ have a distance from $\frac{1}{2}$ which is always $\geq 14$, hence the residual series provide a negligible contribution for large and even values of $k=2m$. }s^{n}$$ where : Feb 2009 1 0. $$\lim\sup_{n\rightarrow\infty}\left(\frac{\delta_{n}}{n}\right)^{\frac{1}{n}}$$. How do open-source projects prevent disclosing a bug while fixing it? Your plot seems to suggest that they are repelled from $+1$, is there any further evidence of this conjecture? At zero, one has = − = − + = −At 1 there is a pole, so ζ(1) is not finite but the left and right limits are: → ± (+) = ± ∞ Since it is a pole of first order, its principal value exists and is equal to the Euler–Mascheroni constant γ = 0.57721 56649+.. ZEROS OF THE DERIVATIVES OF THE RIEMANN ZETA-FUNCTION 53 T~WOREM 9. I've recomputed that table using internal precision, Many thanks Yuriy for the integral and to wake up this all thread! (Those thoughts should not stop you from checking, though.). I can only show you the 'brainy' picture obtained for values of $n$ from $1$ to $250$ : The largest value obtained is near $2.047$ but this doesn't seem to stop. Here is a formula for Pari/GP: sum(k=0,max_k,Stieltjes[k+n]/k!) Concerning the limit : To elaborate a little more, here's some Mathematica code: Ghaith Hiary has computed fairly large tables of zeros at large height, which are at, http://sage.math.washington.edu/home/hiaryg/page/index.html. This looks like the question of whether there is any asymptotic relation between the zeros of the zeta function and gram points, and I think that the answer is expected to be no, but I don't know if that is a theorem. If I calculate correctly (I hope I'm not embarrassing myself), then and we got rid of $(2n)!$ in $K_{2n}$. How to make this illumination effect with CSS, Reference request: Examples of research on a set with interesting properties which turned out to be the empty set, 200 mA output from the Arduino digital output. Why does my character have such a good sense of direction? It's a curious fact that for $n>0$, $\zeta^{(n)}(0)\approx -n!$. This, it seems, is inconsistent with your above statement "the derivatives of odd order just depend on $\pi, \gamma, \log2, \log\pi, G,\zeta(1/2)$". 60 & 62714.1067695718525498151218611523939474897844785985218047818417901 \\ Why is "hand recount" better than "computer rescan"? What does [? Apostol gave a table for $\frac{\zeta^{(n)}(0)}{n! Asking for help, clarification, or responding to other answers. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. $ \small \begin{matrix} }s^{2n}$$ \right|$$. By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. 40 & -2.99127389405887676303274513146663241574504274783600393720076526420 \\ Did a computer error lead to 6,000 votes switching from Joe Biden to President Trump? Could it be a 'added' function? Values of the Riemann zeta function at integers Roman J. Dwilewicz, J an Min a c 1 Introduction The Riemann zeta function is one of the most Leonhard Euler important and fascinating functions in mathematics. In case you don't have it, here's a link to a table of zeros: Mathematica has about $10^7$ zeros built in as the function ZetaZero[n]. 195 & 146090125339857661850314283330560855583771401129477483038.196790939 \\ Isn't "2+2" correct when answering 'What is "2+2"'? . I am trying to get at whether these numbers tend to $-1$. @Gottfried: I think that the real interest of the OP is the convergence of $$\sum_{k=1}^\infty K_{2k} \frac{B_{2k}}{(2k)!} My interest for the n-th derivative of $\zeta$ at $0$ started with this, Your formula for the $n$'th derivative of the zeta at zero seems very nice. 3D plot of the function = ( ψ ( α , s )) with α = 0 . I am not conjecturing that the imaginary parts of the zeros are not transcendental (unless I am missing something implicitly). Thanks. }{(1/2+n)^{k+1}} = (-1)^{k+1} k! 16 & 0.0000834588058255016817276488047155531844625161484345203508967032195293 \\ MathJax reference. By the reflection formula: By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy. $$\begin{eqnarray*} \tfrac{\zeta'}{\zeta}(s)+\tfrac{\zeta'}{\zeta}(1-s)&=&\log\pi-\tfrac{1}{2}\left[\psi\left(\tfrac{s}{2}\right)+\psi\left(\tfrac{1-s}{2}\right)\right]\\&=&\log\pi+\gamma+\sum_{n\geq 1}\left[\frac{1}{2n-1-s}+\frac{1}{2n-2+s}-\frac{1}{n}\right].\tag{A}\end{eqnarray*} $$ Z. Ziaris. My interest for the n-th derivative of $\zeta$ at $0$ started with this thread by the same OP (cf the $\;K_n:=\frac{n!+\zeta^{(n)}(0)}n\,$ coefficients and with the observation that $\zeta^{(n)}\left(\frac 12\right)\approx -2^{n+1}n!$ in conformity with your general expression). 13 & 0.000407241232563033143432121366810273073439244495052894296377049143472 \\ How can a chess game with clock take 5 hours? In fact, uniform distribution would be much stronger, though still a possibility. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. The Riemann zeta function at 0 and 1. Set Theory, Logic, Probability, Statistics, Researchers 3-D print biomedical parts with supersonic speed, New technique may revolutionize accuracy and detection of biomechanical alterations of cells, Jacky dragon moms' time in the sun affects their kids, Functional equation Riemann Zeta function. 4 & 0.00289681198629204101278047225899433810886006507829657502399066695362 \\ $$ Use MathJax to format equations. Edit: 5 & -0.000232907558454724535985837795819747892057172470502296621517290052364 \\ (I think may already be precision issues in the picture in Stopple's answer.) Why is it wrong to answer a question with a tautology? }$ The curve seems to come out a bit better, see, Since few people search questions in answers don't hope too many reactions (it breaks the Q->A principle favored here !). 70 & 369710251.754342613761487189243065702707445997550023978646801198349 \\ "Fast calculation of the Riemann zeta function, Identities inspired from Ramanujan Notebooks, Identities inspired by Ramanujan Notebooks part 2, https://en.wikipedia.org/w/index.php?title=Particular_values_of_the_Riemann_zeta_function&oldid=980034117, All Wikipedia articles written in American English, Creative Commons Attribution-ShareAlike License, This page was last edited on 24 September 2020, at 06:44. I updated my answer with two pictures concerning your limit. But how often are these number $\pm 1$? Use MathJax to format equations. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. The computation to 120 good coefficients took only a few seconds with that given precision of 256 dec digits . See Andrew Odlyzko's website for their tables and bibliographies. hence with the interesting 'saturation' near $0.4646$. 175 & -219609544533102325798714608918968968215179933676.462881353291615996 \\ 3. 200 & 21761038288742061134507006188990514804372485347492068735353.4677389 \\ Positive integers Even positive integers. $$f(s)=-K_{0}-\frac{\zeta^{(1)}(0)}{2}s+\ln(-s)-\sum_{n=1}^{\infty}\frac{B_{2n}}{(2n)! Our primary focus is math discussions and free math help; science discussions about physics, chemistry, computer science; and academic/career guidance. What is the upper bound of $\delta_{n}$ ?? 18 & 0.00102622728654085400217701415546883787759831069743902026886240548348 \\ 0.500000000000 \\ There are tables of derivatives of Z'(t) at the zeros, but not zeta'(1/2 + it). 235 & 2.44222335896278620212620252751346268294589748965319118864768279107E73 \\ 35 & -0.263594454732269692589658594912151283515046273581182559219921957221 \\ This is probably a better way to compute the quotient. Approximation of $\sum_{\rho}\frac{1}{|\rho|^2}$, over the non-trivial zeros of the Ramanujan's zeta function. 205 & 448206643590051608263691568113493984443540648811947725902790.626596 \\ and that around k=256 see Raymond's answer. I'm trying to evaluate the derivative of the Riemann zeta function at the origin, \\zeta'(0), starting from its integral representation \\zeta(s)=\\frac{1}{\\Gamma(s)}\\int_0^\\infty t^{s-1}\\frac{1}{e^t-1}. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. 30 & -0.0264657041470797526937304048599592953393370731885768642502823064627 \\ Our com- polcoeffs] return on your system ? rev 2020.11.11.37991, Sorry, we no longer support Internet Explorer, The best answers are voted up and rise to the top, MathOverflow works best with JavaScript enabled, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, Learn more about hiring developers or posting ads with us. I had noticed later that my cubic root formula failed for larger values => a correct asymptotic formula should of course be very welcome! seems to be fast decreasing. Here is a short routine in Pari/GP how the above coefficients can be computed to high accuracy by a very simple procedure: The first 12 coefficients Is $\zeta^{(k)}(s)$, $k\geq 0$, negative for $s\in (0,1)$? I fear that this will grow without bounds even if much slower than $n!$ but an asymptotic formula could be conjectured from these values ! 130 & 326172379219132017786027255436.163662671728811426407157377065370050 \\ Perhaps it should be. What does "worm of yellow convicts" mean? (D). I had guessed that the imaginary parts would get small quite quickly. 190 & 124737730975894951649278632325321300323483372940042824.738271112913 \\ Turns out it doesn't approach a constant, but starts to fall for $n>200$ almost linearly. Why discrete time signals are defined as sequence of numbers? 23 & 0.00305138562124162713884543738615856563404395363868348883899894968459 \\ What are the values of the derivative of Riemann's zeta function at the known non-trivial zeros? (A), evaluating it at $s = 1/2$, and isolating $\zeta^{(3)}(1/2)$ will make the latter depend on $\zeta^{(2)}(1/2)$ which in turn depends on the sum over the nontrivial zeros, your Eq. represented here (for $n$ from $17$ to $250$) : I tried to divide $\delta_n$ by different expressions in your limit and found : -\frac{\zeta'(1/2 + i\gamma)}{\zeta'(1/2 - i\gamma)} = e^{-2 i \theta(\gamma)} We may also notice that

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