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It's a curious fact that for $n>0$, $\zeta^{(n)}(0)\approx -n!$. Apostol gave a table for $\frac{\zeta^{(n)}(0)}{n!}$, among other results on $\zeta^{(n)}(0)$ . the sequence :

$\delta_{n}=\left | \zeta^{(n)}(0)+n! \right|$

seems to be fast decreasing. What is the upper bound of $\delta_{n}$ ??

Edit: Following the logic in Apostol's paper, $\zeta(s)-\frac{1}{s-1}$ is holomorphic. Thus:

$\zeta(s)-\frac{1}{s-1}=:A(s)=\sum_{n=0}^{\infty}\frac{A^{(n)}(0)}{n!}s^{n}$ where : $\left|A^{(n)}(0)\right|=\delta_{n}=\left|\zeta^{(n)}(0)+n!\right|$ the expansion converges everywhere. Therefore : $\limsup_{n\rightarrow\infty}\left(\frac{\delta_{n}}{n!}\right)^{\frac{1}{n}}=0$ To be exact, I am interested in the limit : $\limsup_{n\rightarrow\infty}\left(\frac{\delta_{n}}{n}\right)^{\frac{1}{n}}$ hence the question !!

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    I updated my answer with two pictures concerning your limit. I hope it will help more,2012-08-24

4 Answers 4

1

consider the original problem : $f(s)=-\psi\left(1-\frac{1}{s}\right)-K_{0}-\frac{K_{1}}{2}s-\sum_{n=1}^{\infty}\frac{K_{2n}B_{2n}}{(2n)!}s^{2n}$ Where :

$K_{0}=1.825593297777$

$K_{1}=1+\zeta^{(1)}(0)$

$K_{n}=\frac{n!+\zeta^{(n)}(0)}{n}$

We use the asymptotic expansion of $\psi(x)$: $\psi(1+x)=\frac{1}{x}+\psi(x)=\ln(x)+\frac{1}{2x}-\sum_{n=1}^{\infty}\frac{B_{2n}}{2nx^{2n}}$ therefore-naively speaking-: $f(s)=-K_{0}-\frac{\zeta^{(1)}(0)}{2}s+\ln(-s)-\sum_{n=1}^{\infty}\frac{B_{2n}}{(2n)!}\left(\frac{\zeta^{(2n)}(0)}{2n} \right )s^{2n}$ and we got rid of $(2n)!$ in $K_{2n}$. a couple of questions remain: is the asymptotic expansion of the digamma function exact!? what's the domain of convergence of the expansion!? and what's the radius of convergence of our last expansion !?

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    Since few people search questions in answers don't hope too many reactions (it breaks the Q->A principle favored here !). Anyway the two series are only asymptotic (from my previous [link](http://en.wikipedia.org/wiki/Bernoulli_number#Asymptotic_approximation)) so that $=$ should be replaced by $\sim$ and the radius of convergence will be $0$ (removing the $2n!$ will make the convergence worse as in $\sum \frac {\zeta(kx)}k$). Asymptotic expansions are very powerful when we use only the first terms! Your problem seems to be asymptotic versus analytic. You could ask a question clarifying this.2012-08-25
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UPDATED:

Since Apostol's table is imprecise for the latest values let's exhibit this partial table of $n!+\zeta^{(n)}(0)$ values obtained with the method proposed by Gottfried Helms in the comments :

$ \small \begin{array} {r|l} n&\qquad n!+\zeta^{(n)}(0)\\ \hline 0 & 0.500000000000000000000000000000000000000000000000000000000000000000 \\ 1 & 0.0810614667953272582196702635943823601386025263622165871828484595172 \\ 2 & -0.00635645590858485121010002672996043819899491016091988116986828085776 \\ 3 & -0.00471116686225444776106081336637528546180766829598013289308154130860 \\ 4 & 0.00289681198629204101278047225899433810886006507829657502399066695362 \\ 5 & -0.000232907558454724535985837795819747892057172470502296621517290052364 \\ 6 & -0.000936825130050929504283508545398558763852909268098676811811642454272 \\ 7 & 0.000849823765001669151706027602351218392176760368993802245821950220545 \\ 8 & -0.000232431735511559582855690063716869861547455605351528951730144900587 \\ 9 & -0.000330589663612296445256127250159219129163115391201597238597920006568 \\ 10 & 0.000543234115779708472231988943120310085619430025648031886746513765534 \\ 11 & -0.000375493172907263650467030884105539552908523317127333739022948360384 \\ 12 & -0.0000196035362810139197664840250843355865881821335996260346542408699771 \\ 13 & 0.000407241232563033143432121366810273073439244495052894296377049143472 \\ 14 & -0.000570492013281777715641291383838137142317654464393538891561665994592 \\ 15 & 0.000393927078981204421827660818939487435931013173319003367358811853101 \\ 16 & 0.0000834588058255016817276488047155531844625161484345203508967032195293 \\ 17 & -0.000660943729628596896169402998134057724748414684628214724260392025847 \\ 18 & 0.00102622728654085400217701415546883787759831069743902026886240548348 \\ 19 & -0.000865575776779282991576072414036571104593129616540810229322531122882 \\ 20 & 0.0000192936717837051401063299760357760104805477068753543599966583874264 \\ 21 & 0.00135690605213454946114913783265117619902887065782808784758635491569 \\ 22 & -0.00269215645875329128403425710948994793671854878855377935283522438652 \\ 23 & 0.00305138562124162713884543738615856563404395363868348883899894968459 \\ 24 & -0.00142429184941854585322218679179524558923410706804512920069410425063 \\ 24 & -0.00142429184941854585322218679179524558923410706804512920069410425063 \\ 25 & -0.00270778921288600678819748219175554231288488376985887236498730634210 \\ 30 & -0.0264657041470797526937304048599592953393370731885768642502823064627 \\ 35 & -0.263594454732269692589658594912151283515046273581182559219921957221 \\ 40 & -2.99127389405887676303274513146663241574504274783600393720076526420 \\ 45 & -37.8116918598476995713457928854407359489376750764425231304638226967 \\ 50 & -484.410856973911340196834881321159996957875322777427689682560124377 \\ 55 & -4532.79225770921715189195122554511879361201057777310708972171082184 \\ 60 & 62714.1067695718525498151218611523939474897844785985218047818417901 \\ 65 & 7023172.71452427788836637890070922964875579872202726818830758507697 \\ 70 & 369710251.754342613761487189243065702707445997550023978646801198349 \\ 75 & 16153042555.8916006284817291830191070360270906645699878986789707549 \\ 80 & 615738270543.419763620055014818673603045117612121993882170431591493 \\ 85 & 18734769337973.5357476254698119630570458879847958412519956399551375 \\ 90 & 236370935383452.039427873106518081170156120659521416134138380827174 \\ 95 & -26002457205974856.1210597020683157222183992446452182712157359931706 \\ 100 & -3067048412469082717.13203493456872773456001456014271660974790930507 \\ 105 & -208147105464557539810.933105520613946023324136236314019489461672300 \\ 110 & -9181100257482418076527.78433198963677385024539967354760263208242840 \\ 115 & -51947662171852808135142.6566163041506055684371473226227514782141120 \\ 120 & 40156333121359621232445103.0657969214804033377921435547843142396453 \\ 125 & 4885455264162691954362582051.19629295409841919596706506250394536303 \\ 130 & 326172379219132017786027255436.163662671728811426407157377065370050 \\ 135 & 5681896814647267766788984138309.92777649447549648680365985502173310 \\ 140 & -1823873410669202891713087061952487.29233810951837725134296601143730 \\ 145 & -287161238605183347710570327381611857.621502693613616741540893113635 \\ 150 & -21305861581790622498949173421790799625.1089486390817454023538053647 \\ 155 & 41341935656925531212500416560539095352.2344118482658208289324353056 \\ 160 & 247591097041903905305863994419088881629306.695276383394597551295589 \\ 165 & 35417487509305790307439844806554155410647762.2818942813077054202595 \\ 170 & 1939388852429349721510180790653718054320127522.76657886070312620767 \\ 175 & -219609544533102325798714608918968968215179933676.462881353291615996 \\ 180 & -64398214417872662764963987879167602127249665707913.3748997726013799 \\ 185 & -6471529441461413822723169640664516218513802097544790.17826333568557 \\ 190 & 124737730975894951649278632325321300323483372940042824.738271112913 \\ 195 & 146090125339857661850314283330560855583771401129477483038.196790939 \\ 200 & 21761038288742061134507006188990514804372485347492068735353.4677389 \\ 205 & 448206643590051608263691568113493984443540648811947725902790.626596 \\ 210 & -436802309714509751568738654004051406952276718382033685343775072.767 \\ 215 & -87517428053442479414927505641545087908985720235451301367834785555.4 \\ 220 & -3.84724299091446288828137723409916186345658241907462046206042911305E66 \\ 225 & 1.78354688800770241687161303825386645838232647101391084926254576406E69 \\ 230 & 4.39266696650096770242083480719428532550626963368237730956507167675E71 \\ 235 & 2.44222335896278620212620252751346268294589748965319118864768279107E73 \\ 240 & -1.01131768916824854497126506938489065442700604328045419557651761065E76 \\ 245 & -2.76259374758593015959757159949637125866476599264626984421242976978E78 \\ 250 & -1.51090342799297835940857060215282929045189635533361177391732022698E80 \end{array} $
(updated 3.12.2016, values from index 200 to 250 had to be corrected)

I fear that this will grow without bounds even if much slower than $n!$ but an asymptotic formula could be conjectured from these values !


Concerning the limit : $\lim\sup_{n\rightarrow\infty}\left(\frac{\delta_{n}}{n}\right)^{\frac{1}{n}}$

I can only show you the 'brainy' picture obtained for values of $n$ from $1$ to $250$ :

brainy

The largest value obtained is near $2.047$ but this doesn't seem to stop.

Note that this is nearly the same picture than for $\ \lim\sup_{n\rightarrow\infty}\left(\delta_{n}\right)^{\frac{1}{n}}\ $ (division by $n$ doesn't matter much).

If we observe that the real takeoff of $\delta_n$ waits until $n=25$ then a not too bad approximation of the previous curve is : $f(n)=\frac{\sqrt[3]{n-17}}3$ represented here (for $n$ from $17$ to $250$) :

(d-17)

I tried to divide $\delta_n$ by different expressions in your limit and found : $\ \lim\sup_{n\rightarrow\infty}\left(\dfrac{\delta_{n}}{\sqrt[3]{n!}}\right)^{\frac{1}{n}}\ $

n!^1/3

with the interesting 'saturation' near $0.4646$.


//Scripts used (pari/gp) :  //Method proposed by Gottfried Helms (precomputed Stieltjes table)  zs(n)=(-1)^n*sum(k=0,#Stieltjes-n-1,Stieltjes[k+n+1]/k!)  //Direct evaluation of the nth derivative at z (ep= 1E-50 or less) zp(z,n,ep)=sum(k=0,n,(-1)^k*binomial(n,k)*zeta(z+(n-2*k)*ep))/(2*ep)^n 
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    @GottfriedHelms: Please feel free to update my answer with the corrected values. I had shortly the hope that the divergence itself was ficticious but the first "signs of divergence" resisted well to increasing precision... Other things are wrong in this answer like the approximation $\,f(n)=\frac{\sqrt[3]{n-17}}3\,$ but I didn't take the time to investigate this further. Many thanks for the interest and your work about this anyway,2016-12-03
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This should go as another comment to @Raymond Manzoni.
Here is a short routine in Pari/GP how the above coefficients can be computed to high accuracy by a very simple procedure:

   \p 400    \\ \p 200      \\ set precision for dec digits    \ps 256   \\ \ps 128     \\ set number of terms for taylor-series expansion    taylor_eta = sumalt(k=0,taylor((-1)^k*1/(1+k)^x,x))     laurent_zeta = taylor_eta/(1-2*2^-x)         \\-- coeffs = polcoeffs( laurent_zeta + 1/(1-x),256)  \\ extract coeffic    coeffs = Vec ( laurent_zeta + 1/(1-x) )  \\ extract coeffs (update dez 16)    vectorv(12,r,coeffs[r]*(r-1)!)           \\ display the first few coefficients 

The first 12 coefficients
$ \small \begin{matrix} 0.500000000000 \\ 0.0810614667953 \\ -0.00635645590858 \\ -0.00471116686225 \\ 0.00289681198629 \\ -0.000232907558455 \\ -0.000936825130051 \\ 0.000849823765002 \\ -0.000232431735512 \\ -0.000330589663612 \\ 0.000543234115780 \\ -0.000375493172907 \\ \vdots \end{matrix} $
and that around k=256 see Raymond's answer. Possibly we should increase the internal num-precision even higher to get meaningful digits below the decimal point for that high coefficients.
The computation to 120 good coefficients took only a few seconds with that given precision of 256 dec digits . For 256 good coefficients we need decimal precision \p 400 and much more memory and a couple of seconds more time


obsolete due to update dez 16 having the most simple precudere by "Vec()"
Pari/GP-script for "polcoeffs"

\\ lp: the polynomial or series, local; maxd: option to force length of result-vector  {polcoeffs(lp, maxd=0) = local(llp, lpd, lv, lv1);   llp=Pol(lp);lpd=poldegree(llp);  if(lpd<0,return(vector(maxd)));  lv=vector(lpd+1,k,polcoeff(llp,k-1));  if(maxd>0,lv1=vector(maxd,k,if(k>lpd+1,0,lv[k]));lv=lv1);  return(lv);} addhelp("polcoeffs","uses a scalar entry containing a polynomial, converts it into a vector of coefficients.") 
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    I was trying it and, with your initial settings, found that the value for $n=120$ was correct to $26$ digits (on the $28$ returned) : not bad. I tried with $500$ digits of precision and this time I got $326$ digits for $n=120$ (no idea how many are right :-)). It is a really excellent method ! Many thanks to share it !!2012-08-24