Derivatives of the Riemann zeta function at $s=0$

Solution 1:

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 $250$ :

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 $250$) :

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

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

Solution 2:

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

\\ 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.")

Solution 3:

Might be interesting to use this integral expression valid for $n>0$:

$$\zeta^{(n)}(s)=\frac{(-1)^n n!}{(s-1)^{n+1}}- \\ -i \int_0^{\infty } \frac{dt}{e^{2 \pi t}-1} \left(\frac{(1+i t)^s}{\left(1+t^2\right)^s} \log ^n\left(\frac{1+i t}{1+t^2}\right)-\frac{(1-i t)^s }{\left(1+t^2\right)^s}\log ^n\left(\frac{1-i t}{1+t^2}\right) \right)$$

Then we have:

$$\zeta^{(n)}(0)+n!=\frac{1}{i} \int_0^{\infty } \frac{dt}{e^{2 \pi t}-1} \left(\log ^n\left(\frac{1+i t}{1+t^2}\right)-\log ^n\left(\frac{1-i t}{1+t^2}\right) \right)$$

This gives us an expression for any $n$. However, for $n \gg 1$ it would make sense to use asymptotic methods for integrals. I'll look into them and see if it's possible to obtain explicit asymptotics.

As an example, let's check Raymond Manzoni's results for the sequence:

$$ \left(\dfrac{\delta_{n}}{\sqrt[3]{n!}}\right)^{\frac{1}{n}}$$

Turns out it doesn't approach a constant, but starts to fall for $n>200$ almost linearly.

Though numerical integrator in Mathematica complains for large $n$ about the loss of accuracy. On the other hand, increasing WorkingPrecision helps a lot.

For example WorkingPrecision->200 gives us:

$\zeta^{(250)}(0)+250!=$ =-1.5109034279929783594085706021528292904518963553336117739173202269846025525168616941972683605219085437983104127128224870598639347093804399780683205039591444322515764303108542485979750277722211393292010*10^80

Solution 4:

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 !?

Solution 5:

Check following paper on this constants:

https://arxiv.org/abs/1703.08601