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

Question

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 !!

Answer 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 !

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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 $0.4646$.

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//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

Answer 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

---

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.")

Answer 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

Answer 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 !?

Answer 5

Check following paper on this constants:

https://arxiv.org/abs/1703.08601