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Values of the Riesz function

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Numbers
$x$ 
$R(x)$
0:
0
1:
0.04398180468826652940393085628238199704230190282530616903784426398471229296133686658743218170447565465
2:
-0.2677934017216908871370275775421357805943279627739324912544459042268272234922176756319054393574450894
3:
-0.5263556499538739919443703705089949882593232383590087728394374720096416370661878463607290549465689980
4:
-0.6855123833582043260658024857731241807261230838306938421038460572393899218492296627566276122828461155
5:
-0.7705597166420417461971145079108605652443034698155569614091127801998977764841987143558511805798929672
6:
-0.8091371821712181445973081353775704602633996699684285610490087285908059196347458184087071446346180547
7:
-0.8200996112275185806463357403451106697425237462496260364987804520502438397296891283037050210910299922
8:
-0.8149240735297561721912978753771926654353211674077997465923208593101657460101894080658058063554207965
9:
-0.8004231779395943527867740615904306771015469634923188660401868722236967381655703835021987159408890156
10:
-0.7806755812521964725926934005264063200751818610943916264912599338754227316261948265196450214937246669
11:
-0.7581689307332972276646002446525716976851041367785704524309808880449170896697739046046343879314278208
12:
-0.7344438566247881789938371879828116481109501138537546530192449520613692870621603419596197140310840864
13:
-0.7104589421109998516978360479221071512720716163010386580746882339156108591483289212575635457064123393
14:
-0.6868045221984648561817851450832142182530732625299707325330195784822739138966364509969761350665066283
15:
-0.6638333542533053126245887588848397586911456399051504694435101037288236885122676745665226778644746226
16:
-0.6417436533223916563605261716673376524952876027259682300299999023388484768102843599687579387803284937
17:
-0.6206333881166416562280534123574598828910148822685063093940583917053529205895513535219685038964378987
18:
-0.6005363800897754933436188948455635922420659198344639975632033602862723633546301933086283088114931719
19:
-0.5814464465475111124876752012049847024563616657170584311475850174596808828156418361345080440562296650
20:
-0.5633335005432003876168384617329157792596198456577328815154463782396795555665459193540965062087625604
21:
-0.5461541752854722667562619552512864889339400450608607034541285283605587938573185192247663669616328799
22:
-0.5298587107530845933758242595621026976698944626508390267640863088802238135802152903161899466727910543
23:
-0.5143952991841143301774891535196115695545257189715269838742117065680644917142736692624517135394454231
24:
-0.4997127196194704201924175816987739847561210084571759139919232277662603604585732784951121196452366218
25:
-0.4857618375671537874732188311678169874844802116252210197248158144360460430901235313597267510905208688
26:
-0.4724963675646677066172437207045432790596374286936042335383537097577729431145288282305252239462898055
27:
-0.4598731708394765767198103112081229931534455803190029154685264489581336782607341310026935779914695572
28:
-0.4478522719425396165807381321755856349605859089173627176544166760959880580776294228126041911378063864
29:
-0.4363967164271180661628079131712950149006592694687631030620938044652401252398859980648569778089395861
30:
-0.4254723487526836205165704062644872783107804199145177949191317492760993296165902775783929975606580060
31:
-0.4150475601531777184780447482679424883383695040825953494199760426586695598686197873933118054973771046
32:
-0.4050930362890668137518397462616911361529936103531177177493122603108423425887466084783898873116102735
33:
-0.3955815212728622722858257426763597595151220529075662966378844213357001931076370046371856824132161445
34:
-0.3864876060852065363553862344694003266647689422058423010894310078148517768757415019172262767082052016
35:
-0.3777875440302684062553427504019995866441256754094773666423250972438895355778069591367638417704549118
36:
-0.3694590926829591183285894597752503736421739931119275438490926656925357100928947769188347309377402883
37:
-0.3614813800289933061840773854087596978467452046405892416248382014718828585001976715368134672603789961
38:
-0.3538347916843252938797118303456419580186063118052936005402182587823914003175942758451736132258230059
39:
-0.3465008758527114542736139504255435754167348460578736081771594615809723289551767876785774125093977092
40:
-0.3394622628009681987816757793529195602337235999132159858489487236238882445364121339559119135504816669
41:
-0.3327025959420679782381667764030159568400660077058076153576096318467699462498490404006014240110770938
42:
-0.3262064720131957324387646183270779677446571840376669941702159488582761281944995734876671345666795092
43:
-0.3199593882543391025368124623152449680033424136591232194556556073237687830358104544954282586001201699
44:
-0.3139476948941778863660594086199797982180891518312916707900793258095767275282825873003346496603160931
45:
-0.3081585516127996962847759187776579099517572485105699834872333303779698588436361918666692866987080478
46:
-0.3025798869654101870776757671170743189959166491471505074579711689729958342614282825287796904232232538
47:
-0.2972003600153006481727642784848623008467723497804537597573487708245396072500699890879673977604574894
48:
-0.2920093236398268308913846331104816242306598587787616186752035178124941494607071778652869493437240146
49:
-0.2869967891444070183454282351006984666480652719466813928307139356570715919569206323016988044608243468
50:
-0.2821533919520713992878882999922323277744912251561092395677695542717255660012565941831603860411618880
51:
-0.2774703582357178597235590589333652046246875000665715892506183267146349374259626919646465541426591987
52:
-0.2729394724326040236027602787530844531556662522034571189358707819217312193225687955236450576829390579
53:
-0.2685530456309258839051613295384891515306492506826025308811212474179881699404498425179778030982727294
54:
-0.2643038848511967614151288633800488500627848737296117472466036143834739501673076544167531223082178728
55:
-0.2601852632645007513681827270962720967225833162280745551877117281738816699298990333515595609457934019
56:
-0.2561908913988747837975932183003803487760718405866241358360775482290188965773401039921647159823854179
57:
-0.2523148893868083111349382457475647698655529257401381148933616301325939745728906952410423817826873317
58:
-0.2485517603033492334444181770022591134368937159969178509805805226139736578170645603027979691522796655
59:
-0.2448963646373212809284245806146741158176794398068131375342969721859961887400476797885247871052306645
60:
-0.2413438959290546002140532133918136799873986931277359944245930541887022353055087613163437230472099502
61:
-0.2378898575978453960751935965138066760926965847519915849754862025578058461690839446535781438375498619
62:
-0.2345300409718626978798159602642001908531585128754261038429007096973849461636826656410704028413286009
63:
-0.2312605045229652255038626810927062150353817502852660540615479737069324804504909327197649751911038168
64:
-0.2280775542992615987467804653590702299925315330485594002539441707443002891840171117219198681456911640
65:
-0.2249777255394907983771738729885777361991682596379783519394330048754170472782786745946459576073231727
66:
-0.2219577654455612411441645825104062879945878948141618789111680739410918417322427012759126959234254602
67:
-0.2190146170829322837341955320275116323042263041128917541468715307561052720959775647448830618838299311
68:
-0.2161454043729600515860401131232862842891013068933534131079518159795931337616171954006023465472513685
69:
-0.2133474181368277207800667641382806960487691105387961546537129938669028072242984900614645818019375245
70:
-0.2106181031471781707571679325360430288071680235208550146726056207100964146756575445470862031051693890
71:
-0.2079550461409856583203366376727727334861222488422542199385941782427696784761911985632060113522754769
72:
-0.2053559647454539290229650137850061417158742644413295639051730827646499779812033389554935348077517990
73:
-0.2028186972677175725280109074755890695118461122327656427770034805258956913475459655546801385113526692
74:
-0.2003411932987577910646844598702316711474297390581456079710873438596625479606159241007889404938795785
75:
-0.1979215050821322150462980191302221758964433056737545883690838951042544013923006741030403189128687556
76:
-0.1955577795987749477854471149550315531794343704267764295125663389293151299113434914517447707935389953
77:
-0.1932482513201678067071010564267317711002155721284246869455650957874209724654926755024525855188233724
78:
-0.1909912355835438376926110521353096771043006572353694634090381805910370586269690957457351385472690957
79:
-0.1887851225443939630188167076827026324101963316195456716128900439710626047372866030635949908868609835
80:
-0.1866283716633487195214916542132231506859514726508247869536278604865928844021331188196868314794054890
81:
-0.1845195066864481775998418599799609938498511827134992111214642103987775014431364704786066754920429068
82:
-0.1824571110798497660419181481467942233871133371038810975039629047793006311775157455984312869523436362
83:
-0.1804398238821176136756407399387336474440113639418735519228299890460405315284983218007867326986722896
84:
-0.1784663359393556915858826640420548758671412081401327212855355756481318605181565209798912937254675095
85:
-0.1765353864905632869800364460333710807262310190197858245019772409835370399578682603400904756237446297
86:
-0.1746457600726826681131231905205609801943008656363858505021354147805367206991897641815132526577654582
87:
-0.1727962837168569121254587701983768779103345569208592201781178939708408537443404701326102523287575697
88:
-0.1709858244094061660101870858122751258320227053759001474888683725072173181829757138208350444297795074
89:
-0.1692132867929517285093066950602428857131380815684475409459077490209439666452449554951150726208655994
90:
-0.1674776110849607102428006176046487459504897307268745255090074320249081367457792183183433424854928182
91:
-0.1657777711927434880370045851789255843144058348818098786011030720118539832868220494023181569141237247
92:
-0.1641127730056076017272411458701694930127769531054826866379140557042240150340276904832592388682099406
93:
-0.1624816528464527590716098717863774868187691055134423503658896855089948319741847787590933997524481294
94:
-0.1608834760665812702015380471982185743798445044325850916321829980247797674234725479095577916461761097
95:
-0.1593173357688967710077919201582399416682172970311536310684502024116633227104276997211386078409867694
96:
-0.1577823516459727285755489081647106612075675179593297899762300035195720548131202213229744441694813700
97:
-0.1562776689206929424900570324405742111131136949912487814378818866492204840080817027815916274032394027
98:
-0.1548024573783016662804187898067961619492599700808415396283421191690291663978742667166420212757049818
99:
-0.1533559104797541448610344393393566226867968223589177586091922034158119506619235829255689123177916858
100:
-0.1519372445472327147317797481952298424815105267808753214092823479215264002044365013549826388650587397
Definition
The Riesz function is the entire function $R(x)=x\sum_{j=0}^{\infty}(-x)^j/(j!\zeta(2j+2))$ [1], equation (2). This table holds its values for real $x\geq0$.
Parameters
$x$
—   Argument ($x\geq0$)
Formulas
(1)
$R(0)=0$ and $\lim_{x\to0}R(x)/x=1/\zeta(2)=6/\pi^2$ [3], equation 25.6.1.
(2)
$\displaystyle R(x)=x\sum_{n=1}^{\infty}\frac{\mu(n)}{n^2}e^{-x/n^2}$, where $\mu$ is the usual Möbius function with $\mu(1)=1$ [1], equation (13).
(3)
$\displaystyle R(x)=xe^{-x}\sum_{k=0}^{\infty}c_k\frac{x^k}{k!}$, where $c_k$ are the Báez-Duarte coefficients [1], equation (10). This identity extends to $x=0$ by continuity.
(4)
$\displaystyle \left|\frac{R(k)}{k}-c_k\right|\leq\frac{3\sqrt{\pi}}{16k^{3/2}}+\frac{27e^{-3}}{2k^2}+\frac{15\sqrt{\pi}}{32k^{5/2}}+\frac{128e^{-4}}{k^3}$ for integers $k\geq1$ [1], Lemma 3, equation (33).
Comments
(5)
Here $\zeta$ is the ordinary Riemann zeta function, and $\pi$ is positive. The values retain their signs. In the generalized notation of [2], §1, this normalization is $R(x)=-xS_{2,2}(x)$. The function here is the number-theoretic Riesz function.
(6)
The Riemann hypothesis is equivalent to $R(x)=O(x^{1/4+\varepsilon})$ for every $\varepsilon>0$ as $x\to+\infty$ [1], equation (3). The finite computations here are unconditional and do not establish this asymptotic bound.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField

def R(x, digits=100):
    x = QQ(x)
    if x == 0:
        return 0
    assert x > 0
    F = RealBallField(numberdb.bits(digits, losing=2*int(x.ceil())+80))
    total, term, majorant = F(0), F(1), QQ(1)
    j = 0
    while True:
        if j+1 > x:
            tail = x*majorant/(1-x/(j+1))
            if tail < QQ(1)/10**(digits+35):
                return F(x)*total + F(0, tail)
        total += term/F(2*j+2).zeta()
        j += 1
        term *= -F(x)/j
        majorant *= x/j

values = [R(x) for x in range(101)]
References
[1]
J. Cisło and M. Wolf, On the Riesz and Báez-Duarte criteria for the Riemann Hypothesis (2008), equations (2), (10), (13), (33), Lemma 3 and Theorem 1. (arXiv)
[2]
R. B. Paris, The numerical evaluation of the Riesz function (2021), §§1–3. (arXiv)
Links
Similar tables
Báez-Duarte coefficients —   The exponential generating function expresses $R(x)$ through $c_k$, and integer arguments allow the comparison of $R(k)/k$ with $c_k$.
Values of the Riemann zeta function at rational numbers —   Reciprocals of the positive even zeta values are the coefficients of the defining series, before its factorial weights.
Bernoulli numbers —   Euler’s even-zeta identity supplies exact rational coefficients for an independent evaluation.
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds the exact origin value and every integer argument $1\leq x\leq100$ in the comparison of $R(k)/k$ with the Báez-Duarte coefficients [1], Lemma 3 and Theorem 1; the noncanonical cutoff gives consecutive early comparison values at moderate certified-series cost, without claiming coverage of the large-argument oscillations. The intrinsic origin and the documented integer-argument family are assessed separately in the attached argument-selection note)
How they were obtained:

The defining series is summed using Arb real balls at 415–613 working bits. After the terms $0\leq j<M$, the absolute remainder is bounded by $x^{M+1}/(M!(1-x/(M+1)))$ when $M+1>x$, since $0<1/\zeta(2j+2)<1$. The generator chooses this bound below $10^{-135}$ and adds it to the ball radius. Every nonzero result retains at least 406 relative accuracy bits, sufficient for its 100 significant digits; $R(0)=0$ is exact.

more

An independent 900-bit calculation splits the Möbius sum at $n=16$ and expands only its remaining exponential factors through degree 99. Taylor's remainder and the integral bound for $\sum_{n>16}n^{-202}$ give the absolute tail bound $x^{101}/(100!\,201\,16^{201})$. It recomputes the even zeta values from exact Bernoulli numbers and a Machin-series enclosure of $\pi$ [3], equation 25.6.2. All independent values retain at least 662 relative accuracy bits. Both enclosures lie strictly inside each written last-place interval.

The exponential-generating-function identity was checked at all 101 arguments with 553 freshly computed coefficients and a rigorous tail bound. The first 101 coefficients agree with the corresponding stored intervals in Báez-Duarte coefficients, and (4) holds on the stored intervals at every integer argument from 1 through 100. Known-answer and deliberate-error controls precede the comparisons. The attached generator, checks, argument-selection note and numerical evidence reproduce the computation.



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