Báez-Duarte R_N Riemann-sum approximation errors

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Numbers

$N$

$E_N^{(R)}$

1.572641014547909618621335725814296820351898834152300007397476151516188389957502427100316891465713220

1.440717929224739404716667450176373255194796284184421005778273431317566941872940149684516458505908484

1.056059843305514163892152710460117129280116704929277988874307616814131455678399339872971653018633553

1.201995956999458877784299385083408108028556121317290141859282650304533452570328306425572494686393650

0.8507776469101210774027600545746711218117609627601302547560475772126309781450566764521807649430083558

0.8106249013713994449935211223953858121002281093079329496553552631431494355828166492535317317057265920

0.7352780556761488587198575505507641985326091566531040043393294084145015898894577869080693229610206988

10:

0.8551166672967525976294933251146667098942242034872394674442293625194950309857340835931930021493983816

11:

0.5994761968233981362811725441119750237264132181675027172090348191886734428255675795706229169454535720

12:

0.6395224419190232560026537759913922409601821460336981668610232111750399603140029597571109783963967550

13:

0.4763431778410431403074257767386502704721419140481049828586711635998839205367420733003192391988900147

14:

0.5222386364241970524593390470917687037145163429538737273454470356090161796599345988943408034561935783

15:

0.6247815611753835415889033634296036705758600341812575541973842540399528874898229604215363609442901751

16:

0.6086641793160985186542348267946320341224843597932169603811824214208282789004858573348761879003606796

17:

0.4237577943222719883227751096463639141533072134241305820215292481334943426842966707708754430595177855

18:

0.4471687451826605657691272594978282617364179228281429138433266819575469779505039786992616724041084094

19:

0.3419621138383225980601404061482101097240688552172611871646328103794127645756683177885042387589667699

20:

0.3543241713588308966979181562800637362585362601615352988796087412491792874051243431129684344533827267

21:

0.4059301902051723911366169569609091460944375396468060359879124927331739351448340385498205062850959220

22:

0.4760281561180412312149449614000614061300754911348005137853182283748651721357687346713638855365757450

23:

0.3315216620179603828239355394469691072307606912328681548088457648178627354329679371776397013375315521

24:

0.3438785821587198151721768302351933395596546434970918218652969661452848869591545441149246260659487701

25:

0.3173653223867267690696474156657194006402862150395939580472160968728626828995640799852105639253690720

26:

0.3730308031021169933326610231529746388802783398171579810589854817471335515800714382002104938576470758

27:

0.3602123328866856743133424326503402179611786614821034626881358402822994238166784765243217190111826311

28:

0.3901444524835869042387743441663294191212448906137122804124339578995863025602797493763006795029216993

29:

0.2712875315434915976363952822633639668142272357736942962826813919891422830970416808721073984231722141

30:

0.2679680796111296696830821029476929860342958030555572615865551496813698476240704620804856924658627967

31:

0.2240239623643853714245792013274573578401240539960642730913632717845225772166691340085710818621537089

32:

0.2201835735056935081122053410814183845240404160797686558754197631080547491724478907545385720300038640

33:

0.2260415904508966089278906053931170572422699824350049220790722946635349869438388419080407376687701415

34:

0.2510506039703497218091013681384979567765472625802164163940847222140183149813052968774523999031259328

35:

0.3132539554124161058248582260925896551037294859407234390548330405938082654421953119938516055644828873

36:

0.3227411543345399104189628054449710428928475068436306448696465011681021006880621595527666725279580606

37:

0.2229882923375211225991224545404643647064356478796603894984548657044662368768362339412061602674065727

38:

0.2624780432428479182956088900869951847653907808576963702079672235609144590733095813112013481107074945

39:

0.3288570144272616677662277110579773548562278052452652047235762612419942973027241462486756901421961990

40:

0.3430711462004368202303048354206132691077152065356387412790966487383874634456049813359803414220049529

41:

0.2335711082354363863572891824009194393454973184438117361053912973023439146210642813491931713525594381

42:

0.2176200347545317906196115588816545869700779833529024157796433167970312339211194290301204939690117588

43:

0.1672353212659528059478571997610561308419126399773228431303722616056537307464417482538733807748853712

44:

0.1736858351981508094027668182660044075179488784759488398542398782558031883452509213256110937352563783

45:

0.1858384275159003846544865728363434880898587477635702459491480210906678381975509069394526758243138204

46:

0.2087548308423281705292820296597567884151872990248517019817410571998789237249252471093079856459226510

47:

0.1580853574831893360117307264130107810128025210908201747906712216888749699636074212787032181728892116

48:

0.1605991187773748926018407859201961078278429639964161556611253972094057309534079145679404969983921776

49:

0.1515938963594975605608672361523167128946412503819148768383428510840907253330729990922213307181210136

50:

0.1566249673870668448917555100326321367240525439737181152532116490287269578735249445777734741195184721

51:

0.1775051262227558786257458089417423142007981080868857808546889675236888953283015303831558072107463787

52:

0.1937963698584722816479139681026386062129657812849014804349734318824096791208708979622125510207182142

53:

0.1438419516175576494467168693331877827498546837416564957703571422474932192612393287457133896100741791

54:

0.1463274674950450308643337101601221962375860435442042681127163118363358771113759132337342441675339316

55:

0.1735137145241372991830132431500175543251998885656006030550399410223498733678121285774611513967374198

56:

0.1821600485899249287522702486050110102369853522110461701728033234622993209371203508306547704585119614

57:

0.2263389381468850644221012586690549158167647157580468174488846819089436463666300448253903623042348410

58:

0.2688222834073359192409159207809063625100355925187861626926509428139075598448208204221681974537901140

59:

0.1816463495062927389617766619885302368353196840575489785182972536234913883032452200590367927628079656

60:

0.1760940498769674621793535276657406138025521400058008496698441715808337473711039123322966062293325549

61:

0.1294498098149720607659149107352675733791933048821381998592049573279945515885705308110820732377385717

62:

0.1500188988986860670575474433778337861961397877778513490742354406067534692990941440726149873407521040

63:

0.1587245766769932542116612798721198267456674805525412816349486897605289043087561476429868506200198833

64:

0.1576904174737902250246382022485797618589451473080880466266479468554464560933307097993492262085164888

65:

0.2015557559393775061573567673929713118518245084942321152086256111853446881290090956024255049626874571

66:

0.1840386588733572890471959420291764026534823696828301140132497458214121520048097657901227886329698563

67:

0.1296672480582678047309223284074436724317969776022600305318152382789618869986582112995260456982830328

68:

0.1381548749511611886288505735028148495868905095447390510042204808215320108708291927404003138257595620

69:

0.1688441744547386193668165604607331506766293311694664780607471849361558898078587399292368735604913712

70:

0.1572749373814105009247800828135105877087867996822257761339266519984744123798214263130175450763675596

71:

0.1141399732812658145043887274949931223893503507274743803089330043305800712435538953333275014448117816

72:

0.1158886836293285932711853053306656837963126143968116065957244429768424804243737215595854981685746135

73:

0.1014209615718088818463139270092537226299970295733748728697310213160420954651031791183644429028341055

74:

0.1013350121990477101965953985010421763826232044307448532746821581556093757095878053015112867650234582

75:

0.1047420374478167723686111113895507061324765031733001447610632908413425544610666958553898746647901429

76:

0.1098798284753426168085497508430608949462688080223053845369181500813398109569988478020484012350917253

77:

0.1331658139155268345839127565877796030160577738856789558102252959534754132206841213724253994105494156

78:

0.1306755401318613032200267715447869855819632700168459091015612717577281593695124802047815111072935667

79:

0.1019204702569265500745834177571652798667177449603964013497441728923058806279424242336705699624582873

80:

0.1037477555568924664900733472392014327267801521995122397142568108285283466434474271470510413350928989

81:

0.1024773271373961227617129667081128363971852965646499965582511048946012237420787965771132437928305440

82:

0.1073555050603516058910011588289606777694537259199841057095543328564520221356561495423133568656304275

83:

0.09162769910581365297077999782404435940065633504343127795991543316781068225205014225766674536329411058

84:

0.09322063810469558056722612563004972613405716368403788426872264613791760207130141971910548910362213150

85:

0.09433834039635849716249084891014301146634337216703786610575326732534236137912545005222895276408108774

86:

0.1049684416413143448800281720197230727395373752058142953917887592081323446273795270016932872948591357

87:

0.1255970858481535927399293056487275467864619347891086869104502530801525435134913430767441100155753397

88:

0.1319285372317462892020949495716929860710717838035030465508957028811853907170465693755245647190175942

89:

0.09490093433616706837029050794943431272525114305657385968911388314278514127490291876903059248046039381

90:

0.09372248669074575092199654668226767959404567659261024591304120454756315770321496076087041609786801844

91:

0.1152327884582843060987353310642584499688456779810013295959755206482710964210467840340718124062256363

92:

0.1249341223728943828219824966090331749156121580347499982413628226577352811910889890171998648726720086

93:

0.1589302832037173163183980408461643705697499435573593716151258839975815746695566035064302047133875522

94:

0.1910856835973258312258747309556751224335862211047652504969910260115300746207913964867521467380062453

95:

0.2497170376462290046418169710763941281024124618571700089603911422746311192415819339736199657822545769

96:

0.2515864396195399355833113415651907116015564417970473968162378549995060872619120051426995007945282657

97:

0.1739589301721827110360876989573547356460006752800362959644370066272942453832958334858142216686578894

98:

0.1784611543440410087783199541062207700594726998532327620388053592297791946774274669756540755684268800

99:

0.1921714524145275051180004394142342822294350356544124794958378239211849756620458224458214627080809163

100:

0.1957588527790800196384175303962837670595206313210984287403785934180020956371484101741527678203093567

Definition

The squared logarithmic-target error is $E_N^{(R)}=\|R_N-\lambda\|_{L^2((0,\infty),dx)}^2$, where $\lambda(x)=\mathbf1_{(0,1)}(x)\log x$ and $R_N(x)=\sum_{k=1}^{N-1}\frac{M(N/k)}{k}\{k/(Nx)\}$ is Báez-Duarte’s Riemann sum [1], §4.1, immediately before Proposition 4.3.

Parameters

$N$

— Riemann-sum index ($N\geq1$)

Formulas

(1)

Put $w_{N,k}=M(N/k)/k$ and $K_{N;jk}=(j/N)A(k/j)$, where $A(r)=\int_0^\infty\{t\}\{rt\}\,dt/t^2$ is the multiplicative fractional-part autocorrelation. Then $\|R_N\|_2^2=\sum_{j,k=1}^{N-1}w_{N,j}w_{N,k}K_{N;jk}$; the rational values of $A$ have the finite cotangent-sum formula in [2], Proposition 89.

(2)

$C_N=\langle R_N,\lambda\rangle=\frac1N\sum_{k=1}^{N-1}M(N/k)\left[(1-\gamma)\log(k/N)-\frac12\log^2(k/N)\right]$, where $\gamma=\gamma_0$ is Euler’s constant. The constant term of the individual Mellin integrals cancels because $\sum_{k=1}^{N-1}M(N/k)=0$.

(3)

$E_N^{(R)}=2+\sum_{j,k=1}^{N-1}w_{N,j}w_{N,k}K_{N;jk}-2C_N$.

(4)

$E_3^{(R)}=2+\frac{\log2}{6}-\frac{2\log2}{3}\left(1-\gamma+\log3-\frac12\log2\right)$.

Comments

(5)

Logarithms are natural. The upper summation endpoint is $N-1$; $M(N/k)$ includes the floor in the definition of $M$. The values are squared errors in measure $dx$, without division by $\|\lambda\|_2^2=2$. Values at jump endpoints do not affect the integral.

(6)

$R_1=R_2=0$, so $E_1^{(R)}=E_2^{(R)}=2$. These trivial controls are not included as numerical rows.

(7)

Möbius inversion gives $\sum_{k=1}^{N-1}M(N/k)=0$. Consequently $R_N(x)=0$ for $x>1$ and, after $t=1/(Nx)$, the sum is a periodic step function of $t$.

(8)

These are the errors of the specified finite sums, not errors minimized over an approximation space. The notation $R_N$ is distinct from the entire Riesz function $R(x)$.

(9)

Here $M(y)=\sum_{1\leq m\leq\lfloor y\rfloor}\mu(m)$, $\mu$ is the Möbius function with $\mu(1)=1$, and $\{t\}=t-\lfloor t\rfloor$.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField
from sage.rings.complex_arb import ComplexBallField
from sage.arith.misc import moebius

def riemann_sum_error(N, digits=100):
    N=ZZ(N)
    if N<1: raise ValueError('N must be positive')
    if N<=2: return ZZ(2)
    R=RealBallField(numberdb.bits(digits,losing=120))
    C=ComplexBallField(R.precision())
    M=[sum((moebius(j) for j in range(1,n+1)),ZZ(0)) for n in range(N+1)]
    w={k:QQ(M[N//k])/k for k in range(1,N) if M[N//k]}
    assert sum(k*v for k,v in w.items())==0
    jumps={QQ(0):QQ(0),QQ(1):QQ(0)}
    for k,v in w.items():
        for j in range(1,k):
            a=QQ(j)/k; jumps[a]=jumps.get(a,QQ(0))-v
    cuts=sorted(jumps); c=QQ(0); norm=R(0)
    for a,b in zip(cuts,cuts[1:]):
        c+=jumps[a]
        if c: norm+=c*c*(C(b).psi()-C(a).psi()).real()/N
    cross=R(0)
    for k,v in w.items():
        L=R(QQ(k)/N).log()
        cross+=R(v*QQ(k)/N)*((1-R.euler_constant())*L-L*L/2)
    return 2+norm-2*cross

# Stored coverage: every N=3,...,100; the function accepts all positive N.
riemann_sum_error(100)

References

[1]

L. Báez-Duarte, Arithmetical Aspects of Beurling’s Real Variable Reformulation of the Riemann Hypothesis, 2000, version 1, §1.1 and §4.1, pp.16–17, definition preceding Proposition 4.3. (arXiv)

[2]

L. Báez-Duarte, M. Balazard, B. Landreau and E. Saias, Sur l’autocorrélation multiplicative de la fonction « partie fractionnaire », 2003, version 1, §4 and §9, Proposition 89. (arXiv)

Links

[3]

Báez-Duarte: definition of R_N, pp.16–17

[4]

BBLS: rational autocorrelation, Proposition 89

[5]

DLMF §25.11: Hurwitz zeta function

Similar tables

Dual Möbius approximation errors — The dual Möbius approximation uses the same dilation locations with different weights and the indicator target $-\mathbf1_{(0,1)}$.

Multiplicative fractional-part autocorrelation — The fractional-part autocorrelation supplies the Gram kernel.

Bettin–Conrey cotangent sums — Bettin–Conrey cotangent sums are related to the finite arithmetic evaluation of that kernel.

Stieltjes constants — The Stieltjes constants fix the convention for $\gamma=\gamma_0$ in the logarithmic cross term.

Values of the Hurwitz zeta function — Hurwitz zeta values are associated with the deflated Hurwitz-zeta expansion used in the independent endpoint integration.

Values of the digamma function — Digamma values give the exact endpoint expression for the periodic norm integral.

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every index $3\leq N\leq100$ in increasing order, including nonsquarefree indices and indices with $M(N)=0$. The omitted indices $N=1,2$ both give the exact error 2. The consecutive prefix covers successive Mertens floor patterns; 100 is a noncanonical computational bound keeping the quadratic Gram calculation and independent rational partition small, not a restriction of the natural domain $N\geq1$)

How they were obtained:

Computed from exact Möbius and rational weights, the finite cotangent-sum kernel [2], Proposition 89, and the cancelled logarithmic cross term. Arb real balls at 453 working bits retain at least 435 bits of relative accuracy over all 98 entries, sufficient for 100 significant digits. All logarithms and cotangents enclose their exact rational inputs. There is no truncated infinite numerical sum and no rounded companion-table input.

A separate 500-bit calculation integrates the periodic step function using exact rational breakpoints, digamma differences and the linear coefficient of the deflated Hurwitz-zeta expansion at 1; see [5] for the defining special-function identities. Its finite balls retain at least 480 bits of relative accuracy and fit inside all 98 recorded intervals. It constructs Möbius values by an independent divisor recursion and imports no generator code, while sharing Arb and the mathematical definition. Controls include the exact errors for $N=1,2$, the closed form at $N=3$, a deliberately shifted value, NaN and broad balls. The generator, checker, derivation, argument selection and numerical evidence are attached.