Báez-Duarte B_N approximation errors in the Nyman–Beurling criterion

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Numbers

$N$

$E_N$

0.3068528194400546905827678785418234319244998656397447458793199905066063780303052843941366730035813125

0.09861029163443178714836109599428549874823576493106367964368518037024722592133560845909898249444726370

0.06761421711443814768225111844033211273198163438272435087727568183004159304471926673016938348929895177

0.04118933583695433939964119161598877526777220436398019994512160956070124992395860205582373919253680851

0.04158012748956997831482404762848265040228119367533778025895248432190299434414586066154600251160944648

0.03580775013638797182681593963854903183026946945023255693051148358512976898880990983213050137910484072

0.03687741131937063869915692782104285612298839809736741334923267461365783834985090348323176747173336680

0.03697831909801807891317756882369788127636125862941495737073264478885548674719317007397876240924484608

10:

0.03603540405648812592023208069910665189927136310568187741997420919135738374342821469704446988100918663

11:

0.03410697464560355858403583412366901412334926194687276160179973277113291547850482754191056556412277712

12:

0.03416551388279420163364209244481724128134841484276410832424367266201723600273638267742434050413152456

13:

0.03416470156464601201466594726441361935776778433325286040234580450401166363852624758514507691672649316

14:

0.03301748521898086464829203438489097273483598659619725827135910336237826455721726947269837262904005025

15:

0.03275551195324441940988206223250116576275349287804430387177946896742916521337420034202722718122156303

16:

0.03717748574148775256097724558460744693357544844322150801318995738937325575800639395280712315702026050

17:

0.03758724298987055427812483664327340724468528585527681920101825351653756073463842691394505608900752603

18:

0.03735911928862364810527212635026618819451928098101659535512572244834833715575945749617574400567498322

19:

0.03729413244829567895255843807461314534344928278065025678117593327968050122900101071463211885520814141

20:

0.04288032809375354935268224794689304261479148470238388755437691437403226934473031795748006386810917524

21:

0.04000460049440525392593337071134730119776251998684108003170640091711314943950952760252313878427292839

22:

0.03960920910453212566500160089821775188982669476044122840955603335340432964433076521197177304729027038

23:

0.04059605026878643498154897311885255507117159497932890205444679851396722391330345154134597886540161022

24:

0.04064967731488141430386559335105456615317504111120502881137630188444672484197457800436193384948440127

25:

0.04059685836545419980130417730818349813835577069158232907804814962574243958722920510124983001744196390

26:

0.04053581690108616454336247953380103968883093789699318036672635367676048615144623217163992456996252319

27:

0.03971052679293734112417654213190254035643033562255428280158797070315904062155698986819984483681845503

28:

0.03824744931341884803893626966848689110732032226590222377534757680117648294096832701462000795495574659

29:

0.03717898440196280343998817621815532890041253954030019837311863945155746744972933419247735158835338184

30:

0.03671232407933205279776151113046798978237221968602425722789299779934563710742693021954403989242519835

Definition

The squared error of Báez-Duarte’s natural approximation is $E_N=\|\chi+B_N\|_{L^2((0,\infty),dx)}^2$, where $B_N(x)=\sum_{n=1}^N\mu(n)\rho_n(x)-Ng_N\rho_N(x)$, $g_N=\sum_{n=1}^N\mu(n)/n$, $\rho_n(x)=\{1/(nx)\}$ and $\chi=\mathbf1_{(0,1)}$ [1], equation (1.14). Here $\mu$ is the Möbius function and $\{y\}=y-\lfloor y\rfloor$.

Parameters

$N$

— Truncation index ($N\geq1$)

Formulas

(1)

$w_{N,n}=\mu(n)$ for $1\leq n<N$, and $w_{N,N}=\mu(N)-Ng_N=-N\sum_{n=1}^{N-1}\mu(n)/n$. Consequently $\sum_{n=1}^Nw_{N,n}/n=0$.

(2)

$G_{jk}=\langle\rho_j,\rho_k\rangle=A(j/k)/j$, where $A(\lambda)=\int_0^\infty\{t\}\{\lambda t\}\,dt/t^2$ is the multiplicative fractional-part autocorrelation. This follows from $t=1/(jx)$.

(3)

$b_n=\langle\chi,\rho_n\rangle=(\log n+1-\gamma)/n$, since $b_n=n^{-1}\int_{1/n}^\infty\{t\}\,dt/t^2$ and $\int_1^\infty\{t\}\,dt/t^2=1-\gamma$.

(4)

$E_N=1+2\sum_{n=1}^Nw_{N,n}b_n+\sum_{j,k=1}^Nw_{N,j}w_{N,k}G_{jk}$.

(5)

$E_1=1$ and $E_2=1-\log2$. Indeed $B_1=0$, and $B_2=\rho_1-2\rho_2$ gives $E_2=\sum_{m\geq1}(1/(2m)-1/(2m+1))$.

(6)

$E_N\geq d_N^2$, where $d_N^2=\min_{c\in\mathbb R^N}\|\chi-\sum_{n=1}^Nc_n\rho_n\|_2^2$ is the optimal squared approximation distance. Choose $c_n=-w_{N,n}$ in the minimum.

Comments

(7)

The plus sign records approximation to $-\chi$. The norm uses Lebesgue measure $dx$ on the positive half-line, with $\mu(1)=1$ and ordinary fractional parts. All logarithms are natural; $\gamma$ denotes the Euler constant. These finite norm values require no assumption about the Riemann hypothesis.

(8)

The correction at the moving dilation $N$ cancels the exterior tail: $B_N(x)=0$ for $x>1$. Moreover, $B_N(x)=-1$ for $1/N<x<1$, so $E_N=\int_0^{1/N}|1+B_N(x)|^2\,dx$ [1], §1.2. Endpoint values have no effect on the integral.

(9)

Write $F_N=\sum_{n=1}^N\mu(n)\rho_n$ for the untapered sum, following [2], equation (1.1). The same sum is called $S_n$ in [1], equation (1.18), while that paper reserves $F_n$ for a different dual approximation in equation (1.19). Thus $B_N=F_N-Ng_N\rho_N$.

(10)

The multiplicative fractional-part autocorrelation supplies the Gram matrix. Its rational-argument formula [3], Proposition 89, uses finite Vasyunin sums, related by sign and modular inversion to the Bettin–Conrey cotangent sums.

Programs

(P1)

Sage

from sage.all import QQ, ZZ, RealBallField, moebius, gcd
from functools import lru_cache

def baez_duarte_error(N, bits=461):
    N=ZZ(N)
    if N<1:
        raise ValueError("N must be positive")
    if N==1:
        return ZZ(1)
    R=RealBallField(bits)
    C=(2*R.pi()).log()-R.euler_constant()
    @lru_cache(None)
    def V(p,q):
        return sum((R(QQ((m*p)%q)/q)*(R.pi()*R(QQ(m)/q)).cot()
                    for m in range(1,q)),R(0))
    def G(j,k):
        d=gcd(j,k); p=j//d; q=k//d; x=R(QQ(p)/q)
        return (((1-x)*x.log()+(1+x)*C)/2
                -R.pi()*(V(p,q)+V(q,p))/(2*q))/j
    w=[QQ(moebius(n)) for n in range(1,N+1)]
    w[-1]-=N*sum((w[n-1]/n for n in range(1,N+1)),QQ(0))
    b=[(R(n).log()+1-R.euler_constant())/n for n in range(1,N+1)]
    return (1+2*sum((w[j]*b[j] for j in range(N)),R(0))
            +sum((w[j]*w[k]*G(j+1,k+1) for j in range(N)
                  for k in range(N)),R(0)))

# The stored selection is every N=1,...,30; the domain is all N>=1.
baez_duarte_error(4)

References

[1]

L. Báez-Duarte, Arithmetical Aspects of Beurling’s Real Variable Reformulation of the Riemann Hypothesis, 2000, version 1, §§1.1–1.2, equations (1.4), (1.14), (1.18) and (1.19). (arXiv)

[2]

L. Báez-Duarte, A strengthening of the Nyman–Beurling criterion for the Riemann Hypothesis, 2002, version 2, §1, equation (1.1). (arXiv)

[3]

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, Propositions 87 and 89. (arXiv)

Similar tables

Untapered Möbius approximation errors — The untapered sum $F_N$ becomes $B_N$ after subtracting $Ng_N\rho_N$ at the moving endpoint.

Multiplicative fractional-part autocorrelation — The values $A(j/k)/j$ provide the Gram kernel for these squared norms.

Squared Nyman–Beurling approximation distances — The minimum $d_N^2$ bounds $E_N$ from below in the same integer-dilation space.

Dual Möbius approximation errors in the Nyman–Beurling criterion — The dual approximation also targets $-\chi$ and cancels the exterior tail, using dilations $N/k$ and Mertens-difference coefficients. These generally differ from the integer dilations and moving-endpoint correction defining $B_N$.

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every truncation index $1\leq N\leq30$ in increasing order, including nonsquarefree endpoints. Thirty is a practical, noncanonical initial bound for comparison with the existing optimal, Selberg and untapered errors and for independently certifying the shared Gram kernel; the natural domain is all positive integers)

How they were obtained:

Computed with Arb real balls at 461 working bits from exact rational Möbius weights and finite cotangent sums at exact rational arguments. All inexact entries retain at least 443 bits of relative accuracy, with radii below $7.42\times10^{-136}$, sufficient for 100 significant digits. The first entry is the exact integer 1 because $B_1=0$. No rounded companion values are inputs.

A separate 720-bit calculation integrates the piecewise quadratic fractional-part product against the trigamma kernel. It evaluates 16 terms analytically, then sums powers 0 through 125 with an explicit alternating-series remainder bound. All 465 symmetric Gram entries and all 30 norms agree; the independent norm radii are below $6.04\times10^{-151}$. Whole-residual integrals also agree for $N=1,2,3,4,6$. Known-answer, deliberately incorrect, wide-ball and NaN controls test the checker. Exact endpoint cancellation and lower bounds from independently solved least-squares systems hold throughout the computed prefix. The generator, checks, derivation and numerical evidence are attached.