Squared Nyman–Beurling approximation distances

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Numbers

$N$

$d_N^2$

0.8582120513955108846699602429560164483399588459775845212939416366957630731074203573526077492391228668

0.2899645737422027318112005834160466112054138711256138547460910874719743025925830812276884334308104723

0.09383722849698329190428280653064391729132770289268991145970787017702685168500709188815963479312577484

0.06512886865990024957106136367286123661983130106485852754299938606864298358168647498509067698519936981

0.03610159981860540420305303010822015165921677941505960525443380905202876269110630866830567669415876810

0.03598631735357441684874565496739849150057560940236395331290812101523038161195715521196725479248227634

0.02430210054347671988410428742850929429275158778164485748788741763160586877108515471975531258308948369

0.02416142158589668502280894301085133744676548383129550670275237960783523709142253624330201384982214990

0.02401495078279004617671234264738635893651075986792286988642435951749686158080679973312202389059883290

10:

0.02377186555603957374966975040957789209541784886012934583757512802222957241310443600256439411017750339

11:

0.02044444456484509094802749313893338019987154287503380244395562309471538637546288108932279971149923284

12:

0.02044306320754249936177763825750427085665354572602214781127957425439349025049114151251667712056490270

13:

0.01982440645628308113229460064086909127595718536279239003135111991873682445078596578948033760955034980

14:

0.01848642561827221155670274954947651251467766519139312974188583140351366534305110896156914850404733938

15:

0.01833503460075522518174374009253289544138590559940849793946972092683370770254022318894407055509198274

16:

0.01789402347696943509905506314795876127745464180160987804844025903873581522050027285013834892881957455

17:

0.01706923844333575743005586709030729028737936262222226208304982217662527566724131606142438418038038910

18:

0.01705923539565751988223606625688232309628964454513476718163986270630521066793710008823807443183474583

19:

0.01681731800069844373685085110344559928704628689318714712714509593583907528245787799414579222751416458

20:

0.01650358307103018838973256823549361415872156702085623525249486733115295445707342082899299351849126111

21:

0.01609075635431805715379443648408456507487617994469552993017915914551795141842515964826587284190522329

22:

0.01605738248915308017497450614911479615798657368893086753604863639410924827209866932902275956169450318

23:

0.01554729962882193925781253895111196115833954670706660477263616165323720708577699940674953529658312638

24:

0.01554691488841462023756475281903107043470773868076083965799972273042990893765983442049681313169104674

25:

0.01552000930071485852939088995393226853685508406387663109153481296990426298528026068260575393035760564

26:

0.01542340901249896645848246258431018197511603937785841986240131117629805528091168440548923394848793268

27:

0.01531396462752732040957648997393755498063833844681284028978367460259708895852092777144940981042325131

28:

0.01521459097526316009616224810558585530690442686476306607542588218745619625186125969253771556861994800

29:

0.01470294702816850426377278073913782395748213499193156846842538305504515234069533257305889549743431314

30:

0.01453011047413725086961718433983317483868908161988556986833859859378759561653672714763647553675586065

Definition

The squared Nyman–Beurling approximation distance is $d_N^2=\min_{c\in\mathbb R^N}\|\chi-\sum_{n=1}^N c_n\rho_n\|_{L^2((0,\infty),dx)}^2$, where $\chi=\mathbf1_{(0,1)}$, $\rho_n(x)=\{1/(nx)\}$, and $\{y\}=y-\lfloor y\rfloor$ [1].

Parameters

$N$

— Truncation index ($N\geq1$)

Formulas

(1)

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

(2)

$b_n=\langle\chi,\rho_n\rangle=(\log n+1-\gamma)/n$. Indeed, $b_n=\frac1n\int_{1/n}^\infty\{t\}\,dt/t^2$ and $\int_1^\infty\{t\}\,dt/t^2=1-\gamma$.

(3)

$G_Nc^{(N)}=b^{(N)}$ and $d_N^2=1-(b^{(N)})^TG_N^{-1}b^{(N)}$, where $G_N=(G_{jk})_{1\leq j,k\leq N}$ and $b^{(N)}=(b_1,\ldots,b_N)^T$. The finite Gram matrix is positive definite.

(4)

$d_1^2=1-(1-\gamma)^2/C$, with $C=A(1)=\log(2\pi)-\gamma$ [3], Proposition 87.

(5)

$0\leq d_{N+1}^2\leq d_N^2\leq1$, since the approximation spaces are nested and the zero approximant has squared error $\|\chi\|_2^2=1$.

Comments

(6)

The norm uses the entire positive half-line and the unanchored functions $\rho_n$. The coefficients are unrestricted real numbers; no condition such as $\sum c_n/n=0$ is imposed. All logarithms are natural, and $\pi$ is positive. Write $\gamma$ for the Euler constant and $C=\log(2\pi)-\gamma$.

(7)

The Riemann hypothesis is equivalent to $d_N^2\to0$ as $N\to\infty$ [1], Theorem 1.1. The finite minima in this table require no assumption about the Riemann hypothesis.

(8)

If $c^{(N)}$ is the optimal vector in the definition, the Dirichlet-polynomial formulation with residual $1-\zeta(s)\sum_{n=1}^N a_n n^{-s}$ uses $a_n=-c_n^{(N)}$ [4], §1. This coefficient convention gives the same squared minimum. Here $\zeta$ is the Riemann zeta function.

(9)

The multiplicative fractional-part autocorrelation $A$ supplies the Gram matrix of the functions $\rho_n$. Its finite rational-argument formula uses the Bettin–Conrey cotangent sums, equivalently Vasyunin sums [2], equation (13).

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.real_arb import RealBallField
from sage.rings.rational_field import QQ

N = 10
R = RealBallField(numberdb.bits(100, losing=256))
C = (2*R.pi()).log() - R.euler_constant()
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 A(r):
    p, q = int(r.numerator()), int(r.denominator())
    x = R(r)
    return ((1-x)*x.log()+(1+x)*C)/2 - R.pi()*(V(p,q)+V(q,p))/(2*q)
b = [(R(n).log()+1-R.euler_constant())/n for n in range(1,N+1)]
G = [[A(QQ(j)/k)/j for k in range(1,N+1)] for j in range(1,N+1)]
M = [row+[v] for row,v in zip(G,b)]
for k in range(N):
    assert M[k][k] > 0
    for i in range(k+1,N):
        f = M[i][k]/M[k][k]
        for j in range(k+1,N+1):
            M[i][j] -= f*M[k][j]
x = [R(0) for i in range(N)]
for i in range(N-1,-1,-1):
    x[i] = (M[i][N]-sum((M[i][j]*x[j] for j in range(i+1,N)),R(0)))/M[i][i]
d2 = 1-sum((b[i]*x[i] for i in range(N)),R(0))
print(d2)

References

[1]

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

[2]

S. Bettin and B. Conrey, Period functions and cotangent sums, Algebra & Number Theory 7 (2013), no. 1, 215–242, §2. (arXiv) (doi)

[3]

L. Báez-Duarte, M. Balazard, B. Landreau and E. Saias, Sur l’autocorrélation multiplicative de la fonction « partie fractionnaire », 2003, §§8–9, Propositions 85, 87 and 89. (arXiv)

[4]

S. Bettin, J. B. Conrey and D. W. Farmer, An optimal choice of Dirichlet polynomials for the Nyman–Beurling criterion, 2012, §1. (arXiv)

Similar tables

Multiplicative fractional-part autocorrelation — The values $A(j/k)/j$ are the Gram entries for the minimizing problem.

Bettin–Conrey cotangent sums — These finite sums give an arithmetic expression for the rational-dilation Gram kernel.

Keiper–Li coefficients — The nonnegativity of the Li coefficients is a different criterion equivalent to the Riemann hypothesis.

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, a complete initial segment of the nested spaces generated by integer dilates. This retains every successive enlargement, including prime and composite indices. Thirty is a practical, noncanonical initial bound for certifying dense systems and keeping the supporting coefficient vectors inspectable; the argument-selection attachment explains the coverage)

How they were obtained:

Computed using Arb real balls at 589 working bits. The Gram kernel is evaluated from finite Vasyunin sums with exact rational arguments. Interval $LDL^T$ elimination encloses each minimum, and every pivot is strictly positive. The smallest reported accuracy is 537 bits and the largest radius is less than $1.60\times10^{-164}$, sufficient for the 100 significant digits stored. No rounded companion values are used as inputs.

An independent 720-bit calculation evaluates all 465 symmetric Gram entries by integrating the piecewise quadratic fractional-part product against the trigamma kernel. It integrates 16 terms analytically and bounds the omitted tail after a 126-term expansion. Independent Gaussian elimination agrees on all 30 minima and all 465 coefficients across the nested spaces; its largest distance radius is less than $3.67\times10^{-136}$. Known-answer and deliberately perturbed controls precede the comparisons. Positive minima, strict decreases through index 30, normal-equation residuals, quadratic norms, and Schur-complement decreases are checked. The generator, independent checker, derivations and numerical evidence are attached.