Vasyunin approximation errors in the Nyman–Beurling criterion

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Numbers

$N$

$E_N$

0.4801396145800410179370759089063675739433748992298085594094899928799547835227289632956025047526859843

0.1373075776575155993042772713373314035243770115705757481797260240505785862665167282646451650866317935

0.04622688604395133042215056786292439124394713210934761235965078773964864520524152272070379513637222791

0.1280286228185727184635076894478671238123314891696383522920216454601534017141341609538282491212089909

0.03716558805377165665127192695495758905043381214562332099836557874808744852839803427846751226691194486

10:

0.1092300822484763136340623625340605530093860400842074303923803347475047476581835882969894125776844642

11:

0.03415857173564666298547493532022029023511018416642116967017765682787212886633659475073873641608663352

12:

0.03415857173564666298547493532022029023511018416642116967017765682787212886633659475073873641608663352

13:

0.1092505153619197324437309777961906285863750016163300881732435332383801472251771236538529621851356175

14:

0.03293540802213847349716168099444429302526059416868308609504453082789820825439476111224811548401075372

15:

0.09239478826218946527779510250469832545557150879644317353207712393168663825122095021798773898618245846

16:

0.09239478826218946527779510250469832545557150879644317353207712393168663825122095021798773898618245846

17:

0.03733458003096910217254679932245459780435311557801440879547796290525842894301988451169463472334328345

18:

0.03733458003096910217254679932245459780435311557801440879547796290525842894301988451169463472334328345

19:

0.08959163817739640522760133257670286329251200070233613692705743336164975311004374977668398838044819709

20:

0.08959163817739640522760133257670286329251200070233613692705743336164975311004374977668398838044819709

21:

0.03972681425784912471112139680385085287223076791715853743961503220817738234608890301458698192307972336

22:

0.08349870459711744683634590672854396826548630976285145554921223501931060082959637476109207580262177079

23:

0.04067165667911617938666281352149133901397091329104595825570281061600910073397401532320865296509543281

24:

0.04067165667911617938666281352149133901397091329104595825570281061600910073397401532320865296509543281

25:

0.04067165667911617938666281352149133901397091329104595825570281061600910073397401532320865296509543281

26:

0.07695276965496454928048508411087631713470219493086708442039703814411330383960875344726698060506497852

27:

0.07695276965496454928048508411087631713470219493086708442039703814411330383960875344726698060506497852

28:

0.07695276965496454928048508411087631713470219493086708442039703814411330383960875344726698060506497852

29:

0.03705153674321387794306986653581934745319156151927608081125598102747010248280538125068680469139512636

30:

0.06703417596559116325039735345863626959410643566727790594376474069976101512889853465611734573984466984

Definition

The squared error of Vasyunin’s approximation is $E_N=\|\chi+V_N\|_{L^2((0,\infty),dx)}^2$, where $V_N(x)=\sum_{n=1}^N\mu(n)\rho_n(x)-g_N\rho_1(x)$, $g_N=\sum_{n=1}^N\mu(n)/n$, $\rho_n(x)=\{1/(nx)\}$ and $\chi=\mathbf1_{(0,1)}$ [1], equation (1.17). Here $\mu$ is the Möbius function and $\{y\}=y-\lfloor y\rfloor$.

Parameters

$N$

— Truncation index ($N\geq1$)

Formulas

(1)

$w_{N,1}=1-g_N=-\sum_{n=2}^N\mu(n)/n$ and $w_{N,n}=\mu(n)$ for $2\leq n\leq N$. Thus $V_N=\sum_{n=1}^Nw_{N,n}\rho_n$ and $\sum_{n=1}^Nw_{N,n}/n=0$.

(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 $E_2=1-\tfrac34\log2$. Indeed $V_1=0$, while $V_2=\tfrac12\rho_1-\rho_2$ alternates between $0$ and $-1/2$ on consecutive unit intervals of $1/x$.

(6)

If $N\geq2$ is not squarefree, then $E_N=E_{N-1}$, because $\mu(N)=0$ leaves both the untapered sum and $g_N$ unchanged.

(7)

$E_N\geq d_N^2$, 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

(8)

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.

(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 uses $F_n$ for a different dual approximation in equation (1.19). Thus $V_N=F_N-g_N\rho_1$.

(10)

The correction at the fixed dilation $1$ cancels the exterior tail: $V_N(x)=0$ for $x>1$. Hence $E_N=\int_0^1|1+V_N(x)|^2\,dx$. Endpoint values do not affect the integral.

(11)

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

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.rational_field import QQ
from sage.rings.integer_ring import ZZ
from sage.rings.real_arb import RealBallField
from sage.arith.misc import moebius
from math import gcd
from functools import lru_cache

def vasyunin_error(N, digits=100):
    N=ZZ(N)
    if N<1:raise ValueError('N must be positive')
    if N==1:return ZZ(1)
    R=RealBallField(numberdb.bits(digits, losing=128))
    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 gram(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-R.pi()*(v(p,q)+v(q,p))/q)/(2*j)
    w=[QQ(moebius(n)) for n in range(1,N+1)]
    w[0]-=sum((w[n-1]/n for n in range(1,N+1)),QQ(0))
    return (1+2*sum(w[n-1]*(R(n).log()+1-R.euler_constant())/n
                    for n in range(1,N+1))
            +sum(w[j-1]*w[k-1]*gram(j,k)
                 for j in range(1,N+1) for k in range(1,N+1)))

vasyunin_error(30)

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), (1.17)–(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 $V_N$ after subtracting $g_N\rho_1$ at the fixed 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.

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 with repeated values. 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 $8.20\times10^{-136}$, sufficient for 100 significant digits. The first entry is the exact integer 1 because $V_1=0$. 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 $5.88\times10^{-151}$. 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 exterior cancellation, nonsquarefree repetitions, the expanded correction identity, and lower bounds from independently solved least-squares systems hold throughout the computed prefix. The displayed program agrees on all 30 entries. The generator, checks, derivation and numerical evidence are attached.