Dual Möbius approximation errors in the Nyman–Beurling criterion

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Numbers

$N$

$E_N$

0.3068528194400546905827678785418234319244998656397447458793199905066063780303052843941366730035813125

0.3954002119219273831353072474526147559053112506357531414767050215372922729578820387719583372626466104

0.2146018366025516903843391541801242789507076501562235447562638519230458984284477503429912936644707330

0.6025665082557023163715023529256703382188301867198476278379567396981561202374056190917080812627528117

0.09310031788289107470296087117892213385796687595362971221505753230593840943682305815793750589396991557

0.4131530124679475713408715768427346024188753776034464671411149790362076232773712207932620249971805287

0.3168886680428644201105251192167598484291774629427476836961049667973760386005450250094203822170121269

0.2967885702019016055832268957129879545272320927671171134650258731271905893161563564071434315001836610

10:

0.07998733138039551203399768133408380395770254419322305964622619588108866670993937143792191728067865289

11:

0.2953602901135822171650022901205150866029143304226089121963457535488160825995976283938569052707677399

12:

0.1728315614869692654273957964288311648455724670401818761913231762665747722588034717236198806838125103

13:

0.5189055749863725975189142561378890813780913427058739476581249089454654658519592749829154659427431873

14:

0.2046804473503202263457814133029240187305718537765127868228078769278567545667740672914142062817979470

15:

0.09751575961239178133910339984558368012053527535829566532373800886899849096013668837766510609377307320

16:

0.07906568490138235704965980947552369336084545108394818295814076525167023917967406766899067218893721327

17:

0.2251650454559669410076666471662720313335505806728062582019239773594192341854254869871645775786964513

18:

0.1378391482066703881745388265726169089386403975863611917898936791456917347351282291060893758055387291

19:

0.4049443376208311975072555334305859650294243132433652685474715893760101351058910709713040827427941904

20:

0.3222310750637622805512275281223508600617260195297633131029933112731967369184400181933766290433591379

21:

0.1609551425265916711671971559971606269725676532035125942827968107428570941904765108246413733154530435

22:

0.07641136468070097680840731038019560796289188505524823403265335198376586038537301937793974464077048962

23:

0.1928225656048420874210336874884235329241289237357267608274635298627387240714661199427231976973991873

24:

0.1165630122261705396081473613397732795962488237280631709917506147512181410825825644265114796849393438

25:

0.1731170438645821355471175255219675909561802559061869828404816826915169090001084319995814328838269298

26:

0.1006391316953058056885187354345131614080931838086322279893063715404056486168772871210990373526080223

27:

0.1430262326925621639716431372925272328701949698539367908671210216634491868515549860097598260099411432

28:

0.07789950789816339076251762542689036626429990869006456984501078227261572532786921239995725220032788241

29:

0.1738751928155157092408274918183370540660469518146863895774007488109162688449567945343808841359999805

30:

0.2041328808750333747335992160595529042134115095806531557404235756517386842018345470061659846410757551

Definition

The squared error of the dual Möbius approximation is $E_N=\|\chi+D_N\|_{L^2((0,\infty),dx)}^2$, where $\chi=\mathbf1_{(0,1)}$ and $D_N(x)=\sum_{k=1}^N\bigl(M(N/k)-M(N/(k+1))\bigr)\{k/(Nx)\}-\{1/(Nx)\}$ [1], equation (1.19). Here $M(y)=\sum_{1\leq m\leq\lfloor y\rfloor}\mu(m)$, $\mu$ is the Möbius function with $\mu(1)=1$, and $\{y\}=y-\lfloor y\rfloor$.

Parameters

$N$

— Truncation index ($N\geq1$)

Formulas

(1)

$w_{N,k}=M(N/k)-M(N/(k+1))-\delta_{k,1}$ and $D_N(x)=\sum_{k=1}^N w_{N,k}\{k/(Nx)\}$, where $\delta$ is the Kronecker delta. Möbius inversion gives $\sum_{k=1}^N k w_{N,k}=\sum_{m=1}^N\mu(m)\lfloor N/m\rfloor-1=0$.

(2)

$K_{N;jk}=(j/N)A(k/j)$ and $b_{N;k}=(k/N)(\log(N/k)+1-\gamma)$, where $A(\lambda)=\int_0^\infty\{t\}\{\lambda t\}\,dt/t^2$ is the multiplicative fractional-part autocorrelation. These are respectively the Gram and target inner products for the functions $\{k/(Nx)\}$.

(3)

$E_N=1+2\sum_{k=1}^N w_{N,k}b_{N;k}+\sum_{j,k=1}^N w_{N,j}w_{N,k}K_{N;jk}$.

(4)

$E_1=1$, $E_2=1-\log2$, $E_3=1-\pi/(3\sqrt3)$ and $E_4=1-\pi/4$. The first value follows from $D_1=0$; the other values follow by integrating the periodic step residual after $t=1/(Nx)$.

Comments

(5)

The dual approximation is denoted $F_N$ in [1]. It is distinct from the untapered $F_N=\sum_{n=1}^N\mu(n)\{1/(nx)\}$ in [2], which [1] calls $S_N$. The dilations $N/k$ are prescribed by the dual construction, rather than independently selected arguments.

(6)

The coefficients satisfy $\sum_{k=1}^N k w_{N,k}=0$, so $D_N(x)=0$ for $x>1$. The same-index minimum in the integer-dilation space of squared Nyman–Beurling approximation distances does not automatically bound this error, whose dilation space differs.

(7)

The finite rational-argument formula of [3], Proposition 89, evaluates the autocorrelation kernel using signed Vasyunin cotangent sums, related to the Bettin–Conrey cotangent sums. Logarithms are natural, and $\gamma$ denotes Euler’s constant.

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

def dual_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))
    w=[ZZ(0)]*N
    for m in range(1,N+1):w[N//m-1]+=moebius(m)
    w[0]-=1
    cuts=sorted({QQ(a)/k for k in range(1,N+1) for a in range(k+1)})
    total=R(0)
    for left,right in zip(cuts,cuts[1:]):
        mid=(left+right)/2
        c=1-sum(w[k-1]*(k*mid).floor() for k in range(1,N+1))
        total+=c*c*(R(1+right).psi()-R(1+left).psi())
        start=max(left,QQ(1)/N)
        if right>start:total+=c*c*(1/R(start)-1/R(right))
    return total/N

# Every N=1,...,30 is stored; this function also accepts larger positive N.
dual_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 (1.14), (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, Proposition 89. (arXiv)

Links

[4]

Dual approximation, equation (1.19), version 1

[5]

Rational autocorrelation formula, Proposition 89

Similar tables

Báez-Duarte B_N approximation errors — The natural $B_N$ approximation and the dual construction are presented together in [1]; they use different dilations and arithmetic corrections.

Multiplicative fractional-part autocorrelation — The rational values $A(k/j)$ provide the Gram kernel $(j/N)A(k/j)$ for these dual errors.

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 natural $B_N$ and other classical errors and for independently certifying the shared rational-ratio kernel; the natural domain is all positive integers)

How they were obtained:

Computed from exact integer weights and finite cotangent sums with Arb real balls at 461 working bits. All inexact values retain at least 446 bits of relative accuracy, with radii below $2.35\times10^{-136}$, sufficient for 100 significant digits. The first entry is the exact integer 1 because $D_1=0$. No rounded companion values are inputs.

A separate 720-bit calculation integrates the entire residual as a periodic step function after $t=1/(Nx)$, using exact rational breakpoints and digamma differences for the infinite tail. All 30 norms agree, with independent radii below $9.43\times10^{-215}$. An explicit series remainder independently checks the tail at $N=1,2,3,4,30$. The checker tests four small closed forms and rejects deliberately incorrect values, wide balls, NaN and two altered formulas. Coefficient cancellation and the displayed program are checked throughout the prefix. The generator, executable checks, derivations and numerical evidence are attached.