Selberg approximation errors in the Nyman–Beurling criterion

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Numbers

$N$

$E_N$

3.106230071704746901741123202564027986596316939455796029217001376310928211521526782578645407396668147

2.210233170591801432318611415587786585683932549164101041610993412578469821828556505878231037769176707

1.594144483773250981404335720553528324892143768083416995841331072800064200083752485769650365442501150

1.310759780593444838390340125198735232001808500318478826056545578982988439761370407088233737136384215

1.047946950570719526599160030523779217394658178290048988934445742615876006079410267128368548164138234

0.9434785889358784247246890499192208968550762694453459633290077135651173276032990657152006935433043657

0.8248667748314742614453099485728179050265984007055043409981603399005065029834962230092117781166675137

0.7379799814812002899191476311752106624859448250657232541439103538396549380047699243546216682801612722

10:

0.6714847607843689521323719601223171618962090737801043821658361787121099412098349599047174783343860934

11:

0.6372395595898129695656506721423843035158645329481744384717914854851277340282927242611683684814642903

12:

0.5939215935940767913959000771196020354143403118641610276949056959708760698002502657081908045444390307

13:

0.5579568570729737857628488206806625883513899791426583766124287196967421153732346736862331098480018299

14:

0.5175415922970201013634921464053359213414030816046256792248389660234572358543647118048555442505260967

15:

0.4914180915458341611097290337545026830498194872092606712637307407976611672552220550899687405937734257

16:

0.4757662379347904865566697332923571512350109566830249391354518598617595483032258028183953688047225177

17:

0.4619848353402753787445654056103401363317861938993534997715201605241914825288778080024633045430412993

18:

0.4443033680859663213347789370828108186286353264345121973926854403480490182959451433812056052578443514

19:

0.4285355715720689976610703830052003769997246503304281676664753225403748584891948039656895173812012408

20:

0.4102230065770727854178113418590558326120135238296710555489686593238664828278765003859946242943693438

21:

0.3937670232135924421790856634517680615789024647044525979373867822386264562399611819527027957137673095

22:

0.3820640774950482062587171974568751024789036134818613805732005800291881582010120473500505624399153758

23:

0.3742509468565284569997017001842206848344310604472734361467836170673593340760170361073855286806539532

24:

0.3644331914949753546195868186541756496120754894613299992794604170375654039128175506029186295384256764

25:

0.3553914669961161102348203820125096658138139942883525223874001705956685587327236160461281332261901361

26:

0.3470329310442912050254287545893150410722437330434422576573543990015528730192194542486607153475844806

27:

0.3412418820734806586955105059816749528456558123007720114274586618113533828142071295520762620339166654

28:

0.3358442251085018934616496804917457364137023077345110341502360585394149278799634137003014605517337829

29:

0.3307987770349344256043275413661187439607279406901437262529670554359538859431746132603667749902284990

30:

0.3244903910175220071158390782447350425783584531682886454680426582127518779973025135841722237700715095

Definition

The squared error of the Selberg approximation is $E_N=\|\chi+S_N\|_{L^2((0,\infty),dx)}^2$, where $S_N(x)=\sum_{n=1}^N\mu(n)(1-\log n/\log N)\rho_n(x)$, $\chi=\mathbf1_{(0,1)}$, $\rho_n(x)=\{1/(nx)\}$, and $\mu$ is the Möbius function [1]. The fractional part is $\{y\}=y-\lfloor y\rfloor$.

Parameters

$N$

— Truncation index ($N\geq2$)

Formulas

(1)

$w_{N,n}=\mu(n)(1-\log n/\log N)$ for $1\leq n\leq N$, with $w_{N,1}=1$ and $w_{N,N}=0$. Write $w_N=(w_{N,1},\ldots,w_{N,N})^T$.

(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. The change of variable $t=1/(jx)$ gives this identity.

(3)

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

(4)

$E_N=1+2w_N^Tb^{(N)}+w_N^TG_Nw_N$, where $G_N=(G_{jk})_{1\leq j,k\leq N}$ and $b^{(N)}=(b_1,\ldots,b_N)^T$.

(5)

$E_2=3+\log(2\pi)-3\gamma$, since $S_2=\rho_1$ and $G_{11}=A(1)=\log(2\pi)-\gamma$ [4], Proposition 87.

(6)

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

Comments

(7)

The norm uses the entire positive half-line with measure $dx$ and the unanchored functions $\rho_n$. All logarithms are natural, and $\pi$ is positive. Write $\gamma$ for the Euler constant and $C=\log(2\pi)-\gamma$.

(8)

The plus sign in $\chi+S_N$ follows the convention that the Selberg approximation targets $-\chi$ [1], §1. The weights are the logarithmically tapered Möbius coefficients also used by Bettin, Conrey and Farmer [2], §1. At $N=1$ the logarithmic taper is undefined.

(9)

The multiplicative fractional-part autocorrelation supplies the Gram matrix of the functions $\rho_n$. Its rational-argument formula uses finite Bettin–Conrey cotangent sums, equivalently Vasyunin sums [3], equation (13).

(10)

Each $E_N$ is an unconditional finite norm computation. The optimal squared approximation distance $d_N^2$ gives a lower bound for $E_N$ in the same norm. No assumption about the Riemann hypothesis is needed for these finite values.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from sage.rings.real_arb import RealBallField
from sage.rings.rational_field import QQ
from sage.arith.misc import moebius

R = RealBallField(numberdb.bits(100, losing=128))
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)
def E(N):
    assert N >= 2
    w = [R(0) if n == N else R(moebius(n))*(1-R(n).log()/R(N).log())
         for n in range(1, N+1)]
    b = [(R(n).log()+1-R.euler_constant())/n for n in range(1, N+1)]
    return 1 + 2*sum((w[i]*b[i] for i in range(N)), R(0)) + sum(
        (w[j-1]*w[k-1]*A(QQ(j)/k)/j
         for j in range(1,N+1) for k in range(1,N+1)), R(0))

print(E(10))

References

[1]

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

[2]

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

[3]

S. Bettin and B. Conrey, Period functions and cotangent sums, Algebra & Number Theory 7 (2013), no. 1, 215–242, §2, equation (13). (arXiv) (doi)

[4]

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

Similar tables

Squared Nyman–Beurling approximation distances — The minima $d_N^2$ give lower bounds for these specified-approximant errors at the same truncation indices.

Multiplicative fractional-part autocorrelation — The kernel values $A(j/k)/j$ are the Gram entries in the squared-error formula.

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

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every truncation index $2\leq N\leq30$ in increasing order, a complete initial segment of the defined Selberg approximants. Every prime, composite and nonsquarefree endpoint is retained because changing $N$ changes the taper even when $\mu(N)=0$. Thirty is a practical, noncanonical initial bound for comparing all these errors with the matching finite optimal distances and certifying their shared Gram data; the argument-selection attachment explains the coverage)

How they were obtained:

Computed using Arb real balls at 461 working bits. Finite Vasyunin sums give the Gram entries at exact rational ratios, and interval arithmetic propagates every rounding error through the squared norm. Every value is finite and retains at least 449 bits of relative accuracy, sufficient for the 100 significant digits stored. No rounded companion values are used as inputs.

An independent 640-bit computation evaluates all 465 symmetric Gram entries by analytically integrating the piecewise quadratic fractional-part products against the trigamma kernel. It integrates 16 terms directly, uses 112 terms of a uniformly convergent expansion for the remainder, and adds an explicit bound for the omitted tail. The independent squared-error radii are less than $10^{-130}$. Known-answer and deliberate-error controls precede the comparisons. Both enclosures lie strictly inside each stored last-place interval. All 29 errors, the weight conventions, Gram symmetry and the lower bounds from independently solved least-squares systems are checked; the independent minima also lie inside the matching stored intervals in T641. The displayed Sage program agrees over the whole stored range. The generator, checks, derivation and numerical evidence are attached.