Untapered Möbius approximation errors in the Nyman–Beurling criterion

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Numbers

$N$

$U_N$

3.106230071704746901741123202564027986596316939455796029217001376310928211521526782578645407396668147

1.076210744818494484684871849188509284920090596879948774133892760368435462237487971260121273210043907

0.2143668757778172491681734499375546193587481070247515066689983993561993605139103753173498237362858704

0.04857587456570884487773965270930336807226487754301404616129744647095708330853011821860317341645905099

0.1816853260527228617903812783357108262130978730493851468933774152031084604968563136936559234228992479

0.03739422234720036180993410480521683555226397025348968837914997271794713545325284407490287155674011302

10:

0.1371864713835239794258190434617638316002800552747325366875291274770052645784764563911908498931038205

11:

0.03415951217838289673530843212573891415649523164421900604339779629784285888988478252724320150480486990

12:

0.03415951217838289673530843212573891415649523164421900604339779629784285888988478252724320150480486990

13:

0.1307570929863762127840213138639002289408657984500514726148337663780300941462222810831223873816261020

14:

0.03303433605975004778955938991494351923606024164981433463257157798869621994446171540468696834891775278

15:

0.1064699896454489502720703654819309103637395269038534420046390514017222626305336511132195438240123308

16:

0.1064699896454489502720703654819309103637395269038534420046390514017222626305336511132195438240123308

17:

0.03734473333989613079275481813625914618025623115796719169063112875978165841889346706153967807784484873

18:

0.03734473333989613079275481813625914618025623115796719169063112875978165841889346706153967807784484873

19:

0.09989006830109911458986677017079178901287517975096975336200652640775108199446594508869278754664982310

20:

0.09989006830109911458986677017079178901287517975096975336200652640775108199446594508869278754664982310

21:

0.03975729760935954361538296498772485798006461662658488808258349441487612513935720276712112436563754068

22:

0.09098244291052858500087030245666410085553869559422493761932055791801081452764234007077166687878487424

23:

0.04067573865995944517268085740426421832486127022742974643099416990540714985426976099622581673707236803

24:

0.04067573865995944517268085740426421832486127022742974643099416990540714985426976099622581673707236803

25:

0.04067573865995944517268085740426421832486127022742974643099416990540714985426976099622581673707236803

26:

0.08294732986206539384598400160985330491442833630871226912616031185402642258643666080722545646828099039

27:

0.08294732986206539384598400160985330491442833630871226912616031185402642258643666080722545646828099039

28:

0.08294732986206539384598400160985330491442833630871226912616031185402642258643666080722545646828099039

29:

0.03707901364099820882008262541079028189330143020174222379930063110425757543443039960059566838197723337

30:

0.07122139051812860924706802112128642976033071556318861617419973554370558756315225855007903410622982348

Definition

The untapered Möbius approximation has squared error $U_N=\|\chi+F_N\|_{L^2((0,\infty),dx)}^2$, where $F_N(x)=\sum_{n=1}^N\mu(n)\{1/(nx)\}$, $\chi=\mathbf1_{(0,1)}$, and $\mu$ is the Möbius function [1], equation (1.1).

Parameters

$N$

— Truncation index ($N\geq1$)

Formulas

(1)

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

(2)

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

(3)

$U_N=1+2\sum_{n=1}^N\mu(n)b_n+\sum_{j,k=1}^N\mu(j)\mu(k)G_{jk}$.

(4)

$U_1=3+\log(2\pi)-3\gamma$ and $U_2=2-\gamma-\tfrac12\log2$. The first identity uses $A(1)=\log(2\pi)-\gamma$ [3], Proposition 87.

(5)

If $N\geq2$ is not squarefree, then $U_N=U_{N-1}$, because $\mu(N)=0$.

(6)

$\int_1^\infty |F_N(x)|^2\,dx=g_N^2$, where $g_N=\sum_{n=1}^N\mu(n)/n$, since $F_N(x)=g_N/x$ for $x>1$.

(7)

$U_N\geq d_N^2$, 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. Set $c_n=-\mu(n)$ to obtain the bound.

Comments

(8)

The fractional part is $\{y\}=y-\lfloor y\rfloor$, and $\mu(1)=1$. The norm uses Lebesgue measure $dx$ on the entire positive half-line. The coefficients are the untapered Möbius values, and the plus sign records approximation to $-\chi$. All logarithms are natural; $\gamma$ denotes the Euler constant.

(9)

This $F_N$ is the natural approximation in [1], §1, equation (1.1). The same sum is denoted $S_n$ in [2], §1.2, equation (1.18); that paper uses $F_n$ for a different dual approximation.

(10)

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

(11)

These are unconditional finite norm computations. No assumption about the Riemann hypothesis is needed, and the values alone do not establish a limiting approximation theorem.

Programs

(P1)

Sage

import numberdb.sage as numberdb
from functools import lru_cache
from sage.rings.rational_field import QQ
from sage.rings.real_arb import RealBallField
from sage.arith.misc import moebius
R = RealBallField(numberdb.bits(100, 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))
@lru_cache(None)
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 U(N):
    assert N >= 1
    w = [moebius(n) for n in range(1,N+1)]
    return 1 + 2*sum((w[n-1]*(R(n).log()+1-R.euler_constant())/n
                     for n in range(1,N+1)), 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))
# The stored prefix is every N=1,...,30; U also accepts larger N.
print(U(10))

References

[1]

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

[2]

L. Báez-Duarte, Arithmetical Aspects of Beurling’s Real Variable Reformulation of the Riemann Hypothesis, 2000, §1.2, equation (1.18). (arXiv)

[3]

L. Báez-Duarte, M. Balazard, B. Landreau and E. Saias, Sur l’autocorrélation multiplicative de la fonction « partie fractionnaire », 2003, §§4 and 9, Propositions 87 and 89. (arXiv)

Similar tables

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 $U_N$ from below in the same integer-dilation space.

Selberg approximation errors in the Nyman–Beurling criterion — The Selberg construction replaces $\mu(n)$ by $\mu(n)(1-\log n/\log N)$. Its $N=2$ error equals $U_1$.

Báez-Duarte B_N approximation errors in the Nyman–Beurling criterion — The moving-endpoint correction $B_N=F_N-Ng_N\rho_N$ cancels the exterior tail of $F_N$. The linked values are $\|\chi+B_N\|_2^2$ in the same $L^2((0,\infty),dx)$ space.

Vasyunin approximation errors in the Nyman–Beurling criterion — The fixed-endpoint correction $V_N=F_N-g_N\rho_1$ cancels the exterior tail of $F_N$. Its squared residual norms retain the nonsquarefree-endpoint repetitions because both $F_N$ and $g_N$ then remain unchanged.

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 comparing the full prefix with the existing optimal and Selberg errors and 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. Exact integer Möbius weights and finite cotangent sums at exact rational arguments give the Gram matrix; ball arithmetic propagates the error through every squared norm. All 30 values are finite, retain at least 444 bits of relative accuracy, and have radii below $7.30\times10^{-136}$, sufficient for the 100 significant digits stored. No rounded companion values are used as inputs.

A separate 720-bit calculation integrates the piecewise quadratic fractional-part product against the trigamma kernel, evaluating 16 terms analytically and enclosing the remaining series after powers 0 through 125 with an explicit alternating-series bound. All 465 symmetric Gram entries and all 30 norms agree; the independent norm radii are below $6.05\times10^{-151}$. Direct whole-residual integrals also agree at $N=1,2,3,6$. Known-answer controls and deliberate incorrect, wide-ball and NaN controls test the checker. All nonsquarefree repetitions, exterior-tail bounds, and lower bounds from independently solved least-squares systems pass. The attached generator, checks and derivation preserve the executable evidence.