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Squared Nyman–Beurling distances in Bagchi's sequence space

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Numbers
support
$N$ 
$D_{\mathrm{support}}(N)^2$
all
2:
0.3068528194400546905827678785418234319244998656397447458793199905066063780303052843941366730035813125
all
3:
0.09574778392775402661430811934014709003105690882872867980703311526036140789233593746114189793271551331
all
4:
0.06595169421556965355895907941992572044740051585557043288111835234926688802362616573691878362148726504
all
5:
0.03631901793818696047422749135750981806394372955267061218407079962615681432293049088908527008490912539
all
6:
0.03620803463690562840174885320818928103914413705003535930779895059951197730732720218861993677195206903
all
7:
0.02438836788524400644614166028424866093103525218641079048043746363740762554399495878568648673964785233
all
8:
0.02424452530652267314820583316791120499527793638105566151166887360811219580865051249555713867305065644
all
9:
0.02409507881871232158751270198814401751721271674854830751076173393305719875909960687982801451766320244
all
10:
0.02385255331888299941570681207054595762106380229850926467134133807201769589645260894991706399603257723
all
11:
0.02049960134702381595154389416926353863968330816261533859517952536281201045952954895766344109993922001
all
12:
0.02049814733289416643842476316631318167934403466784663608310066449723905505661095199345693066048531353
all
13:
0.01987424304169146636827677955326208038483065180699386110004272915878576079473304409147449822259776354
all
14:
0.01853181414232141132748004914517563752667099301475807475985200204955120981175572691752761933268245006
all
15:
0.01838021483747243407818428660017724756493074806279340786811544280889637667095144146822947851624352481
all
16:
0.01793626702020254986000384985802453220367561602390076750497931192751049636347641745156835869982546016
all
17:
0.01710664639217534619616287211655356000124262535050719375490947637256147526978556166915701681375653261
all
18:
0.01709653511225306649856568607185100827947374009316938005449304132259407390860261326437381536920657643
all
19:
0.01685306515200006668873206676017269381718717074471140482571095159812119476990844173857000102190555848
all
20:
0.01653825210939251560905834489428395619512617903962245418499450148939009557493640939055823579745774710
all
21:
0.01612418792278515852854417503616027932910458645421428332622845593774861190033158076217079252212833756
all
22:
0.01609082660923739795124163494223065991387058371202483893158781083123430433105158788854286614726477774
all
23:
0.01557815678134527971380675137330227919813314181910345990114158399669306146351535532689251195097285422
all
24:
0.01557777799236068930336597454409388807315031634867325887640361329008598279918722886686044781989134212
all
25:
0.01555084225299854252547620834171309073709258018148454538355672154501529186652554849528059891305530828
all
26:
0.01545398359559796478914739569844286128588003424359721891530179580839777485503181019784920702670191706
all
27:
0.01534401650811366695087734534366622888603339372060205234072381056078274303340074086220996787620210935
all
28:
0.01524421267434023855766758364403549473874337230544186933556101301552376657391787912798373055036800708
all
29:
0.01473027365781969595962710004576530143721970773910463012784376381338254041404402063766006801780834896
all
30:
0.01455673357699699235093427994055474259423083111169116665925499587990921828459497396059067527146661573
squarefree
2:
0.3068528194400546905827678785418234319244998656397447458793199905066063780303052843941366730035813125
squarefree
3:
0.09574778392775402661430811934014709003105690882872867980703311526036140789233593746114189793271551331
squarefree
4:
0.09574778392775402661430811934014709003105690882872867980703311526036140789233593746114189793271551331
squarefree
5:
0.04023860691261084430413126483613629660459901946102065677586293722298398516176520809116366025663749556
squarefree
6:
0.04023010207850747381126163007239142157006345325128317126975075765895109397508506353087588392160018127
squarefree
7:
0.02598729498244347990815597547919449413482195647409494218267843768499229446160812892543068265878891555
squarefree
8:
0.02598729498244347990815597547919449413482195647409494218267843768499229446160812892543068265878891555
squarefree
9:
0.02598729498244347990815597547919449413482195647409494218267843768499229446160812892543068265878891555
squarefree
10:
0.02598657565060827337596741208366672155511948548531169870617862976235627006801768160918716753969778280
squarefree
11:
0.02181608324865721758254853521986979973332388239995874569227222161903901486378449088945101606173084506
squarefree
12:
0.02181608324865721758254853521986979973332388239995874569227222161903901486378449088945101606173084506
squarefree
13:
0.02118662102133146909568080191838384848437412995737192033843458375343795690700013016012393838952262954
squarefree
14:
0.01961548384684870706296155472965664948524707603931513937575177184777482184295935123184854375064226615
squarefree
15:
0.01948812450099691479241331494536997661173598986180682145638267837948686751752802436430412396519011411
squarefree
16:
0.01948812450099691479241331494536997661173598986180682145638267837948686751752802436430412396519011411
squarefree
17:
0.01786864927717206169024482124945678688282928343467915174686007156717092427016071352010247729780725669
squarefree
18:
0.01786864927717206169024482124945678688282928343467915174686007156717092427016071352010247729780725669
squarefree
19:
0.01763281863409592926868862665115709047791892692075400169937948697483331628605404089918803020858257920
squarefree
20:
0.01763281863409592926868862665115709047791892692075400169937948697483331628605404089918803020858257920
squarefree
21:
0.01677136294678169820215527235412457660920815637666148215054778511045234363688624005803143379114504305
squarefree
22:
0.01673017237527825563339356150349294963467571531518098642023085423619372904556644448468921717689767422
squarefree
23:
0.01622460857943068170545718211350604270959307800282535777729732249896536600197858528555186354519112205
squarefree
24:
0.01622460857943068170545718211350604270959307800282535777729732249896536600197858528555186354519112205
squarefree
25:
0.01622460857943068170545718211350604270959307800282535777729732249896536600197858528555186354519112205
squarefree
26:
0.01613423866696323752462863081355252917876955723168974041181973177168502744967632150714275244977739744
squarefree
27:
0.01613423866696323752462863081355252917876955723168974041181973177168502744967632150714275244977739744
squarefree
28:
0.01613423866696323752462863081355252917876955723168974041181973177168502744967632150714275244977739744
squarefree
29:
0.01536728566276687681633642867792050753268694532084703238193252490008869063389192042374327349216120140
squarefree
30:
0.01524296932847705467288662317039599562680055926934389609951917254048436896880488188243508887985611252
Definition
The squared distance $D_{\mathrm{support}}(N)^2$ is $\min_{\alpha\in\mathbb R^{S_{\mathrm{support}}(N)}}\sum_{m=1}^{\infty}(1-\sum_{k\in S_{\mathrm{support}}(N)}\alpha_k\{m/k\})^2/[m(m+1)]$, where $\{t\}=t-\lfloor t\rfloor$, $S_{\mathrm{all}}(N)=\{2,\ldots,N\}$ and $S_{\mathrm{squarefree}}(N)=\{k:2\leq k\leq N,\ \mu(k)^2=1\}$. Here $\mu$ is the Möbius function; the weight and sequences are those of [1], p. 137.
Parameters
support
—   Denominator support ($\mathrm{all}$ or $\mathrm{squarefree}$, with the sets specified in the definition)
$N$
—   Largest allowed denominator ($N\geq2$)
Formulas
(1)
For $j,k\in S_{\mathrm{support}}(N)$, put $K_{jk}=\langle g_j,g_k\rangle$ and $b_k=\langle1,g_k\rangle=(\log k)/k$. Then $K\alpha=b$ and $D_{\mathrm{support}}(N)^2=1-b^TK^{-1}b$. The Gram matrix is positive definite; all logarithms are natural.
(2)
For an $L$-periodic sequence $h$, $\sum_{m=1}^{\infty}h(m)/[m(m+1)]=\frac1L\sum_{r=1}^{L}h(r)\bigl(\psi((r+1)/L)-\psi(r/L)\bigr)$, where $\psi$ is the digamma function. This follows by grouping absolutely convergent terms by their residues modulo $L$. It evaluates $b_k$ with $L=k$ and $K_{jk}$ with $L=\operatorname{lcm}(j,k)$.
(3)
Let $A(\lambda)=\int_0^\infty\{t\}\{\lambda t\}\,dt/t^2$ be the fractional-part autocorrelation and $G_{jk}=A(j/k)/j$. Then $K_{jk}=G_{jk}-G_{j1}/k-G_{1k}/j+G_{11}/(jk)$. The rational evaluations of $A$ are finite cotangent sums [2], Proposition 89.
(4)
For $G=(G_{jk})_{1\leq j,k\leq N}$, $h_k=(\log k+1-\gamma)/k$ and $v_k=1/k$, the unrestricted minimum is $d_N^2=1-h^TG^{-1}h$. Imposing $v^Tc=0$ gives $D_{\mathrm{all}}(N)^2=d_N^2+(v^TG^{-1}h)^2/(v^TG^{-1}v)$. Here $\gamma$ is Euler’s constant. This follows by minimizing the quadratic error subject to the linear constraint.
(5)
$0<D_{\mathrm{support}}(N+1)^2\leq D_{\mathrm{support}}(N)^2<1$ and $D_{\mathrm{all}}(N)^2\leq D_{\mathrm{squarefree}}(N)^2$. If $N\geq3$ is not squarefree then $D_{\mathrm{squarefree}}(N)^2=D_{\mathrm{squarefree}}(N-1)^2$. These follow from the nested approximation spaces; strict positivity at finite $N$ follows because all $g_k$ vanish at every multiple of their common period.
(6)
$D_{\mathrm{all}}(2)^2=D_{\mathrm{squarefree}}(2)^2=1-\log2$: $b_2=(\log2)/2$, $K_{22}=(\log2)/4$, and the minimizing coefficient is $\alpha_2=2$.
Comments
(7)
The parameter $N$ bounds denominators, not the dimension. The zero sequence $g_1(m)=\{m\}=0$ is omitted; the basis $g_k(m)=\{m/k\}$ and weights are unnormalized. Allowing complex coefficients gives the same minimum. The empty-span control is $D_{\mathrm{support}}(1)^2=1$.
(8)
Writing $\rho_k(x)=\{1/(kx)\}$ and $f_k=\rho_k-\rho_1/k$, one has $f_k(x)=g_k(m)$ on $1/(m+1)<x<1/m$, an interval of length $1/[m(m+1)]$, and $f_k(x)=0$ for $x>1$. Thus this is also the squared distance from $\chi=\mathbf1_{(0,1)}$ to the span of these $f_k$ in $L^2((0,\infty),dx)$; see [1], equation (5) and the proof of Theorem 1.
(9)
For all denominators, $D_{\mathrm{all}}(N)$ is Bagchi’s $D(N)$ (p. 146). Convergence of these distances to zero is equivalent to the Riemann hypothesis; the squarefree restriction has the same closure criterion by [1], Remark 8. No asymptotic estimate is inferred from the finite data.
Programs
(P1)
Sage
import numberdb.sage as numberdb
import sage.rings.polynomial.laurent_polynomial_ring
from sage.rings.rational_field import QQ
from sage.rings.integer_ring import ZZ
from sage.rings.real_arb import RealBallField
from sage.rings.complex_arb import ComplexBallField
from sage.matrix.constructor import matrix
from functools import lru_cache
from math import lcm

support, N = "squarefree", 10
R, C = RealBallField(640), ComplexBallField(640)
S = [k for k in range(2, N+1)
     if support == "all" or ZZ(k).is_squarefree()]
@lru_cache(None)
def weights(L):
    p = [C(QQ(r)/L).psi().real() for r in range(1, L+2)]
    return [(p[r]-p[r-1])/L for r in range(1, L+1)]
def inner(j, k):
    return sum((R(QQ((r%j)*(r%k))/(j*k))*w
                for r, w in enumerate(weights(lcm(j,k)), 1)), R(0))
K = matrix(R, [[inner(j,k) for k in S] for j in S])
b = matrix(R, len(S), 1, [R(k).log()/k for k in S])
print(1-(b.transpose()*K.solve_right(b))[0,0])
# Stored selection: support in ("all", "squarefree"), every N=2..30.
References
[1]
B. Bagchi, On Nyman, Beurling and Baez-Duarte’s Hilbert space reformulation of the Riemann hypothesis, Proceedings of the Indian Academy of Sciences (Mathematical Sciences) 116 (2006), 137–146; Theorem 1, Remarks 4 and 8, and p. 146. (arXiv) (doi)
[2]
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)
[3]
L. Báez-Duarte, Arithmetical Aspects of Beurling’s Real Variable Reformulation of the Riemann Hypothesis, 2000, equations (1.14) and (1.17). (arXiv)
Similar tables
Squared Nyman–Beurling approximation distances —   The unrestricted minima use $\rho_1,\ldots,\rho_N$. The zero-tail constraint $\sum c_k/k=0$ changes the finite problem by (4).
Báez-Duarte approximation errors —   The moving-endpoint approximant $B_N$ of [3], equation (1.14), satisfies the zero-tail constraint. Hence $D_{\mathrm{all}}(N)^2\leq\|\chi+B_N\|_2^2$. Its endpoint correction need not have squarefree support.
Vasyunin approximation errors —   The fixed-anchor approximant $V_N$ of [3], equation (1.17), lies in the squarefree span, so $D_{\mathrm{squarefree}}(N)^2\leq\|\chi+V_N\|_2^2$.
Multiplicative fractional-part autocorrelation —   The kernel $A(j/k)/j$ gives the Gram matrix before anchoring.
Bettin–Conrey cotangent sums —   These supply the finite cotangent evaluation of the rational-dilation autocorrelation.
Values of the digamma function $\psi(x)$ at rational numbers —   For a period $L$, the differences $[\psi((r+1)/L)-\psi(r/L)]/L$ sum the weights $1/[m(m+1)]$ over $m\equiv r\pmod L$, as in (2). Applied to $g_k$ and $g_jg_k$, they evaluate the target inner products and Gram matrix directly from the sequence-space definition.
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds both denominator supports for every integer $2\leq N\leq30$, including squarefree-support plateaus. These matched prefixes cover the two sourced arithmetic spaces before extending their depth; thirty is a practical, noncanonical cutoff for certified systems of at most 29 dimensions)
How they were obtained:

Computed with exact rational inputs and Arb real balls at 513 bits, using the finite cotangent formula for the autocorrelation and a real-ball linear solve. Every distance retains at least 461 bits of relative accuracy, with absolute radius less than $7.00\times10^{-142}$. The linear independence of the finite sequence basis makes the Gram matrix positive definite. These enclosures support the 100 significant digits stored; no rounded companion values are inputs.

more

An independent 640-bit check groups the defining sequence sum by residue classes and evaluates the resulting digamma differences. Separate interval $LDL^T$ elimination verifies all 58 entries, 435 distinct Gram entries, 29 target inner products and 45 distinct optimizer systems. All pivots are positive. The independent distances retain at least 565 accuracy bits and have radius less than $5.51\times10^{-173}$. Both calculations lie inside each stored last-place interval. Known-answer, perturbed-value, NaN and excessively wide interval controls are included.

The attached checks also test normal equations, nested-space inequalities, squarefree plateaus, the constrained correction to the unrestricted minimum, both constructive comparison bounds, and four finite residual sums with explicit remainder bounds. The generator, checking programs, Gram matrices, coefficient enclosures, argument selection and numerical evidence are attached. These checks do not establish any asymptotic assertion.



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