A fixed-probe second-order correction for the anchored Selberg–Möbius approximant

Problem and contribution

Let

$$ \mathcal H=L^2((1,\infty),dy/y^2) $$

and, for an integer $n\geq 1$, put

$$ \rho_n(y)=\left\lfloor\frac yn\right\rfloor-\frac{\lfloor y\rfloor}{n}. $$

This is the anchored image of the usual Beurling fractional-part function under $x=1/y$. In particular, $\rho_1=0$, and the anchor is part of the definition. Write $L=\log N$ and

$$ g_N=\sum_{n\leq N}\mu(n)\left(1-\frac{\log n}{L}\right)\rho_n. $$

The logarithmic Möbius coefficients are established Selberg weights; their use is prior art [1,2,3]. The result here is a fixed-coordinate expansion for the complete anchored inner product, including its infinite physical tail.

Theorem

For every fixed integer $m\geq2$ and every fixed $B>0$,

$$ \left\langle 1-g_N,\rho_m\right\rangle =\frac{C_m}{\log N}+O_{m,B}((\log N)^{-B}), $$

where

$$ C_m=\int_1^\infty \bigl(\lfloor y\rfloor-\psi(\lfloor y\rfloor)\bigr) \rho_m(y)\,\frac{dy}{y^2}. $$

Here $\psi(x)=\sum_{k\leq x}\Lambda(k)$ is the Chebyshev function. The integral is absolutely convergent as an $L^1$ pairing. The same constant has the absolutely convergent representation

$$ C_m=T_m(1)-2\gamma a_m-\sum_{k\geq1}\Lambda(k)K_m(k), $$

where

$$ a_m=\frac{m-1}{2m},\qquad f_m(y)=\rho_m(y)+a_m,\qquad K_m(t)=\int_t^\infty f_m(y)\frac{dy}{y^2}, \qquad T_m(n)=\sum_{j\geq1}K_m(jn). $$

As an unconditional consequence, the prescribed, $N$-independent correction

$$ g_N^\star=g_N-\frac{\rho_2}{\log N} $$

strictly decreases the complete squared error for all sufficiently large $N$:

$$ \|1-g_N^\star\|_{\mathcal H}^2-\|1-g_N\|_{\mathcal H}^2 =\frac{\log 2/4+2C_2}{\log^2N}+o(\log^{-2}N), $$

and

$$ \frac{\log 2}{4}+2C_2 \leq \frac{9\log2-2}{8}-\gamma<-0.0474. $$

This is a second-order directional improvement of a specified approximant. It is not a bound on the original approximation error and does not imply the Riemann hypothesis.

Related work and distinction

Báez-Duarte records the same Selberg/Cesàro coefficient profile and its role in the natural Nyman–Beurling approximation [1]. Conrey–Myerson study the corresponding weighted fractional-part sums and cite this baseline [2]. Bettin–Conrey–Farmer use the same Dirichlet polynomial in a conditional leading-norm theorem [3]. Ehm gives exact unanchored Gram kernels and related Möbius moment limits [4]. Those works do not state the anchored fixed-probe expansion above, its full-tail divisor calculation, or the explicit signed $\rho_2$ descent in this form. This comparison does not establish priority; the baseline and its anchoring are claimed as prior art.

The distinction between anchored and unanchored functions matters. If $\{\,\cdot\,\}$ denotes fractional part, then

$$ \rho_n(1/x)=\left\{\frac1{nx}\right\}-\frac1n\left\{\frac1x\right\}. $$

Thus the calculation below keeps the scalar anchor that is absent from an unanchored Gram entry.

Proof of the fixed-probe expansion

The function $f_m$ is $m$-periodic and has mean zero. Its primitive is bounded, so

$$ K_m(t)=O_m(t^{-2}),\qquad t\geq1. $$

For an integer $t$, summation by parts over the intervals $[j,j+1)$ gives the exact formula

$$ K_m(t)=\frac{f_m(t)}t+ \frac{H_t-H_{\lfloor t/m\rfloor}-\log m}{m}, \tag{1} $$

where $H_0=0$. Indeed,

$$ f_m(j)-f_m(j-1)={\bf1}_{m\mid j}-\frac1m, $$

and the finite sum of these differences tends to $(H_t-H_{\lfloor t/m\rfloor}-\log m)/m$. Formula (1) also makes the $O_m(t^{-2})$ cancellation explicit.

Let

$$ G(n,m)=\langle\rho_n,\rho_m\rangle_{\mathcal H}. $$

Since $\rho_m=f_m-a_m$ and

$$ \int_1^\infty\rho_n(y)\frac{dy}{y^2}=-\frac{\log n}{n}, \tag{2} $$

it remains to calculate the centered part. For an integer $Y$,

$$ \begin{aligned} \int_1^Y\left\lfloor\frac yn\right\rfloor f_m(y)\frac{dy}{y^2} &=\sum_{j\leq Y/n}\int_{jn}^Y f_m(y)\frac{dy}{y^2}\\ &=\sum_{j\leq Y/n}K_m(jn)-\left\lfloor\frac Yn\right\rfloor K_m(Y). \end{aligned} \tag{3} $$

The last term is $O_m((nY)^{-1})$. Subtracting $n^{-1}$ times the same formula with $n=1$, then taking $Y\to\infty$, proves

$$ G(n,m)=\frac{a_m\log n}{n}+T_m(n)-\frac{T_m(1)}n. \tag{4} $$

The same computation at $n=1$ gives

$$ \langle1,\rho_m\rangle=-\frac{\log m}{m}. \tag{5} $$

The $K_m$ decay makes all divisor regroupings below absolute. For $r=0,1$,

$$ \sum_{n,j\geq1}|\mu(n)|(\log n)^r|K_m(jn)| \ll_m\sum_{k\geq1}\frac{\tau(k)(1+\log k)}{k^2}<\infty. \tag{6} $$

Consequently, using $\sum_{d\mid k}\mu(d)={\bf1}_{k=1}$ and

$$ \sum_{d\mid k}\mu(d)\log d=-\Lambda(k), \tag{7} $$

we obtain

$$ \sum_{n\geq1}\mu(n)T_m(n)=K_m(1), \qquad \sum_{n\geq1}\mu(n)\log n\,T_m(n) =-\sum_{k\geq1}\Lambda(k)K_m(k). \tag{8} $$

We use the standard consequence of the classical zero-free region

$$ M_\mu(x)=\sum_{n\leq x}\mu(n) \ll x\exp(-c\sqrt{\log x}). \tag{9} $$

Partial summation shows that for each fixed $r,A\geq0$,

$$ \sum_{n\leq N}\frac{\mu(n)(\log n)^r}{n} =\ell_r+O_{r,A}((\log N)^{-A}). \tag{10} $$

The constants needed here are

$$ \ell_0=0,\qquad \ell_1=-1,\qquad \ell_2=-2\gamma. \tag{11} $$

After (9) supplies ordinary convergence, their values follow by Abel's theorem from

$$ \frac1{\zeta(1+z)}=z-\gamma z^2+O(z^3). \tag{12} $$

The tails of $T_m(n)$ and $\log n\,T_m(n)$ are $O_m(N^{-1})$ and $O_m(\log N/N)$, respectively. Inserting (4), (8), (10), and (11) therefore gives

$$ \sum_{n\leq N}\mu(n)G(n,m) =\langle1,\rho_m\rangle+O_{m,A}((\log N)^{-A}) \tag{13} $$

and

$$ \sum_{n\leq N}\mu(n)\log n\,G(n,m) =T_m(1)-2\gamma a_m-\sum_{k\geq1}\Lambda(k)K_m(k) +O_{m,A}((\log N)^{-A}). \tag{14} $$

Since

$$ \langle1-g_N,\rho_m\rangle =\langle1,\rho_m\rangle -\sum_{n\leq N}\mu(n)G(n,m) +\frac1{\log N}\sum_{n\leq N}\mu(n)\log n\,G(n,m), $$

(5), (13), and (14) prove the theorem's first formula.

Identification with the prime-counting discrepancy

Put

$$ h(y)=\lfloor y\rfloor-\psi(\lfloor y\rfloor). $$

The classical PNT remainder implies $\int_1^\infty |h(y)|\,dy/y^2<\infty$. It does not imply that $h$ belongs to $\mathcal H$, and no such assertion is used. On each cell,

$$ h(y)=\sum_{k\leq y}(1-\Lambda(k)). $$

Using $K_m(k)=O_m(k^{-2})$ and summing first over the cells gives

$$ \int_1^\infty h(y)\rho_m(y)\frac{dy}{y^2} =-a_m\int_1^\infty h(y)\frac{dy}{y^2} +\sum_{k\geq1}(1-\Lambda(k))K_m(k). \tag{15} $$

The first integral equals $2\gamma$. To see this, its finite cell sum is

$$ \sum_{k\leq X}(k-\psi(k))\left(\frac1k-\frac1{k+1}\right) =\sum_{k\leq X}\frac{1-\Lambda(k)}k-\frac{X-\psi(X)}{X+1}. $$

The PNT form

$$ \sum_{k\leq X}\frac{\Lambda(k)}k=\log X-\gamma+o(1) $$

and $H_X=\log X+\gamma+o(1)$ give the limit $2\gamma$. Equation (15) is exactly the prime-weighted expression for $C_m$.

The explicit $m=2$ descent

For $m=2$,

$$ a_2=\frac14,\qquad f_2(k)=\frac{(-1)^k}{4}, $$

so

$$ K_2(k)=\frac14\sum_{j\geq k}\frac{(-1)^j}{j(j+1)}, \qquad |K_2(k)|\leq\frac1{4k(k+1)}. \tag{16} $$

Absolute summation gives

$$ T_2(1)=\frac14\sum_{k\geq1}\frac{(-1)^k}{k+1} =\frac{\log2-1}{4}. \tag{17} $$

Directly summing the odd cells gives

$$ G(2,2)=\frac14\sum_{\substack{k\geq1\\k\ {\rm odd}}} \left(\frac1k-\frac1{k+1}\right)=\frac{\log2}{4}. \tag{18} $$

Because $k\mapsto\log k/k^2$ decreases for $k\geq2$,

$$ \left|\sum_{k\geq1}\Lambda(k)K_2(k)\right| \leq\frac14\sum_{k\geq2}\frac{\log k}{k^2} \leq\frac{3\log2+2}{16}. \tag{19} $$

Using (17) in the expression for $C_2$ gives

$$ C_2\leq\frac{7\log2-2}{16}-\frac{\gamma}{2}<-0.1103. \tag{20} $$

If $e_N=1-g_N$, then

$$ \|e_N+\rho_2/L\|^2-\|e_N\|^2 =\frac{G(2,2)+2C_2}{L^2}+o(L^{-2}). $$

Equations (18) and (20) give

$$ G(2,2)+2C_2 \leq\frac{9\log2-2}{8}-\gamma<-0.0474, $$

which proves the stated strict descent.

For reference, an independent finite computation through $10^6$ cells gives $C_2=-0.35066058$ from the prime-weighted sum and $-0.35065804$ from the direct discrepancy pairing. An Arb enclosure through prime powers $16384$, with the explicit tail in (19), gives $C_2\approx-0.35067$ and certifies the stronger prescribed correction $-2\rho_2/\log N$, whose asymptotic change coefficient is approximately $-0.7096$. These numerical values are supporting evidence; the strict $-\rho_2/\log N$ result uses only (19)–(20).

Limitations and scope

The theorem is a statement about a fixed finite-dimensional direction around one canonical approximant. It gives no upper bound for $\|1-g_N\|$, no convergence of $g_N$, no improved leading constant for the conditional theorem of [3], and no information that excludes zeros off the critical line. It also does not claim that the fixed-probe expansion is absent from earlier literature. Its value is the explicit signed full-tail calculation and the resulting prescribed, coefficient consistent descent.

Reproducibility

The supplement contains an independent double-precision audit of the $m=2$ formulas and a standalone Arb certificate for the prime-power tail. The audit uses a least-prime-power sieve and the exact cell formula (16). Running the scripts from the supplement reports the finite pairing, the prime-weighted pairing, the elementary sign bound, and the interval for the stronger rational correction. No GPU or external data is needed.

References

[1] L. Báez-Duarte, “A strengthening of the Nyman–Beurling criterion for the Riemann hypothesis, 2,” 2002. https://arxiv.org/abs/math/0205003

[2] J. B. Conrey and G. Myerson, “On the Balazard–Saias criterion for the Riemann hypothesis,” 2000. https://arxiv.org/abs/math/0002254

[3] S. Bettin, J. B. Conrey, and D. W. Farmer, “An optimal choice of Dirichlet polynomials for the Nyman–Beurling criterion,” 2012. https://arxiv.org/abs/1211.5191

[4] W. Ehm, “On certain Gram matrices and their associated series,” 2024. https://arxiv.org/abs/2405.06349

[5] H. Davenport, Multiplicative Number Theory, 3rd ed., revised by H. L. Montgomery, Springer, 2000. The zero-free-region form of the prime number theorem used in (9) is standard.