bibkey: ehm2024grammatrices
authors: Werner Ehm
year: 2024
title: On certain Gram matrices and their associated series
doi: null
url: https://arxiv.org/abs/2405.06349v2
claim: "Theorem 2.1 and Proposition 3.1 give full-space fractional-part Gram kernels; Section 8 separates moment products from a signed inversion error, and Proposition 8.1 centers moments by modifying the Selberg coefficients. The known modification has a paid comparison with the actual balanced residual, but no complete residual estimate is supplied."
strata_touched: []
license: citation-only
triage: anchor

Fractional-part Gram kernels and known moment centering

The pinned source is arXiv:2405.06349v2, revised 13 May 2024. The introductory approximation claims, the selected kernel statements and the argument of Proposition 8.1 were inspected. No independent whole-paper proof audit, Lean certification or originality claim is made. The citation records the author version without asserting a journal DOI.

Optimized coefficients and the fixed-array claim

The Introduction defines $d_q^2(N;a)$ for $q=1,2$. Its Báez-Duarte criterion optimizes over all real coefficient arrays:

$$\mathrm{RH}⟺\underset{N\to \infty }{\mathrm{liminf}}\underset{a\in {\mathbb{R}}^N}{\inf }{d}_q^2(N;a)=0.$$

Fix instead ${\lambda }_{n,N}=\mu (n)(1-\log n/\log N)$ and write $Q_{\lambda,N}^{(q)}=\sum_{m,n\le N}\lambda_{m,N}\lambda_{n,N} G_{m,n}^{(q)}$. The square expansion and Proposition 1.1 give

$$d_q^2(N;\lambda )=q-2\sum _{n\le N}{\lambda }_{n,N}{F}_n^{(q)}+Q_{\lambda ,N}^{(q)},\sum _{n\le N}{\lambda }_{n,N}{F}_n^{(q)}⟶q.$$

Consequently $Q_{\lambda ,N}^{(q)}\to q$ is equivalent to $d_q^2(N;\lambda )\to 0$. The optimized criterion then implies RH. The sentence following Proposition 1.1 also states the reverse implication from RH. The displayed optimization and the mixed-term limit do not supply that reverse implication for the fixed array. This note therefore does not use it as an RH-only supplier. This applicability limitation does not establish that the reverse claim is false.

The preceding paragraph of the source explicitly retains extra hypotheses when citing the fixed-array result of Bettin–Conrey–Farmer, arXiv:1211.5191v1, Theorem 1. For $q=1$ that theorem assumes RH and, for some fixed $\delta >0$,

$$\sum _{|\Im \rho |\le T}|{\zeta }^{\prime }(\rho ){|}^{-2}\ll T^{3/2-\delta }.$$

The derivative condition includes simplicity of the zeros. With these complete assumptions the theorem supplies the known $1/\log N$ approximation asymptotic. Neither that conditional result nor the mixed-term limit removes the fixed-array convergence condition in the FIB source, Section 391.4.

Multiplicity calibration for the complete $q=1$ residual

The distinction has a concrete paper-level consequence, obtained by directly applying (I11), (I15) and (I16) of the existing Selberg analysis. Define on the half-line

$${\mathcal{F}}_N(x)=1_{(0,1)}(x)+\sum _{n\le N}{\lambda }_{n,N}\{1/(nx)\},{\epsilon }_N=\Vert {\mathcal{F}}_N{\Vert }_{L^2(0,\infty )}.$$

Mellin–Plancherel identifies ${\epsilon }_N^2=d_1^2(N;\lambda )$. If ${\epsilon }_N\to 0$ over all integer $N$, the known natural-dilation criterion gives RH. In fact this convergence also forces every nontrivial zero to be simple, as follows.

Retain $S_y(t)=\log y+{\sum }_{n\le y}\mu (n)\log (y/n)\{t/n\}$. For $N=\lfloor y\rfloor$, (I11) gives $|S_y(t)-S_N(t)|\le 2$ uniformly in $t$. On $J=(5/4,7/4)$, $S_N(yq)=\log N{\mathcal{F}}_N(1/(yq))$ for large $y$. The substitution $x=1/(yq)$ gives

$$\Vert S_N(y\cdot ){\Vert }_{L^2(J)}\le \frac{7}{4}\sqrt{y}\log N{\epsilon }_N,\Vert S_y(y\cdot ){\Vert }_{L^2(J)}=o(\sqrt{y}\log y).$$

This passes from all integer cutoffs to all real $y$, without replacing the original residual by a sampled step residual. Fix a nontrivial zero $\rho$, now on $\Re \rho =1/2$. Equation (I16) provides $h\in C_c^{\infty }(J)$ with $H_h(\rho )\ne 0$. Set $A_h(y)={\int }_{J}h(q)S_y(yq)dq$. Then $A_h(y)=o(\sqrt{y}\log y)$, and (I15) identifies

$${\mathcal{M}}_h(s)={\int }_1^{\infty }{A}_h(y)y^{-s-1}dy=\frac{{\int }_{J}h(q)dq}{s^2}+\frac{H_h(s)}{\zeta (s)}(\Re s>1/2).$$

For real $u\downarrow 0$, the little-oh bound gives $u^2{\mathcal{M}}_h(\rho +u)\to 0$. Indeed, for every $\eta >0$ choose $Y$ so that $|A_h(y)|\le \eta \sqrt{y}\log y$ for $y\ge Y$. The finite initial integral vanishes after multiplication by $u^2$, while the tail is bounded by

$$\eta u^2{\int }_1^{\infty }\log y{y}^{-u-1}dy=\eta .$$

The meromorphic right side has a genuine pole of order $m_{\rho }$ at $\rho$, since $H_h(\rho )\ne 0$ and $\rho \ne 0$. Its radial asymptotic is $c{u}^{-m_{\rho }}$ with $c\ne 0$. The little-oh limit excludes $m_{\rho }\ge 2$. Thus the complete fixed-array convergence implies RH together with zero simplicity.

This is a direct application of the existing cutoff and Mellin identities, with an explicit little-oh integral estimate; it is not a new convergence supplier or a Lean-certified statement. No converse from RH, convergence along a subsequence, $q=2$ simplicity conclusion or derivative-moment bound is asserted. A further simplicity consequence alone does not prove that this convergence condition is strictly stronger than RH.

Existing kernels and their remaining signed term

For $u,v>0$, Theorem 2.1, equation (9), states

$$G_{u,v}^{(1)}=\frac{1}{v}\left(K+\frac{1}{2}\log \frac{v}{u}\right)+\frac{1}{u}{S}_1(v/u),K=\frac{\log (2\pi )-\gamma +1}{2},$$

where $S_1(x)={\sum }_{k\ge 1}{R}_1(kx)$ and $R_1(x)=\log x+\gamma -H(x)-(\{x\}-1/2)/x$. Here $H(x)={\sum }_{n\le x}1/n$. Proposition 2.1 supplies the decay $R_1(x)=O(x^{-2})$, and hence $S_1(x)=O(x^{-2})$, at infinity. The paper explicitly identifies the $q=1$ kernel with Báez-Duarte–Balazard–Landreau–Saias, Proposition 90, rather than claiming this underlying kernel as new.

Proposition 3.1, equation (18), gives

$$G_{u,v}^{(1)}=\frac{1}{uv}{\int }_0^{\infty }\{tu\}\{tv\}\frac{dt}{t^2}={\int }_0^{\infty }\{t/u\}\{t/v\}\frac{dt}{t^2}.$$

The second equality is the substitution by $uv$ and symmetry of the two factors. Thus $u=n,v=k$ matches the full Gram kernel for the fractional-part probes in the actual Selberg residual. It is not the truncated, weighted step kernel on $N\le m\le N^2$.

Section 8.1, equation (50), already decomposes the full quadratic form. For arbitrary finite real coefficients $a_n$, write $L_j={\sum }_{n\le N}{a}_n{\log }^j(n)/n$ and $M_j={\sum }_{n\le N}{a}_n{\log }^j(n)$. Then

$$\begin{array}{rl}{Q}_{a,N}^{(1)}& =M_0\left(K{L}_0+\frac{L_1+1}{2}\right)-\frac{M_1{L}_0}{2}+(\gamma -1)L_0-L_1+{\mathfrak{E}}_{a,N}^{(1)},\\ {\mathfrak{E}}_{a,N}^{(1)}& =\sum _{m\le N}\frac{a_m}{m}\left(\sum _{n\le N}{a}_n{S}_1(n/m)-R_1(1/m)\right).\end{array}$$

The paper calls estimating this inversion error a major challenge and sets it aside in that discussion. Convergence of the centered $L_j$ factors alone does not control their products with $M_j$; the source states this limitation explicitly. The inversion error is signed, and is not the project’s nonnegative local energy $E_N$. These formulas are existing decompositions, not a supplied upper bound for the complete quadratic residual.

The source’s coefficient modification

Section 8.2, equation (53), uses the known coefficients

$${\nu }_{n,N}=\mu (n)\left(1-\frac{\log n}{L}-\frac{A\log n+B}{nL}\right)-\frac{C}{nL},L=\log N.$$

Proposition 8.1 is conditional on fixed real $A,B,C$ solving the specified system. With ${\eta }_j=(-1{)}^j(1/\zeta {)}^{(j)}(2)$, that system is

$$\begin{array}{rl}A{\eta }_1+B{\eta }_0+C\zeta (2)& =1,\\ A{\eta }_2+B{\eta }_1-C{\zeta }^{\prime }(2)& =2\gamma ,\\ A{\eta }_3+B{\eta }_2+C{\zeta }^{\prime \prime }(2)& =6({\gamma }^2+{\gamma }_1).\end{array}$$

Here ${\gamma }_1$ is the Stieltjes constant. Under this condition, equations (54)–(55) match the modified centered moments with the source’s differences of rescaled Möbius moments, up to $O(N^{-1}{\log }^{j}N)$ and $O({\log }^{j}N)$, respectively, for $j=0,1,2$. Each source difference has the form $\Delta X_j=X_j-X_{j+1}/L$; the $L$ moments are centered at their stated limits before taking this difference. This moment matching does not assert that the complete-period Fourier mean $d_N$ vanishes. The construction and its centering property are reused from the paper. They do not bound its inversion error or the complete approximation norm.

Paid comparison with the balanced residual

There is a simple application to the exact balanced dictionary of the Selberg note, without using a full-space Gram formula as a truncated kernel. For any fixed real $A,B,C$, let $a_{N,n}=\mu (n)\log (N/n)/L$ as there, and put

$${\Delta }_{N,n}={\nu }_{n,N}-a_{N,n}=-\frac{\mu (n)(A\log n+B)+C}{nL}.$$

This comparison does not require solving the centering system. On the same dictionary $b_n=\{1/(nx)\}-\{1/x\}/n$, let $v_N=-{\sum }_{2\le n\le N}{a}_{N,n}{b}_n$ and ${\tilde{v}}_N=-{\sum }_{2\le n\le N}{\nu }_{n,N}{b}_n$. For $m\ge N$ their step residuals differ by ${\sum }_{n\le N}{\Delta }_{N,n}\{m/n\}$. Consequently

$$\underset{m\ge N}{\sup }|{\tilde{R}}_N(m)-R_N(m)|\le \frac{1}{L}\sum _{n\le N}\frac{|A|\log n+|B|+|C|}{n}{\ll }_{A,B,C}L(N\ge 3).$$

Use the actual telescoping weight ${\sum }_{m=N}^{N^2}1/[m(m+1)]\le 1/N$. If ${\tilde{E}}_N$ is the same local energy with the modified coefficients, then

$$\left|\sqrt{{\tilde{E}}_N}-\sqrt{E_N}\right|{\ll }_{A,B,C}\frac{\log N}{\sqrt{N}}.$$

The global comparison is also paid. For ${\rho }_n(x)=\{1/(nx)\}$ on the half-line, scaling gives $\Vert {\rho }_n{\Vert }_2=n^{-1/2}\Vert {\rho }_1{\Vert }_2$, and $\Vert b_n{\Vert }_2\le 2n^{-1/2}\Vert {\rho }_1{\Vert }_2$. Hence

$$\Vert {\tilde{v}}_N-v_N{\Vert }_{L^2(0,1)}\le \frac{2\Vert {\rho }_1{\Vert }_2}{L}\sum _{n\ge 2}\frac{|A|\log n+|B|+|C|}{n^{3/2}}{\ll }_{A,B,C}\frac{1}{\log N}.$$

Thus a genuinely supplied complete bound for the modified family can be compared with the original one, with these independent allowances. The allowances themselves supply no such bound. These are paper applications of elementary norm and harmonic estimates, not new supplier theorems or Lean-certified statements.

The modified family is not silently substituted into the original Fourier formulas. For example, when $C\ne 0$ the additional $-C/(nL)$ term can give non-squarefree denominators nonzero aliases. The general alias formula still applies, but the original squarefree support and short logarithmic Möbius formula must be recalculated. Nor do these norm comparisons supply a same-integer signed Robin budget, a FIB approximation rate or a proof of RH.