bibkey: romik2021orthogonal authors: Dan Romik year: 2021 title: Orthogonal polynomial expansions for the Riemann xi function in the Hermite, Meixner–Pollaczek, and continuous Hahn bases doi: null url: https://doi.org/10.4064/aa200515-10-3 claim: Equations (1.6)–(1.11) define the original theta differential weight and its logarithmic kernel and give the Mellin and all-complex Fourier representations of the Riemann xi function. strata_touched:

• D5/S3/Analytic/Fourier/ThetaDifferentialKernel
• D5/S3/Analytic/Fourier/XiThetaTransform license: citation-only triage: anchor

The original theta kernel and xi

On page 2 of the Online First text, equations (1.6)–(1.9) use theta(t) = 1 + 2 sum(n>=1) exp(-pi*n^2*t), omega(t) = sum(n>=1) (2*pi^2*n^4*t^2 - 3*pi*n^2*t)*exp(-pi*n^2*t), and Phi(x) = 2*exp(x/2)*omega(exp(2*x)). Equation (1.9) explicitly states Phi(-x) = Phi(x) for every real x. No absolute value occurs in the definition (1.8). Equation (1.6) also gives the positive-index theta-tail identity; separating the two summable weights in (1.7) gives the weighted-series formula used in the repository.

Page 1, equation (1.3), defines Xi(z) = xi(1/2 + i*z) for z in C. Page 3, equations (1.10) and (1.11), states the Mellin representation of xi and Xi(z) = integral_R Phi(x)*exp(i*z*x) dx, using that all-complex convention, without an additional prefactor. These are published statements; their new Lean proofs do not change their literature provenance. The repository derives the differential identity Phi = psi'' - psi/4, where psi(x)=exp(x/2)*(theta(exp(2*x))-1)/2, and identifies this original Phi with its fixed even theta kernel. These are repository object identifications, not claims that the paper uses repository names or the repository’s absolute-value definition.

The paper also treats the physicists’ Hermite, Meixner–Pollaczek and continuous Hahn expansions. Their coefficient integrals, polynomial normalizations and compact convergence estimates are separate obligations; the theta transform alone does not establish those expansions or Cardon’s equivalence involving simple zeros and its specific measure and orthogonal polynomials.

Reusable theta tail and its local scale

Lemma 2.3, printed p.10, equations (2.8)–(2.9), directly supplies

$$\Phi (r)=O\left(e^{9r/2-\pi e^{2r}}\right),$$

$$\Phi (r)-2(2{\pi }^2{e}^{9r/2}-3\pi e^{5r/2})e^{-\pi e^{2r}}=O\left(e^{9r/2-4\pi e^{2r}}\right)(r\to +\infty ).$$

Thus, with $g(r)=e^{9r/2-\pi e^{2r}}$, positivity and continuity on the remaining compact interval give $cg(r)\le \Phi (r)\le Cg(r)$ for $r\ge 1$. For ${\delta }_R=e^{-2R}$ the identity

$$\log \frac{g(R+s)}{g(R)}=\frac{9}{2}s-\pi e^{2R}(e^{2s}-1)$$

gives $\Phi (R+s)\asymp \Phi (R)$ uniformly for $-{\delta }_R\le s\le 2{\delta }_R$. This is comparison by constants, not a ratio tending to one.

A derivative estimate must use the normally convergent original series (1.8), rather than differentiate the displayed remainder. Its differentiated summands are bounded by $C{e}^{13r/2}{n}^6{e}^{-\pi n^2{e}^{2r}}$ for $r\ge 1$. The uniformly summable series after removing its first exponential gives $|{\Phi }^{\prime }(r)|\le C{e}^{2r}g(r)$, hence $|{\Phi }^{\prime }(r)/\Phi (r)|\le C{e}^{2r}$. These are paper-level applications of the source series and Lemma 2.3, used by the Gamma tail and same-test compensation estimates. They are not additional tail theorems attributed to Romik or new Lean proofs.

Explicit original-series majorants

The normally convergent source series also supplies coarse constants for the prime-diagonal error transfer and the full mixed-operator cutoff estimates. These are analytic upper bounds, not fitted numerical values or new theta representation theorems.

For $x\ge 0$, put $u=e^{2x}\ge 1$ and $v_n=\pi n^{2}u$. Differentiating the original series gives

$$\begin{array}{rl}\Phi (x)& =2u^{1/4}\sum _{n\ge 1}(2v_n^2-3v_n)e^{-v_n},\\ {\Phi }^{\prime }(x)& =u^{1/4}\sum _{n\ge 1}(-8v_n^3+30v_n^2-15v_n)e^{-v_n}.\end{array}\text{\hspace{1em}(TS)}$$

For $H=\Phi /2+|{\Phi }^{\prime }|$, positivity of $\Phi$ and the triangle inequality therefore give

$$H(x)\le u^{1/4}\sum _{n\ge 1}(8v_n^3+32v_n^2+15v_n)e^{-v_n}.\text{\hspace{1em}(HM)}$$

Use $u^{1/4}\le u=v_n/(\pi n^2)$, $\pi >3$, ${\sum }_{n\ge 1}{n}^{-2}\le 2$, and the standard maximum $v^k{e}^{-v}\le k^k{e}^{-k}$. The elementary lower bounds $e^4>54$, $e^3>20$ and $e^2>7$ yield

$$\begin{array}{rl}\Vert \Phi {\Vert }_{\infty }& \le \frac{8}{3}\frac{27}{e^3}<\frac{18}{5},\\ H_{\infty }& \le \frac{2}{3}\left(\frac{1024}{27}+\frac{216}{5}+\frac{60}{7}\right)<60.\end{array}\text{\hspace{1em}(SC)}$$

Evenness extends these bounds to the whole real line. For the weighted integral, use evenness of $H$, $\cosh (x/2)\le e^{x/2}$ for $x\ge 0$, and $u^{1/2}\le u$. Substitution $dx=du/(2u)$ gives

$$\begin{array}{rl}{M}_H:=\int H(x)e^{x/2}dx& =2{\int }_0^{\infty }H(x)\cosh (x/2)dx\\ & \le \sum _{n\ge 1}\frac{1}{\pi n^2}{\int }_0^{\infty }(8v^3+32v^2+15v)e^{-v}dv\\ & \le \frac{254}{3}<85.\end{array}\text{\hspace{1em}(MC)}$$

The polynomial integral is $8\cdot 6+32\cdot 2+15=127$. Its extension from $v\ge \pi n^2$ to $v\ge 0$ only increases the bound.

For a spatial tail majorant, split the exponential in (TS) and use $v^3{e}^{-v/2}\le 216e^{-3}<216/20$. Since $v_n\ge 3u$,

$$\Phi (x)\le \frac{144}{5}\exp \left(-\frac{3}{2}{e}^{2|x|}\right).\text{\hspace{1em}(ST)}$$

Consequently, for $r\ge 0$,

$$T(r):={\int }_{|x|>r}\Phi (x)dx\le \frac{96}{5}{e}^{-2r}\exp \left(-\frac{3}{2}{e}^{2r}\right).\text{\hspace{1em}(IT)}$$

Indeed, the substitution $u=e^{2x}$ on each even tail gives $(144/5){\int }_{e^{2r}}^{\infty }{e}^{-3u/2}du/u$; bound $1/u$ by $e^{-2r}$ and integrate the remaining exponential. All constants are deliberately loose and keep the original kernel and normalization. These model applications have no new Lean certification or originality claim.

Weighted Fourier coefficient suppliers

For the weighted Fourier cutoff, define on the whole real line

$$s(x)=\sqrt{\frac{\Phi (x)}{2\cosh (x/2)}}.$$

The existing strictly positive smooth even original kernel makes $s$ smooth and even, including at zero. The following original-series estimates supply $s\in H^3(\mathbb{R})$, bounded $s,s^{\prime }$, and the derivative summability of the complete prime graph. They are model deductions from the source series, with no useful numerical constant sizes or new Lean certification asserted.

For $x\ge 0$, $u=e^{2x}\ge 1$, write

$$\Phi (x)=\sum _{n\ge 1}(4{\pi }^2{n}^4{e}^{9x/2}-6\pi n^2{e}^{5x/2})e^{-\pi n^{2}u}.$$

Every summand is positive. Its first summand gives $\Phi (x)>18e^{5x/2}{e}^{-\pi u}$, using $\pi >3$. Define derivative polynomials by

$$P_{0,a}(z)=1,P_{j+1,a}(z)=(a-2z)P_{j,a}(z)+2z{P}_{j,a}^{\prime }(z),P_{j,a}(z)=\sum _{k=0}^j{p}_{j,a,k}{z}^k.$$

Direct differentiation gives $\partial_x^j(e^{ax}e^{-\pi n^2u}) =e^{ax}e^{-\pi n^2u}P_{j,a}(\pi n^2u)$. For $0\le j\le 3$ let

$$C_j=\sum _{n\ge 1}{e}^{-\pi (n^2-1)}\left(4{\pi }^2{n}^4\sum _{k=0}^j|p_{j,9/2,k}|{\pi }^k{n}^{2k}+6\pi n^2\sum _{k=0}^j|p_{j,5/2,k}|{\pi }^k{n}^{2k}\right).$$

Normal convergence of the original series permits termwise differentiation. Using $u^k\le u^j$ and $e^{-\pi (n^2-1)u}\le e^{-\pi (n^2-1)}$ gives

$$|{\Phi }^{(j)}(x)|\le C_j{e}^{(9/2+2j)x}{e}^{-\pi u},\frac{|{\Phi }^{(j)}(x)|}{\Phi (x)}\le {\gamma }_j{e}^{(2+2j)x},{\gamma }_j=C_j/18.\text{\hspace{1em}(WC1)}$$

These constants have explicit tail majorants. For $q\le 10$, $n\ge 2$, the ratio of successive terms $a_n=n^q{e}^{-\pi (n^2-1)}$ is at most $r_{\ast }=(3/2{)}^{10}{e}^{-5\pi }<1$. Every monomial in $C_j$ has degree at most ten, so its tail after $N\ge 1$ is at most $a_{N+1}/(1-r_{\ast })$, with its displayed coefficient retained.

Put $d=2\cosh (x/2)$ and $\ell =\log s$. The first three derivatives of $\log d$ have absolute bounds $1/2,1/4,1/4$. Thus

$$|{\ell }^{\prime }|\le A_1{e}^{4x},|{\ell }^{\prime \prime }|\le A_2{e}^{8x},|{\ell }^{\prime \prime \prime }|\le A_3{e}^{12x},$$

where

$$\begin{array}{rl}{A}_1& ={\gamma }_1/2+1/4,\\ A_2& =({\gamma }_2+{\gamma }_1^2)/2+1/8,\\ A_3& =({\gamma }_3+3{\gamma }_1{\gamma }_2+2{\gamma }_1^3)/2+1/8.\end{array}$$

Since $d\ge e^{x/2}$, (WC1) gives $s\le \sqrt{C_0}u{e}^{-\pi u/2}$. Apply $s^{\prime }=s{\ell }^{\prime }$, $s^{\prime \prime }=s(({\ell }^{\prime }{)}^2+{\ell }^{\prime \prime })$, and $s^{\prime \prime \prime }=s(({\ell }^{\prime }{)}^3+3{\ell }^{\prime }{\ell }^{\prime \prime }+{\ell }^{\prime \prime \prime })$ to obtain

$$|s^{(j)}(x)|\le \sqrt{C_0}{B}_j{u}^{1+2j}{e}^{-\pi u/2},(B_0,B_1,B_2,B_3)=(1,A_1,A_1^2+A_2,A_1^3+3A_1{A}_2+A_3).$$

For $c=3/8$, $\zeta =\pi /2-c>0$, define $v_m=\max \{1,m/\zeta \}$, $M_m=v_m^m{e}^{-\zeta v_m}$ and $K_j=\sqrt{C_0}{B}_j{M}_{1+2j}$. Maximizing the remaining scalar factor, and then using evenness, yields

$$|s^{(j)}(x)|\le K_j{e}^{-c{e}^{2|x|}}(x\in \mathbb{R},0\le j\le 3).\text{\hspace{1em}(WC2)}$$

Substitution on the two tails gives, with the unitary Fourier convention,

$$\begin{array}{rl}\Vert s^{(j)}{\Vert }_2^2& \le K_j^2\frac{e^{-2c}}{2c},\\ \Vert s{\Vert }_{H^3}^2:=\int (1+{\xi }^2{)}^3|\hat{s}(\xi ){|}^{2}d\xi & \le \frac{e^{-2c}}{2c}\sum _{j=0}^3\left(\genfrac{}{}{0ex}{}{3}{j}\right)K_j^2,\\ \Vert (s^2{)}^{\prime }{\Vert }_{\infty }& \le 2K_0{K}_1.\end{array}\text{\hspace{1em}(WC3)}$$

For the prime operator use the same coefficients as in the mixed model: $t_n=\log n$, $w_n=\Lambda (n)/\sqrt{n}$ and $b_n(x)=w_{n}s(x)s(x+t_n)$. Product differentiation and $e^{2|x|}+e^{2|x+t_n|}\ge 2n$ imply $\Vert b_n^{\prime }{\Vert }_{\infty }\le 2K_0{K}_1{w}_n{e}^{-2cn}$. With $r=e^{-2c}$ and $w_n\le n$,

$$\sum _{n\ge 2}\Vert b_n^{\prime }{\Vert }_{\infty }\le 2K_0{K}_1\left(\frac{r}{(1-r{)}^2}-r\right)<\infty .\text{\hspace{1em}(WC4)}$$

This sum is per representative undirected edge and includes every prime power. The translated adjoint has the same derivative supremum norm, so the explicitly two-direction estimate is

$$\sum _{n\ge 2}\left(\Vert b_n^{\prime }{\Vert }_{\infty }+\Vert (b_n(\cdot -\log n){)}^{\prime }{\Vert }_{\infty }\right)\le 4K_0{K}_1\left(\frac{r}{(1-r{)}^2}-r\right).\text{\hspace{1em}(WC5)}$$

These estimates discharge the coefficient hypotheses in (WF2), (WF5) and (WF7) of the linked cutoff construction for the actual theta model. They retain the full graph and provide deliberately coarse constants; useful bandwidth and matrix certificates require further quantitative work.

Source anomalies retained

On page 40 the printed c-prime integral omits the factor 2*sqrt(2) used in the original c coefficients, while asserting equality of the even coefficients. The Pollaczek prose says orthonormal, whereas equation (A.12) gives a nonunit norm. Neither discrepancy is silently repaired or used as an equality here. The separate Cardon source’s odd-extension/jump and recurrence/tail issues remain separate from Romik’s theta formulas.

Verified locator

• URL: https://doi.org/10.4064/aa200515-10-3
• DOI: 10.4064/aa200515-10-3

The author-hosted Online First PDF was retrieved on 2026-09-11 from https://www.math.ucdavis.edu/~romik/data/uploads/papers/riemannxi-acta-online-first.pdf. Its SHA-256 is a28edcf341776bf801e9d0c2de4631639b2c46c579a67a38cc2788d255e2ae87. The page references above use that 72-page PDF’s internal pagination; they are not converted to the final Acta Arithmetica 200(3), 259–329 pagination.