Laguerre signs at the height of an off-line zero: a window lemma, a crossing index, and a two-sided depth meter

Author: span_of_dilates

Status. Every result is unconditional [U]. Nothing here implies or approaches RH. All statements in §§2–5 hold verbatim for the completed Davenport–Heilbronn function; see §8.

1. Introduction and results

For a real entire function $f$, the generalized Laguerre expression of order $p$ is $$\mathcal L_p[f](t)=[y^{2p}]\,f(t+iy)f(t-iy),$$ as in [P41, (1)]. Let $\Xi(t)=\xi(\tfrac12+it)$. Jensen's criterion [J] (see also [CV]) says that RH holds iff $\mathcal L_p[\Xi](t)\ge0$ for all $p$ and all real $t$. Paper [P41] proves positivity for all orders $2p\ge c_0\ell$ with any fixed $c_0>1/2$, where $\ell=\log(t/2\pi)$. Its local lemma [P41, §4.6] lowers the threshold to $(D+\delta)\ell$ near heights where $\zeta$ has no zero in $\operatorname{Re}s>\tfrac12+D$.

This note asks the converse, local question: what do the Laguerre signs look like at the height of an off-line zero? The answer is a short dictionary.

1. Window lemma (Lemma 1). All orders are nonnegative at $t$ unless $t$ lies within $\beta-\tfrac12$ of the ordinate of some zero $\beta+i\gamma$ with $\beta>\tfrac12$. With the Balasubramanian–Conrey–Heath-Brown mean value and Conrey's $\theta=4/7$, this gives all-order nonnegativity on a proportion at least $0.839$ of $[T,2T]$ (Corollary 1).
2. Crossing index (Lemmas 2–3). At the height $\gamma_0$ of a simple off-line zero $\rho_0$, the normalized coefficients are the second differences of a sequence $g_k$ summing to 1. The first descent of $g$ is a named negative order.
3. Neighbour-count law (§4). If no other off-line zero is close to $\gamma_0$, $(g_k)$ is the law of an independent sum of Bernoulli and $\{0,1,2\}$-valued variables. Its mean is exactly $d\,\operatorname{Re}\frac{\xi''}{2\xi'}(\rho_0)-\tfrac12$, where $d=\beta_0-\tfrac12$.
4. Two-sided depth meter (Theorem 1). Combined with the local lemma of [P41], the largest negative order $p$ at $\gamma_0$ satisfies $2p/\ell\to d$ under explicit local hypotheses. So the Laguerre sign pattern at a zero height reads off the depth of the zero.

2. Setup and the window lemma

A zero $\rho=\tfrac12+a+i\gamma$ of $\xi$ with $|a|<\tfrac12$ corresponds to the zero $w=\gamma-ia$ of $\Xi$. The zero set is invariant under $w\mapsto-w$ and $w\mapsto\bar w$. For real $t,y$, the functional equation and $\xi(\bar s)=\overline{\xi(s)}$ give $$P_t(y):=\Xi(t+iy)\Xi(t-iy)=|\xi(\tfrac12+y+it)|^2\ge0 .$$ Write $\mathcal L_m(t)=[y^{2m}]P_t$, $Y=y^2$, and $b=\gamma-t$. Hadamard's product, paired over $\pm w$, gives the locally uniformly convergent factorization [P41, (18f)–(18g)] $$P_t=C_t\,Y^{m_0}\prod_j\Big(1+\frac{Y}{b_j^2}\Big)\prod_\rho\frac{F_\rho(Y)}{F_\rho(0)},\qquad F_\rho(Y)=Y^2+2(b^2-a^2)Y+(a^2+b^2)^2,\tag{2.1}$$ with $C_t>0$.

• The first product runs over on-line zeros with $b_j\ne0$, including the reflected ordinates.
• $m_0$ is the multiplicity of an on-line zero at ordinate $t$.
• The second product runs over conjugate off-line pairs $\gamma\mp ia$, $a>0$, including reflected ones.

Lemma 1 (window) [U]. If every zero $\beta+i\gamma$ with $\beta>\tfrac12$ satisfies $|t-\gamma|\ge\beta-\tfrac12$, then $\mathcal L_m(t)\ge0$ for every $m$.

Proof. $F_\rho$ has coefficients $1,\,2(b^2-a^2),\,(a^2+b^2)^2$, which are nonnegative iff $b^2\ge a^2$, i.e. $|t-\gamma|\ge a$. On-line factors have nonnegative coefficients. Coefficients of a locally uniform limit of entire functions are limits of the coefficients. ∎

The global all-order criterion is classical [J, CV], and [P41, §4.3] uses the same near/far split of quartic factors. The point here is the $t$-local support of the obstruction.

Corollary 1 [U]. For large $T$, $\mathcal L_m(t)\ge0$ for every $m$ on a subset of $[T,2T]$ of measure at least $0.839\,T$.

Proof. By Lemma 1 the exceptional set is covered by intervals of length $2(\beta-\tfrac12)$ about the ordinates of zeros with $\beta>\tfrac12$. Littlewood's lemma and Jensen's inequality, applied to $\zeta M$ with a mollifier $M$ of length $T^\theta$, give $$\sum_{\substack{\beta>1/2\\ T<\gamma\le2T}}(\beta-\tfrac12)\le\frac T{4\pi}\log\Big(\frac1T\int_T^{2T}|\zeta M(\tfrac12+it)|^2dt\Big)+o(T).$$ For $M(s)=\sum_{n\le y}\mu(n)n^{-s}\frac{\log(y/n)}{\log y}$ the mean value is $(1+1/\theta+o(1))T$ [BCHB], valid for $\theta<4/7$ [C89]. Thus the exceptional measure is at most $\frac{\log(11/4)}{2\pi}T+o(T)\le0.1610\,T$. The $O(\log T)$ zeros with ordinates within $1/2$ of the endpoints contribute $O(\log T)$. ∎

3. The crossing index at an off-line zero

Fix a zero $\rho_0=\tfrac12+d+i\gamma_0$ with $0<d<\tfrac12$, and take $t=\gamma_0$ and $D=d^2$. The pair $\gamma_0\mp id$ has $b=0$ and contributes $F=(Y-D)^2$, with $F(0)=D^2$. Hence $$P_{\gamma_0}=C\,(Y-D)^2B(Y),\qquad C=C_{\gamma_0}/D^2>0,$$ where $B(Y)=Y^{m_0}\times$(the remaining normalized factors of (2.1)) $=\sum_kh_kY^k$ and $h_{m_0}=1$. The hypotheses are used separately:

• (S) $\rho_0$ is simple.
• (I) every other off-line zero has $b^2\ge a^2$, i.e. $\gamma_0$ lies in no other window of Lemma 1.
• (C) every other off-line zero has $b^2\ge3a^2$.

(C) implies (I). Zeros with $|b|\ge1$ satisfy both automatically, since $a<\tfrac12$.

Positivity of $B(D)$. For real $Y$, $F_\rho(Y)=|Y+(b-ia)^2|^2\ge0$, with equality only if $b=0$ and $Y=a^2$. At $Y=D$ that would be a second pair at height $\gamma_0$ and depth $d$, which (S) excludes. On-line factors are positive. So $B(D)>0$. Put $$g_k=\frac{h_kD^k}{B(D)},\qquad\sum_kg_k=1 .$$ The series converges absolutely since $B$ is entire, so $g_k\to0$. The lowest nonzero index is $j_0=m_0$, with $g_{j_0}=D^{m_0}/B(D)>0$.

Lemma 2 (second difference) [U, (S)]. For $m\ge1$, with $g_{-1}=0$, $$\frac{D^{m-2}}{C\,B(D)}\,\mathcal L_m(\gamma_0)=g_m-2g_{m-1}+g_{m-2}.$$ Proof. $[Y^m](Y-D)^2B=h_{m-2}-2Dh_{m-1}+D^2h_m$. Multiply by $D^{m-2}/B(D)$. ∎

So the negative orders at $\gamma_0$ are exactly the strict concavity points of $(g_k)$, shifted by one.

Lemma 3 (crossing) [U, (S)]. Let $k^*=\min\{k\ge j_0:g_{k+1}\le g_k\}$. It exists, and $\mathcal L_{k^*+1}(\gamma_0)<0$.

Proof. Otherwise $g$ would increase strictly from $g_{j_0}>0$, contradicting $g_k\to0$.

• If $k^*>j_0$, then $g_{k^*}>g_{k^*-1}$, so $(g_{k^*+1}-g_{k^*})-(g_{k^*}-g_{k^*-1})<0$.
• If $k^*=j_0$, then $g_{j_0-1}=0$ and the second difference is at most $g_{j_0}-2g_{j_0}<0$.

Apply Lemma 2. ∎

No sign information on $g_k$ beyond $[j_0,k^*]$ is used. Note that $P_{\gamma_0}(d)=0<P_{\gamma_0}(0)$ already forces some negative coefficient; Lemma 3 names its order, $m^*=k^*+1$.

4. The neighbour-count law and its mean [U, (S)+(I)]

Under (I), every $h_k\ge0$, and $B(Dz)/B(D)$ factors into probability generating functions. Thus $(g_k)$ is the law of $$S=m_0+\sum_j\varepsilon_j+\sum_\rho X_\rho,\qquad \varepsilon_j\sim\mathrm{Bern}(p_j),\quad p_j=\frac D{D+b_j^2},\qquad \Pr(X_\rho=i)\propto\big(F_\rho(0),\,2(b^2-a^2)D,\,D^2\big)_i ,$$ with independent summands.

• $X_\rho$ is log-concave iff $(2(b^2-a^2)D)^2\ge(a^2+b^2)^2D^2$, i.e. iff $b^2\ge3a^2$. This holds for every $D$.
• Mean identity. $\mathbb ES=D(\log B)'(D)=d\operatorname{Re}\frac{\xi''}{2\xi'}(\rho_0)-\tfrac12.$

Proof. $\log B=\log P-2\log(Y-D)-\log C$ and $\frac d{dY}=\frac1{2y}\frac d{dy}$ give $(\log B)'=\frac1y\operatorname{Re}\frac{\xi'}{\xi}(\tfrac12+y+i\gamma_0)-\frac2{y^2-d^2}$. Insert $\frac{\xi'}{\xi}(\rho_0+u)=\frac1u+\frac{\xi''}{2\xi'}(\rho_0)+O(u)$ and let $y\to d$. The limit is $-\frac1{2d^2}+\frac1d\operatorname{Re}\frac{\xi''}{2\xi'}(\rho_0)$. ∎

• Termwise form. $\mathbb ES=d\sum_{\rho\ne\rho_0,\,1-\bar\rho_0}\operatorname{Re}\frac1{\rho_0-\rho}$. For an on-line zero the term is $p_j$. For an off-line pair $\{\rho,1-\bar\rho\}$ the two terms sum to $\mathbb EX_\rho$, because $((d-a)^2+b^2)((d+a)^2+b^2)=F_\rho(D)$. Under (I) every term is nonnegative.
• Asymptotic form. Since $\xi''/(2\xi')=\zeta''/(2\zeta')+G'/G$ at $\rho_0$, where $\xi=G\zeta$, we get $\mathbb ES=\frac{d\ell}2+d\operatorname{Re}\frac{\zeta''}{2\zeta'}(\rho_0)-\frac12+O(d/\gamma_0)$, with $\ell=\log(\gamma_0/2\pi)$.

5. Locating the crossing [U]

Lemma 4 (bracket). Assume (S) and (I).

(a) If there is no off-line zero other than $\rho_0$ and $1-\bar\rho_0$, then $k^*\in[\mathbb ES-2,\mathbb ES+1]$, and $|k^*-\mathbb ES|\le1$ when $g$ has a unique mode.

(b) Under (C), $|k^*-\mathbb ES|\le\sqrt{3\operatorname{Var}S+\tfrac14}+\tfrac12\le\sqrt{6\,\mathbb ES}+1$.

Proof of (a). $S-m_0$ is Poisson-binomial with $\sum p_j<\infty$. Let $g^{(n)}$ be the law of the truncation to $n$ Bernoulli variables, with mean $\mu_n\to\mu=\mathbb ES$ and $g^{(n)}\to g$ pointwise.

• By Darroch [D], every mode of $g^{(n)}$ lies in $\{\lfloor\mu_n\rfloor,\lceil\mu_n\rceil\}$. A subsequence of modes is constant, equal to some $M$, so $M$ is a mode of $g$ with $|M-\mu|\le1$.
• Newton's inequalities pass to the limit as $g_k^2\ge(1+\frac1{k-m_0})g_{k-1}g_{k+1}$. So $g$ has at most two modes, and they are adjacent.
• Since $g$ is log-concave, $k^*$ is its smallest mode, so $k^*\in\{M-1,M\}$.
• If $k^*>\mu+1$, then for large $n$ every mode of $g^{(n)}$ is below $k^*$, so $g^{(n)}_{k^*-1}\ge g^{(n)}_{k^*}$. In the limit $g_{k^*-1}\ge g_{k^*}$, contradicting the definition of $k^*$.

Proof of (b).

• Each $X_\rho$ and $\varepsilon_j$ is log-concave, so $g$ is log-concave and $k^*$ is a mode.
• The step density $f(x)=g_{\lfloor x\rfloor}$ is unimodal with mode $k^*$. It is the density of $S+U$, where $U$ is uniform on $[0,1)$ and independent of $S$.
• Johnson–Rogers [JR]: $|\text{mode}-\text{mean}|\le\sqrt3\,\sigma$ for unimodal laws. Here the mean is $\mathbb ES+\tfrac12$ and $\sigma^2=\operatorname{Var}S+\tfrac1{12}$.
• Finally, $\operatorname{Var}X_\rho\le2\,\mathbb EX_\rho$ (values in $\{0,1,2\}$) and $\operatorname{Var}\varepsilon\le\mathbb E\varepsilon$. ∎

6. Two-sided depth meter at zero heights [U]

The input is the local lemma of [P41, §4.6]. For fixed $0<\delta<\tfrac18$ there are $H_\delta$ and $T_\delta$ such that, uniformly for $0\le D'\le\tfrac12-\delta$: if $t\ge T_\delta$ and $\zeta$ has no zero with $\operatorname{Re}s>\tfrac12+D'$ and $|\operatorname{Im}s-t|\le H_\delta+2$, then $\mathcal L_p(t)>0$ for every $2p\ge(D'+\delta)\ell$. Its proof also gives [P41, (18p)] $$|(\log\zeta)^{(j)}(\sigma+iv)|\ll_\delta\ell^{a_\delta},\qquad j\le4,\ \sigma\ge\tfrac12+D'+\tfrac\delta4,\ |v-t|\le H_\delta,$$ for some $a_\delta<1$.

Theorem 1. Fix $d\in(0,\tfrac12)$ and $\varepsilon>0$. There are $\delta=\delta(\varepsilon,d)>0$ and $T(\varepsilon,d)$ such that the following holds for every zero $\rho_0=\tfrac12+d+i\gamma_0$ with $\gamma_0\ge T(\varepsilon,d)$ satisfying (S), (C) and

• (Z) no zero has $\operatorname{Re}\rho>\tfrac12+d$ and $|\operatorname{Im}\rho-\gamma_0|\le H_\delta+2$.

Then $\mathcal L_p(\gamma_0)>0$ for all $2p\ge(d+\varepsilon)\ell$, and $\mathcal L_{m^*}(\gamma_0)<0$ for some $m^*$ with $2m^*\in[(d-\varepsilon)\ell,(d+\varepsilon)\ell]$. In particular, the largest order $p$ with $\mathcal L_p(\gamma_0)<0$ satisfies $|2p/\ell-d|\le\varepsilon$.

Proof. Let $C_0$ be the absolute constant of step 3, and take $\delta=\min\big(\varepsilon d/(4C_0),\ \tfrac1{10},\ (\tfrac12-d)/2\big)$.

1. Upper side. Apply the local lemma with $D'=d$; then $\delta\le\varepsilon$.
2. Mean at a shifted point. Put $f_x(\rho)=\frac{x-a}{(x-a)^2+b^2}$ for $\rho=\tfrac12+a+i(\gamma_0+b)$. Then $\sum_{\rm all}f_x(\rho)=\operatorname{Re}\frac{\xi'}{\xi}(\tfrac12+x+i\gamma_0)$. At $x=d+\delta$, (18p) and Stirling give $\tfrac\ell2+O_\delta(\ell^{a_\delta})$. Removing $\rho_0$ and $1-\bar\rho_0$ subtracts $\frac1\delta+\frac1{2d+\delta}$.
3. Lipschitz shift to $x=d$. $|\partial_xf_x|\le((x-a)^2+b^2)^{-1}$.
   • For $|b|\le1$, (Z) gives $a\le d$, so $x-a\ge0$ on $[d,d+\delta]$. The denominator is at least $d^2$ if $a\le0$. If $0<a\le d$, (C) gives $(d-a)^2+3a^2\ge\tfrac34d^2$, with the minimum at $a=d/4$.
   • For $|b|>1$ the bound is $b^{-2}$.
   • With $N(\gamma_0+1)-N(\gamma_0-1)\ll\ell$, this gives $|\mathbb ES-\tfrac{d\ell}2|\le C_0\delta\ell/d+O_\delta(\ell^{a_\delta})$.
4. Bracket. By Lemma 4(b), $|k^*-\mathbb ES|\le\sqrt{6\,\mathbb ES}+1=O(\sqrt\ell)$. Hence $|2m^*-d\ell|\le2C_0\delta\ell/d+o(\ell)\le\tfrac\varepsilon2\ell+o(\ell)$, where $m^*=k^*+1$ is negative by Lemma 3. Take $\gamma_0$ large. ∎

Remark (the $\sqrt\ell$ scale). Moving the unconditional threshold to $\ell/2+O(\sqrt\ell)$ needs $|(\zeta'/\zeta)'(1+\eta+it)|\ll\ell$ at $\eta\asymp\ell^{-1/2}$. The crude bound $\eta^{-2}$ is exactly at that edge. Certifying a crossing at resolution $\sqrt\ell$ near the 1-line needs $O(\log\ell)$ zeros within $\ell^{-1/2}$ of $1+i\gamma_0$, whereas Richert's bound [R] gives only $\ll\ell^{1/4}$ there. Both requirements sit beyond the Vinogradov–Korobov range. This is a correspondence between thresholds, not an equivalence.

7. Numerical illustrations

These finite checks illustrate the identities. None verifies a uniform estimate. Scripts and outputs are in the supplement.

• Davenport–Heilbronn bench. $f=\frac{1-i\kappa}2L(s,\chi)+\frac{1+i\kappa}2L(s,\bar\chi)$ with $\chi\bmod5$, $\chi(2)=i$, $\kappa=0.28407904384$, and $\Lambda(s)=(5/\pi)^{(s+1)/2}\Gamma(\frac{s+1}2)f(s)$.
  • At a test point, $|\Lambda(s)-\Lambda(1-s)|/|\Lambda(s)|=1.493\times10^{-40}$.
  • Four off-line zeros were refined at 40 digits: $0.8085171825+85.6993484854i$, $0.6508300806+114.163342731i$, $0.5743560505+166.479305913i$, $0.7242576946+176.702461243i$.
  • At each, $\mathcal L_1(\gamma_0)<0$ and $m^*=1$, as Lemma 3 predicts. Here $d\ell/2\in\{0.651,0.34,0.182,0.555\}$, with $\ell=\log(5\gamma_0/2\pi)$.
  • The mean identity of §4 holds to working precision: at the first zero, $\mu=0.0802223$ both from the formula and numerically.
• Window scan about the first DH zero, at 52 heights in $[84,86.9]$ and orders $m\le8$.
  • Negative coefficients occur only at $m=1$, for $t\in(85.4129,85.9835)$, of half-width $0.2853$.
  • That interval lies inside the window of Lemma 1, $(85.3908,86.0079)$, of half-width $0.3085$. No negatives occur outside it.
• Synthetic backgrounds. A jittered on-line lattice of density $\ell/2\pi$ plus one off-line zero gives $2m^*/\ell=0.1010,\,0.2500,\,0.3980$ at $\ell=2000$, for $d=0.10,0.25,0.40$.
• Exact algebra. In 20000 rational cases the log-concavity threshold $b^2\ge3a^2$ had 0 mismatches. In 4000 random configurations the bracket of Lemma 4(b) had 0 failures, with maximal ratio $0.444$.

8. Scope

• §§2–5 use only Hadamard's product, the strip and local zero counts. They hold verbatim for the completed Davenport–Heilbronn function, whose computed zeros satisfy (S). In the sense of the Davenport–Heilbronn test these results are dictionaries, not routes to RH.
• Theorem 1 uses the local lemma of [P41], whose logarithm bounds rest on the Euler product to the right of 1. Corollary 1 uses a mean value of $\zeta M$.
• Separation between $\zeta$ and DH enters through density inputs only. For $\zeta$, classical zero density bounds the measure of heights with negative orders $\ge c\ell$ [P41, Thm 1c]. DH has $\gg T$ off-line zeros in fixed strips [DH].
• Nothing is claimed between zero heights beyond the upper side, or uniformly as $d\to0$.

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