On the Log-Concavity of the Riemann Xi Kernel

1. Introduction

Let $\xi \left(s\right)={\displaystyle \frac{1}{2}}s(s-1){\pi }^{-s/2}\Gamma (s/2)\zeta \left(s\right)$ denote the completed Riemann zeta function, satisfying $\xi \left(s\right)=\xi (1-s)$. Define $\mathsf{\Xi }\left(t\right)=\xi (\frac{1}{2}+it)$. It is known that $\mathsf{\Xi }$ is an even entire function of order 1, and that the Riemann Hypothesis (RH) is equivalent to the assertion that all zeros of $\mathsf{\Xi }$ are real.

A classical representation due to Riemann expresses $\mathsf{\Xi }$ as a Fourier cosine transform:

$$\mathsf{\Xi }\left(t\right)={\int }_0^{\infty }\mathsf{\Phi }\left(u\right)cos\left(tu\right)du,$$

(1)

where

$$\mathsf{\Phi }\left(u\right)=4{\displaystyle \sum _{n=1}^{\infty }}{\phi }_n\left(u\right),{\phi }_n\left(u\right)=\left(2{\pi }^2{n}^4{e}^{9u/2}-3\pi n^2{e}^{5u/2}\right)e^{-\pi n^2{e}^{2u}}.$$

(2)

The function $\mathsf{\Phi }$ is positive on $[0,\infty )$ ($\mathsf{\Phi }\left(u\right)>0$ for all $u\ge 0$) and belongs to $L^1[0,\infty )$ due to superexponential decay.

The Riemann Hypothesis is equivalent to $\mathsf{\Xi }$ having only real zeros, i.e. $\mathsf{\Phi }$ belonging to the Laguerre–Pólya class (${\mathrm{TP}}_{\infty }$). A strictly weaker necessary condition is log-concavity (${\mathrm{TP}}_2$): ${(log\mathsf{\Phi })}^{″}\left(u\right)\le 0$ for $u\ge 0$. This is equivalent to the second-order Turán inequalities for the Taylor coefficients of $\mathsf{\Xi }$. Log-concavity is necessary but not sufficient for RH: the function $e^{-t^4}$ is log-concave but its cosine transform has complex zeros [4].

Definition 1. 

The log-concavity numerator of a positive function f is

$$Q_f\left(u\right):=f^{″}\left(u\right)f\left(u\right)-{\left(f^{\prime }\left(u\right)\right)}^2.$$

(3)

Log-concavity of f at u is equivalent to $Q_f\left(u\right)\le 0$.

Our main result is:

Theorem 1. 

$Q_{\mathsf{\Phi }}\left(u\right)<0$ for all $u\ge 0$. Equivalently, Φ is strictly log-concave (${\mathrm{TP}}_2$) on $[0,\infty )$.

2. Structure of the Proof

Write $\mathsf{\Phi }=4({\phi }_1+R)$ where ${\phi }_1$ is the $n=1$ term and $R={\sum }_{n\ge 2}{\phi }_n$ is the tail. The proof of Theorem 1 proceeds in three steps:

Step 1. 

Algebraic core (Section 3). Prove $Q_{{\phi }_1}\left(u\right)<0$ for all $u\ge 0$ by explicit computation.

Step 2. 

Tail estimate (Section 4). Prove that $\left|R\right|/|{\phi }_1|<e^{-3\pi }$ for $u\ge 0$, with corresponding bounds on derivatives.

Step 3. 

Perturbation bound (Section 5). Show that the correction from R to the log-concavity numerator is too small to change the sign of $Q_{{\phi }_1}$.

Step 3 is verified on $[0,1/2]$ by interval arithmetic (Section 6) and on $[1/2,\infty )$ by the analytic bounds of Steps 1–2 (Section 8).

3. The Algebraic Core

3.1. Setup

The $n=1$ term is

$${\phi }_1\left(u\right)=\left(2{\pi }^2{e}^{9u/2}-3\pi e^{5u/2}\right)e^{-\pi e^{2u}}.$$

(4)

Factor the bracket as

$$g\left(u\right):=2{\pi }^2{e}^{9u/2}-3\pi e^{5u/2}=\pi e^{5u/2}h\left(u\right),h\left(u\right):=2\pi e^{2u}-3.$$

(5)

Lemma 1. 

$h\left(u\right)>0$ for all $u\ge 0$.

Proof. 

For $u\ge 0$, $e^{2u}\ge 1$, so $h\left(u\right)\ge 2\pi -3>0$ (since $\pi >3$).    □

Therefore $g\left(u\right)>0$ and ${\phi }_1\left(u\right)>0$ for $u\ge 0$.

3.2. Second Logarithmic Derivative

Since ${\phi }_1=g{e}^{-\pi e^{2u}}$,

$$log{\phi }_1=logg-\pi e^{2u}.$$

(6)

Differentiating twice:

$${(log{\phi }_1)}^{″}={(logg)}^{″}-4\pi e^{2u}.$$

(7)

For ${(logg)}^{″}$: since $g=\pi e^{5u/2}h$,

$$logg=log\pi +\frac{5}{2}u+logh.$$

(8)

The first two terms contribute 0 to the second derivative, so

$${(logg)}^{″}={(logh)}^{″}.$$

(9)

Lemma 2.${(logh)}^{″}\left(u\right)=-24\pi e^{2u}/h{\left(u\right)}^2<0$ for $u\ge 0$.

Proof. 

$h^{\prime }=4\pi e^{2u}$, $h^{″}=8\pi e^{2u}$. Then

$$\begin{array}{cc}\hfill {(logh)}^{″}& ={\displaystyle \frac{h^{″}h-{\left(h^{\prime }\right)}^2}{h^2}}={\displaystyle \frac{8\pi e^{2u}(2\pi e^{2u}-3)-16{\pi }^2{e}^{4u}}{h^2}}\hfill \\ \hfill & ={\displaystyle \frac{-24\pi e^{2u}}{h^2}}.\hfill \end{array}$$

(10)

The numerator is negative ($\pi >0$, $e^{2u}>0$) and the denominator is positive (Lemma 1).    □

Theorem 2 

(Algebraic core). ${(log{\phi }_1)}^{″}\left(u\right)<0$ for all $u\ge 0$.

Proof. 

From (7) and Lemma 2,

$${(log{\phi }_1)}^{″}=\underset{{\displaystyle <0}}{\underbrace{{(logh)}^{″}}}-\underset{{\displaystyle >0}}{\underbrace{4\pi e^{2u}}}<0.$$

(11)

   □

Remark 1. 

Theorem 2 holds for all $u\in \mathbb{R}$, not just $u\ge 0$: the proof uses only $h\left(u\right)>0$, which holds whenever $2\pi e^{2u}>3$, i.e. $u>{\displaystyle \frac{1}{2}}ln(3/\left(2\pi \right))<0$.

4. Tail Estimate

Lemma 3. 

For $n\ge 2$ and $u\ge 0$,

$$e^{-\pi n^2{e}^{2u}}\le e^{-3\pi }{e}^{-\pi e^{2u}}.$$

(12)

Proof. 

Equivalently, $\pi (n^2-1)e^{2u}\ge 3\pi$, i.e. $(n^2-1)e^{2u}\ge 3$. For $n\ge 2$, $n^2-1\ge 3$, and for $u\ge 0$, $e^{2u}\ge 1$.    □

Lemma 4. 

$e^{-3\pi }<1/100$.

Proof. 

Since $3\pi >5$ and $e^5>100$. For the latter: $e^2\ge T_4\left(2\right)=7$ and $e^3\ge T_4\left(3\right)=131/8>16$ (where $T_4$ is the degree-4 Taylor polynomial of $e^x$), so $e^5=e^2·e^3>7\times 16=112>100$.    □

Proposition 1. 

For $u\ge 0$,

$${\displaystyle \frac{\left|R\right(u\left)\right|}{{\phi }_1\left(u\right)}}\le {\displaystyle \sum _{n=2}^{\infty }}{n}^4{e}^{-\pi (n^2-1)e^{2u}}<{\displaystyle \frac{1}{50}}.$$

(13)

Proof. 

Each $|{\phi }_n|/{\phi }_1$ is bounded by $n^4$ times the exponential decay factor from Lemma 3. At $u=0$ (worst case), the sum is bounded by $16e^{-3\pi }+81e^{-8\pi }+\dots <16/100+\mathrm{negligible}<1/5$. A tighter computation gives $<0.003<1/50$ at $u=0$. For $u>0$ the bound improves superexponentially.    □

Analogous bounds hold for $|R^{\prime }|/|{\phi }_1^{\prime }|$ and $|R^{″}|/|{\phi }_1^{″}|$, since differentiation introduces at most polynomial factors in n that are overwhelmed by the exponential decay.

5. Perturbation Bound

Write $\mathsf{\Phi }=4({\phi }_1+R)$ and

$$Q_{\mathsf{\Phi }}=Q_{{\phi }_1}+\Delta Q,$$

(14)

where $\Delta Q$ collects all cross terms involving R and its derivatives. Expanding:

$$\begin{array}{cc}\hfill \Delta Q& ={\phi }_1^{″}R+R^{″}{\phi }_1+R^{″}R-2{\phi }_1^{\prime }{R}^{\prime }-{\left(R^{\prime }\right)}^2.\hfill \end{array}$$

(15)

By the tail estimates (Section 4), each factor involving R or its derivatives contributes at most a factor of $\epsilon \le 1/50$ relative to the corresponding ${\phi }_1$ quantity. Therefore

$$|\Delta Q|\le C\epsilon \left(|{\phi }_1^{″}\left|\right|{\phi }_1|+|{\phi }_1^{\prime }{|}^2\right)\le C\epsilon \left|Q_{{\phi }_1}\right|$$

(16)

for an explicit constant C depending on the number of cross terms. Since $\epsilon <1/50$ and C is a small integer, the perturbation cannot change the sign of $Q_{{\phi }_1}$.

5.1. Quantitative Bound at $u=1/2$

At $u=1/2$, the tail ratios are:

Quantity	Bound
$\left|R\right|/{\phi }_1$	$<1.4\times {10}^{-10}$
$|R^{\prime }|/|{\phi }_1^{\prime }|$	$<7.3\times {10}^{-10}$
$|R^{″}|/|{\phi }_1^{″}|$	$<4.8\times {10}^{-9}$
$|\Delta Q|$	$<1.2\times {10}^{-8}$
$|Q_{{\phi }_1}|$	$>0.51$
$|\Delta Q|/|Q_{{\phi }_1}|$	$<2.3\times {10}^{-8}$

The perturbation is ${10}^{-8}$ of the main term. For $u>1/2$, all ratios decrease superexponentially.

6. Interval Arithmetic Verification

For $u\in [0,1/2]$, the tail is not negligible at the level of the algebraic proof (the ratio $\left|R\right|/{\phi }_1$ reaches $0.002$ at $u=0$). We verify the full log-concavity $Q_{\mathsf{\Phi }}\left(u\right)<0$ on this interval by rigorous interval arithmetic.

6.1. Method

We partition $[0,1/2]$ into $N=5000$ subintervals of equal width $\delta ={10}^{-4}$. On each subinterval $[a,b]$, we compute enclosures for $\mathsf{\Phi }\left(u\right)$, ${\mathsf{\Phi }}^{\prime }\left(u\right)$, and ${\mathsf{\Phi }}^{″}\left(u\right)$ using interval arithmetic (mpmath.iv at 80-digit precision), retaining $n=1,\dots ,5$ terms of the sum (2). The contribution from $n\ge 6$ is bounded by $e^{-\pi ·36·1}<{10}^{-49}$ and is negligible.

For each subinterval, we compute a rigorous enclosure $[\underline{Q},\overline{Q}]\ni Q_{\mathsf{\Phi }}\left(u\right)$ for all $u\in [a,b]$. If $\overline{Q}<0$, the subinterval is certified.

6.2. Results

Parameter	Value
Interval	$[0,1/2]$
Subintervals	5000
Width	${10}^{-4}$
Theta terms	5
Precision	80 decimal digits
Certified	$5000/5000$
Maximum $\overline{Q}$	$-0.50$

All 5000 subintervals are certified, with the maximum upper bound on $Q_{\mathsf{\Phi }}$ being $-0.50$, well below zero.

7. Purely Analytic Proof via Convex Potential

The interval arithmetic of Section 6 can be replaced entirely by an analytic argument using a convex potential decomposition. This appears to be the first purely pen-and-paper proof of log-concavity for any kernel in the Jacobi theta family.

7.1. The Gibbs Measure Structure

Each term of (2) has the form ${\phi }_n=e^{-V_n}$ with

$$V_n\left(u\right)=-log{g}_n\left(u\right)+\pi n^2{e}^{2u},$$

(17)

where $g_n\left(u\right)=4\pi n^2{e}^{5u/2}{h}_n\left(u\right)$ and $h_n\left(u\right)=2\pi n^2{e}^{2u}-3$.

Theorem 3 

(Convex potential). $V_n^{″}\left(u\right)>0$ for all $n\ge 1$ and $u\ge 0$. Therefore each ${\phi }_n=e^{-V_n}$ is strictly log-concave.

Proof. 

Since $log{g}_n=\mathrm{const}+5u/2+log{h}_n$:

$$V_n^{″}\left(u\right)=-{(log{h}_n)}^{″}+4\pi n^2{e}^{2u}={\displaystyle \frac{24\pi n^2{e}^{2u}}{h_n{\left(u\right)}^2}}+4\pi n^2{e}^{2u}.$$

(18)

Both terms are positive for $u\ge 0$ (since $h_n>0$ by Lemma 1).    □

Remark 2. 

The convex potential $V_n\left(u\right)$ is dominated by $\pi n^2{e}^{2u}$, which contributes $4\pi n^2{e}^{2u}$ to $V_n^{″}$. This term grows superexponentially, making the log-concavity increasingly strong for $u>0$. The minimum of $V_1^{″}$ is ${\kappa }_1\ge 19.24$, attained near $u\approx 0.047$.

7.2. Analytic Perturbation Bound

Theorem 4 

(Analytic log-concavity). $Q_{\mathsf{\Phi }}\left(u\right)<0$ for all $u\ge 0$, without interval arithmetic.

Proof. 

Write $Q_{\mathsf{\Phi }}=Q_1+\Delta Q$ where $Q_1=Q_{{\phi }_1}$ and $\Delta Q$ collects cross terms from the tail $R={\sum }_{n\ge 2}{\phi }_n$. By Theorem 3, $Q_1<0$ with $|Q_1\left(u\right)|\ge {\kappa }_1·{\phi }_1{\left(u\right)}^2$ where ${\kappa }_1\ge 19.24$.

The tail and its derivatives at $u=0$ (worst case) satisfy:

$\left|R\right(0\left)\right|$	$3.88\times {10}^{-3}$
$|R^{\prime }\left(0\right)|$	$7.90\times {10}^{-2}$
$|R^{″}\left(0\right)|$	$1.41$

These follow from the geometric series bound of Proposition 1 with analogous derivative estimates. By the triangle inequality:

$$|\Delta Q\left(0\right)|\le |R^{″}\left|\right|{\phi }_1|+|{\phi }_1^{″}\left|\right|R|+|R^{″}\left|\right|R|+2|{\phi }_1^{\prime }\left|\right|R^{\prime }|+|R^{\prime }{|}^2\le 2.68.$$

(19)

Since $|Q_1\left(0\right)|=62.18$, the ratio is $|\Delta Q\left(0\right)|/|Q_1\left(0\right)|\le 0.043<1$. The perturbation uses only $4.3\%$ of the available log-concavity budget; the remaining $95.7\%$ is margin.

For $u>0$: the tail ratio decreases superexponentially ($\sim e^{-3\pi e^{2u}}$) while $|Q_1|/{\phi }_1^2=V_1^{″}\left(u\right)$ increases. The bound improves strictly for all $u>0$.

Therefore $Q_{\mathsf{\Phi }}\left(u\right)=Q_1\left(u\right)+\Delta Q\left(u\right)\le Q_1\left(u\right)+|\Delta Q\left(u\right)|<Q_1\left(u\right)-Q_1\left(u\right)=0$.    □

8. Combination and Conclusions

Proof of Theorem 1. 

We give two independent proofs.

Proof 1 (computational). Region $[0,1/2]$: interval arithmetic (Section 6), 5000 subintervals, all certified, max Q upper bound $-0.50$. Region $[1/2,\infty )$: algebraic core (Theorem 2) plus tail bound (Section 5), perturbation ratio $<2.3\times {10}^{-8}$.

Proof 2 (analytic). Convex potential decomposition (Theorem 3) plus analytic perturbation bound (Theorem 4). No interval arithmetic. The perturbation ratio is $<0.043$ at $u=0$ (worst case), with $95.7\%$ margin.    □

Remark 3 

(Relation to the Riemann Hypothesis).   Theorem 1 establishes ${\mathrm{TP}}_2$ (log-concavity), which is a necessary condition for RH. The full RH is equivalent to ${\mathrm{TP}}_{\infty }$ (all Turán inequalities), which is strictly stronger. The gap between ${\mathrm{TP}}_2$ and ${\mathrm{TP}}_{\infty }$ is genuine: $e^{-t^4}$ is ${\mathrm{TP}}_2$ but not ${\mathrm{TP}}_{\infty }$ [4]. Establishing the higher-order Turán inequalities for Φ remains an important open problem.

9. Formal Verification

The algebraic core (Theorem 2) and the exponential decay estimates (Lemmas 3–4) have been formalised in the Lean 4 proof assistant (version 4.29.0) using the Mathlib library. The formalisation compiles with zero sorry declarations. The following table summarises the status of each component:

Result	Status	Method
$h\left(u\right)>0$ for $u\ge 0$	Machine-checked	nlinarith, $\pi >3$, $e^{2u}\ge 1$
${(logh)}^{″}<0$ for $u\ge 0$	Machine-checked	Sign of quotient
${(log{\phi }_1)}^{″}<0$ for $u\ge 0$	Machine-checked	Sum of negatives
$e^{-\pi n^2{e}^{2u}}\le e^{-3\pi }{e}^{-\pi e^{2u}}$	Machine-checked	$n^2-1\ge 3$, $e^{2u}\ge 1$
$e^{-3\pi }<1/100$	Machine-checked	Taylor bound $e^5>100$
$e^x\ge T_4\left(x\right)$ for $x\ge 0$	Axiomatised	Standard (integration of $1+x\le e^x$)
$\mathsf{\Xi }={\int }_0^{\infty }\mathsf{\Phi }cos$ representation	Axiomatised	[5]
$\mathsf{\Phi }>0$ and $\mathsf{\Phi }\in L^1[0,\infty )$	Axiomatised	Standard

The Lean 4 formalisation and the interval arithmetic verification script (verify.py) are available at https://github.com/gershonavi/xi-log-concavity.

10. Discussion

10.1. Relation to the de Bruijn–Newman Constant

The de Bruijn–Newman constant $\mathsf{\Lambda }$ is defined so that ${\mathsf{\Xi }}_{\lambda }\left(z\right):=\int \mathsf{\Phi }\left(u\right)e^{\lambda u^2}{e}^{izu}du$ has only real zeros for $\lambda \ge \mathsf{\Lambda }$. De Bruijn [2] proved $\mathsf{\Lambda }\le 1/2$; Rodgers and Tao [3] proved $\mathsf{\Lambda }\ge 0$. RH is equivalent to $\mathsf{\Lambda }=0$.

Log-concavity (${\mathrm{TP}}_2$) is a necessary condition for $\mathsf{\Lambda }\le 0$ but does not by itself imply it. The passage from ${\mathrm{TP}}_2$ to $\mathsf{\Lambda }=0$ requires the full ${\mathrm{TP}}_{\infty }$ condition (all Turán inequalities).

10.2. The ${\mathrm{TP}}_2$/${\mathrm{TP}}_{\infty }$ Gap

Log-concavity establishes total positivity of order 2 (${\mathrm{TP}}_2$). The Riemann Hypothesis requires total positivity of infinite order (${\mathrm{TP}}_{\infty }$). The gap is genuine: $e^{-t^4}$ is ${\mathrm{TP}}_2$ but not ${\mathrm{TP}}_{\infty }$ [4]. Bridging this gap for the specific kernel $\mathsf{\Phi }$, using the Euler product structure, the functional equation, or the Hecke eigenform properties of the theta function, remains an important open problem.

10.3. Why the $n=1$ Term Dominates

The superexponential decay $e^{-\pi n^2{e}^{2u}}$ ensures that higher-order terms are negligible for $u\ge 0$. Quantitatively:

u	$|{\phi }_2|/{\phi }_1$	$|{\phi }_3|/{\phi }_1$
0	$2.2\times {10}^{-3}$	$3.2\times {10}^{-9}$
$0.5$	$1.4\times {10}^{-10}$	$2.5\times {10}^{-29}$
1	$9.6\times {10}^{-30}$	$1.1\times {10}^{-85}$

By $u=0.5$, the $n=2$ term is ${10}^{-10}$ of $n=1$, and by $u=1$ it is ${10}^{-30}$. The only nontrivial verification is the interval $[0,0.5]$, where the sum of the first five terms suffices.

10.4. Reliability of the Interval Arithmetic

The computation uses the mpmath.iv module (version 1.3.0) for rigorous interval enclosures at 80-digit precision. Each arithmetic operation produces an interval $[a,b]$ that is guaranteed to contain the true value. The implementation follows IEEE 754 directed rounding conventions.

The computation is reproducible: the Python source code is provided alongside the Lean formalisation. The total runtime is under 10 minutes on a standard workstation.

10.5. Axiomatised Components

Three components are axiomatised rather than proved:

1.

The Taylor lower bound $e^x\ge {\sum }_{k=0}^4{x}^k/k!$ for $x\ge 0$. This follows from four applications of the identity ${\int }_0^x(1+t)dt\le {\int }_0^x{e}^{t}dt$, and is provable in Mathlib using the integration API.

2.

The representation (1). This is a standard result in analytic number theory (see e.g. Titchmarsh [5], Chapter 2).

3.

Positivity and integrability of $\mathsf{\Phi }$ on $[0,\infty )$. These are immediate from the explicit formula (2).

Each of these is a published theorem with a complete proof in the literature. Their formalisation in Lean/Mathlib is a valuable but separate project.