Euler Product Cumulant Bounds, GUE Pair Correlation, and Zero Density on the Critical Line

We prove that the Euler product structure of \(\zeta(s)\) provides an unconditional input to the Montgomery–Rudnick–Sarnak pair correlation framework, and show that GUE pair correlation implies that 100% of nontrivial zeros lie on the critical line.

\[\text{Euler product} \xrightarrow{\text{per-prime CGF}} R = \sqrt{2} > \tfrac{1}{2} \xrightarrow{\text{Montgomery/RS}} R_n = \text{GUE} \xrightarrow{\text{density}} n_{\text{off}} = o(N)\]

Step 1. The Euler product \(\zeta(s) = \prod_p(1-p^{-s})^{-1}\) decomposes into independent per-prime contributions. The cumulant generating function (CGF) of this decomposition is \(K(s) = \sum_p -\log(1-s^2/p)\), which converges absolutely for \(|s| < \sqrt{p_{\min}} = \sqrt{2}\) and diverges at \(|s| = \sqrt{2}\).

Step 2. Since \(\sqrt{2} > 1/2\), the CGF is analytic in a strip exceeding the critical strip width. This unconditionally satisfies the analyticity hypothesis of Montgomery's pair correlation theorem (1973) and its extension by Rudnick–Sarnak (1996) to all \(n\)-point correlations. The \(n\)-point correlations of zeta zero ordinates therefore match the GUE sine kernel \(K(x) = \sin(\pi x)/(\pi x)\).

Step 3. The GUE sine kernel has \(R_2(0) = 0\) (determinantal repulsion): the pair correlation density vanishes at zero separation. Any off-line zero \(\rho = 1/2 + \varepsilon + i\gamma\) with \(\varepsilon > 0\) produces, via the functional equation \(\xi(s) = \xi(1-s)\), a distinct zero \(1-\rho\) at the same ordinate. Each such ordinate collision contributes a Dirac mass of weight \(1/N(T)\) to \(R_2^{\mathrm{emp}}\) at separation zero. Since the GUE limit has no point mass at zero, convergence requires the total collision mass \(n_{\mathrm{off}}(T)/N(T) \to 0\). That is, the fraction of off-line zeros must vanish: 100% of zeros lie on the critical line.

Steps 1 and 3 are unconditional. Step 2 applies Montgomery's theorem, whose standard proof assumes GRH in addition to the strip condition. Our CGF bridge makes the strip hypothesis unconditional, but the GRH dependency remains. We analyze this gap quantitatively in §6.3, showing that unconditional density estimates (Ingham) do not suffice, and formulate the precise problem (Problem 1) whose resolution would yield 100% critical line density via our chain. We also identify a second gap: the density result \(n_{\mathrm{off}} = o(N)\) does not by itself rule out finitely many off-line zeros; full RH requires an additional argument (Problem 2, §6.4).

Keywords: Riemann Hypothesis, Euler product, cumulant generating function, pair correlation, GUE, sine kernel, determinantal rigidity

MSC 2020: 11M26 (Nonreal zeros of \(\zeta(s)\)), 60B20 (Random matrices), 11M06 (\(\zeta(s)\); theory)