A Schwarzian inequality for the Riemann zeta function and its real dynamics

Abstract

We prove a two-sided moment inequality for the Riemann zeta function: for $A_j(s)=\sum_{n\ge2}(\log n)^j n^{-s}$, one has $A_2(s)^2< A_1(s)A_3(s)<\tfrac32 A_2(s)^2$ for every $s>1$. The upper bound is equivalent to negativity of the Schwarzian derivative, whose endpoint limits are $6\gamma_1$ and $-\tfrac12(\log2)^2$. Our proof separates three discrete atoms from a truncated gamma integral and bounds the Euler--Maclaurin remainder explicitly; all finite inequalities used in the proof are certified by ball arithmetic. A convexity argument then shows that zeta composed with itself has exactly one fixed point on $(1,\infty)$, and every other orbit alternates toward $1$ and infinity. This resolves Conjecture~(i) of OEIS A344428. On the negative axis, we construct expanding two-branch horseshoes in every band $[-4k-2,-4k]$, $k\ge5$, and finite horseshoes with arbitrarily large symbolic alphabets. Consequently the supremum of entropy over compact negative-real invariant sets is infinite. We also enumerate exactly the two-cycles meeting $(0,1)$, one per trivial zero $-2m$ for $m\ge9$, and derive their asymptotics. For the Hurwitz family we prove a uniform numerator inequality for parameters in $(0,1]$, positive Schwarzian points for $1 MSC 2020:Primary 11M06, 37E05; Secondary 26D15, 11M35, 37B40. Version 2.0.0 (2026-09-05): Section 5 (Hurwitz family) upgraded from numerical observations to proved/certified propositions — uniform numerator inequality for parameters in (0,1], Schwarzian sign regions tied to the two positive roots of gamma_1(a), certified saddle-node and period-doubling parameters, explicit multiplier theorem with limiting constant gamma - 1/2. Archive contents (zip): LaTeX source, abstract, verification scripts with recorded logs and manifests (python-flint 0.9.0 / Arb; mpmath 1.3.0), expository proof notes, change log. See README.md and CHECKSUMS.txt.