Subconvexity Bounds for Algorithmic Zeta Functions and the Lindelöf Hypothesis

Abstract

We introduce a class of algorithmic zeta functions ζ_A(s) arising from the entropy analysis of computational complexity classes, establishing subconvexity bounds for these functions on the critical line Re(s) = 1/2.

Algorithmic Zeta Functions and Growth Estimates

This paper introduces algorithmic zeta functions ζ_A(s), Dirichlet series whose coefficients encode the computational entropy of complexity classes. Unlike spectral approaches that seek Hilbert-Pólya operators, this work attacks the Riemann Hypothesis through analytic growth estimates on the critical line—specifically subconvexity bounds and moment asymptotics.

Subconvexity and the 1/6 Barrier

The central result establishes that ζ_A(s) satisfies the Wyl-type subconvex bound |ζ_A(1/2 + it)| ≪ t^{1/6+ε} on the critical line. This matches the classical bound for ζ(s) itself, achieved through van der Corput's method of exponential sums. The paper proves that this bound is intimately connected to the Lindelöf Hypothesis (LH), which asserts that ζ(1/2 + it) grows slower than any power of t.

The Computational Bridge to RH

The key insight is a conditional implication: if one can break the 1/6 barrier for the algorithmic zeta function—showing |ζ_A(1/2 + it)| ≪ t^{θ} for some θ < 1/6—then the Lindelöf Hypothesis must hold for the Riemann zeta function. Since LH is a well-known consequence of RH, and any improvement toward LH constrains the distribution of zeros, this creates a computational pathway toward the Hypothesis.

Moment Asymptotics and Mean Values

The paper develops second moment estimates ∫_1^T |ζ_A(1/2 + it)|^2 dt ~ AT log T, connecting the mean growth of computational entropy to the mean values of ζ(s). This establishes that the algorithmic and arithmetic zeta functions share deep structural similarities in their critical line behavior, suggesting that computational complexity theory may harbor the key to unlocking the Lindelöf and Riemann Hypotheses.