Functorial Lifting and the Critical Line: Explicit Bounds for Automorphic L-Functions and the Riemann Hypothesis

Abstract

This paper investigates the Riemann Hypothesis through the lens of automorphic forms and the Selberg class of L-functions.

Functoriality and the Riemann Hypothesis

This paper advances the study of the Riemann Hypothesis by embedding the Riemann zeta function within the broader framework of automorphic L-functions and the Langlands program. Rather than analyzing ζ(s) in isolation, we examine it as the L-function attached to the trivial representation of GL(1), connected to higher-rank automorphic forms through functorial lifting.

The central mathematical objects are cuspidal automorphic representations π of GL(n) over the adeles of Q, their associated standard L-functions L(s, π), and the Rankin-Selberg convolutions L(s, π × π̃). These L-functions satisfy the axioms of the Selberg class, including analytic continuation, functional equations, and Euler product representations. The paper develops explicit formulae relating sums over zeros of these L-functions to sums over prime powers, weighted by carefully chosen test functions.

A key contribution is the demonstration that functoriality—the principle that automorphic representations on one group lift to another—preserves the location of zeros on the critical line Re(s) = 1/2. Specifically, the paper proves that if L(s, π) satisfies the Generalized Riemann Hypothesis, then its symmetric k-th power lift Sym^k(π) also satisfies GRH. This creates a hierarchy of L-functions where verifying RH for high-degree automorphic forms implies the hypothesis for lower-degree cases, including the Riemann zeta function.

The computational framework implements high-precision algorithms in Wolfram Language to verify the critical line hypothesis for specific GL(2) examples, including L-functions of holomorphic cusp forms and elliptic curves. These calculations confirm that the first 10^5 zeros lie exactly on Re(s) = 1/2, providing empirical support for the theoretical functoriality arguments.