Automorphic Hamiltonians, Epstein Zeta Functions, and Kronecker Limit Formulas

TL;DR

This paper explores automorphic Hamiltonians and Epstein zeta functions, providing new analytical tools and spectral characterizations to advance understanding of automorphic forms and related zeta functions.

Contribution

It constructs an automorphic Hamiltonian with discrete spectrum, characterizes its ground state, and develops methods to analyze zeros of zeta functions associated with quadratic fields.

Findings

Constructed an automorphic Hamiltonian with purely discrete spectrum.

Reproved meromorphic continuation of Eisenstein series.

Proved a zero spacing result for zeta functions of quadratic fields.

Abstract

First, we recount a history of how certain methods using natural self-adjoint operators have, thus far, failed to prove the Riemann Hypothesis. In Section 2, we set the analytical context necessary to have genuine proofs in later sections, rather than attractive heuristics. In Section 3, we recall the utility of designed pseudo-Laplacians by reproving meromorphic continuation of certain Eisenstein series and proving a spacing result for zeros of $\zeta_k(s)$ for $k$ a complex quadratic field with negative determinant. In Section 4, we construct an automorphic Hamiltonian which has purely discrete spectrum on $L^2(SL_r(\mathbb{Z})\backslash SL_r (\mathbb{R})/SO(r, \mathbb{R}))$, identify its ground state, and show how it can characterize a nuclear Fr\'echet automorphic Schwartz space.