Source Reconstruction in Neuroimaging and the Explicit Formula: An Exploratory Negative Assessment

Abstract

This essay examines the structural parallel between the electromagnetic inverse problem in electroencephalography and magnetoencephalography (EEG/MEG) and the Hardy-Littlewood explicit formula linking the Riemann zeros to the distribution of primes.

Overview

This paper explores why the linear inverse problem of electromagnetic source reconstruction in brain imaging fails to provide a meaningful bridge to the Riemann Hypothesis via the explicit formula. The source material (Leahy & Medani, arXiv:2607.17602v1) describes how EEG and MEG measurements require solving a forward problem (computing sensor fields from neural currents) and an inverse problem (reconstructing sources from noisy sensor data). The forward operator is a compact integral operator (the lead field matrix) that smooths high-frequency spatial information, making the inverse problem ill-posed and requiring regularization.

The suggested angle for this essay was to map this structure onto the explicit formula of prime number theory, which reconstructs the Chebyshev function ψ(x) (describing the distribution of primes) as a superposition of oscillatory terms x^ρ/ρ associated with the non-trivial zeros ρ of ζ(s). Superficially, both problems involve "reconstructing sources" (neural currents or zeta zeros) from "observable data" (sensor readings or prime counts).

We find that this analogy is a structural mismatch. The EEG inverse is ill-posed because the forward operator is compact (smoothing); the explicit formula is an exact identity, and recovering zeros from ψ(x) is a well-posed spectral estimation problem (Fourier inversion). The source domains differ fundamentally: continuous spatial fields versus a discrete set of complex frequencies. Consequently, regularization techniques from neuroimaging (Tikhonov, minimum-norm) have no direct counterpart in the theory of the Riemann zeta function. The essay rates the correspondence as a suggestive metaphor only and details the specific failure modes that prevent any substantive transfer of methodology.

This essay was produced by an automated research pipeline and has not been peer reviewed; conjectures herein are unproven.