Heuristic formula for logarithm of the Frobenius morphism

TL;DR

The paper proposes a heuristic formula for the logarithm of the Frobenius morphism, linking it to the natural logarithm, and discusses its implications for the eigenvalues related to the zeta function's zeros.

Contribution

It introduces an explicit heuristic formula for the Frobenius morphism's logarithm, suggesting a connection to Hilbert's conjecture on zeta zeros.

Findings

The logarithm of Frobenius is given by x log x, independent of q.

The formula relates to eigenvalues of an operator conjectured by Hilbert.

Implications for understanding zeros of the zeta function.

Abstract

We show that the logarithm $\log_q$ of the Frobenius morphism $x\to x^q$ is given by the formula $x\to x\log x$ (the natural logarithm). In particular, it does not depend on $q$. This is the explicit (although heuristical) formula for the operator conjectured by Hilbert whose eigenvalues coincide with the zeroes of the zeta function.