The Prime Sum Identity Characterizes L-Function Zero Sequences

Abstract. We prove that the prime sum identity of Wright (2026c) — which expresses the trace Tr(LN) of the Riemann zero log-gas Laplacian as a sum over prime powers — is not merely a consequence of the zeros lying on the critical line, but rather characterizes sequences arising from L-function zeros among all sequences with the correct asymptotic density. The key is a scaling dichotomy: the exponential sum FN(p) = Σj=1 N p−iγj grows as O(N/log N) for L-function zero sequences (governed by the Guinand–Weil explicit formula) but as O(√N) for equidistributed sequences (by Weyl’s theorem). Since these two growth rates are asymptotically incompatible with the prime sum identity, the identity characterizes exactly those sequences satisfying the explicit formula. This places the connection between the Laplacian framework and the Riemann Hypothesis on structural rather than motivational footing: the prime sum identity is specific to L-function zeros, not a property of arbitrary real sequences. We identify the remaining step toward RH as a constraint from the functional equation (Open Problem 5.1), which is attacked in the companion paper Wright (2026e).