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THE CROSS-ENERGY OF DIRICHLET PARTIAL SUMS AND THE ZEROS OF THE RIEMANN ZETA FUNCTION

A mix of human-written and AI-generated textHuman understanding: all partsmath.NT — Number Theory

Contributed by Pedro Caceres

Version 2 / Oct 09, 2026 / CC BY 4.0

Abstract

Let X_n(s) denote the sum of k^(-s) for k from 1 to n, and define the cross energy C_2(n,s) as |X_n(s)|^2 − |X_(n−1)(s)|^2 − n^(−2σ). At every point of the critical strip, the limit of C_2(n,s) is 1/(1/4 + t^2), zero, or positive infinity, according as σ = 1/2, σ > 1/2, or σ < 1/2. This trichotomy reflects the pole at s = 1. On the critical line, the associated helix has a natural orientation whose signed radius is Hardy’s Z(t), and the zeros of the truncated functions spiral toward the zeros of the Riemann zeta function at rate n^(−β). We prove that the mean energy of the normalized error E(u) in the prime number theorem equals the sum, over zeros ρ, of the limits of C_2(n,ρ), with values in [0,∞]. This energy is at least 2 + γ₀ − log(4π), with equality if and only if the Riemann hypothesis (RH) holds and all zeros are simple. An unconditional local formulation expresses the energy in a Gaussian window as a double sum over zeros. We also study the pointwise inequality S(σ,t) = ∂/∂σ log|ξ(σ + it)| > 0, which is equivalent to RH. An off-line pair reduces S precisely inside a disc; at abscissa σ, the failure set up to height T has measure at most N(σ,T + 1). Moreover, S is Weil’s functional evaluated at a Poisson kernel, and its truncation admits an exact finite expression involving primes up to e^L. We test each criterion against the Davenport–Heilbronn function, which has zeros off the critical line, and against its Euler-product counterpart L(s,χ) modulo 5. The former violates each criterion as predicted, whereas the latter behaves like the zeta function. For zeta, primes up to 10^8 yield a mean energy of 0.0453, compared with 0.0462. Assuming RH, simple zeros, and a spacing condition, we determine the next-order term in the energy: the integral from 1 to X of (ψ(x) − x)^2/x^2 dx equals C_B log X + K + Φ(log X) + o(1), where K = 2.5125… and Φ describes almost-periodic interference between pairs of zeros. Prime data up to 10^7 track this term with correlation 0.999; random models of primes and of the Möbius function μ carry 20 to 150 times more energy. On the circle associated with the Euler factor of a prime p, the zeros favor angles near π, and near π + arg χ(p) for L(s,χ). Restoring the endpoint term (1/2)N^(−s), we prove, with explicit constants, that zeros of the truncations converge to zeros ρ of zeta along helices with radius |ρ| N^(−β−1)/(12|ζ′(ρ)|). For the Davenport–Heilbronn function, the convergence rates for the two zeros in an off-line pair differ by 2(β − 1/2). Finally, Weil’s form, using two windows at the endpoints of its support, detects an isolated zero at β = 1/2 + δ and height γ using prime powers up to (C log γ)^(1/δ). The corresponding positivity holds for zeta up to 3 × 10^6 but fails for a random multiplicative model. We do not prove the Riemann hypothesis.

Provenance statement

The research questions, the cross-energy C₂ (introduced by the author in 2017) and the overall program are the author's. The author used Claude (Anthropic), an AI assistant, in the Cowork environment, as a research and writing aid: to derive and check proofs, run and verify the numerical computations in Python, test claims against counterexamples, search the literature, draft and edit the LaTeX text and figures, and review earlier drafts critically. Other AI assistants (e.g. ChatGPT) were used by the author for exploratory discussion; their outputs were checked and only verified statements were retained. All results were checked by the author, who takes full responsibility for the content. No AI system is an author.

Tools used

Anthropic
Claude Sonnet
OpenAI
ChatGPT
Python
PythonVersion 3.13

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Version history

  1. v1Submitted by Pedro CaceresInitial depositSupersededOct 08, 2026
  2. v2Submitted by Pedro CaceresSubmitted as a new versionCurrentOct 18, 2026