Written through Claude Shannon International University in collaboration with the ColdAI Research Division.
We establish a novel connection between the Riemann Hypothesis (RH), Shannon–Nyquist sampling theory, and the harmonic analysis of almost-periodic functions. Starting from Li's criterion — which asserts that RH is equivalent to the non-negativity of a sequence {λn}n≥1 — we derive a reformulation of RH as a sign-alternation condition on an almost-periodic function F(t) evaluated at odd integers. The central discovery is that the frequencies of F lie in the interval (0, π/2) and the sampling rate at odd integers corresponds to exactly the Shannon–Nyquist critical rate for this bandwidth — a coincidence that is not approximate but algebraically exact, emerging from a fundamental identity in the theory of Li coefficients. This places RH at the precise boundary between signal determination and underdetermination in the sense of information theory, suggesting that the Riemann zeros constitute an optimally encoded representation of the prime distribution with zero redundancy. We further establish: (i) an unconditional proof that λn+1 − λn > 0 for all n ≤ 43 via a manifestly positive sine decomposition; (ii) a bootstrap extension to n ≤ 511 using tail-dominance arguments; (iii) a complete Weil-analogy dictionary mapping every component of Weil's proof of the function-field RH to the noncommutative geometry of the adele class space; (iv) an exhaustive evaluation of all applicable harmonic analysis tools (Beurling–Malliavin, de Branges, Kadec, Toeplitz–Fisher–Hartwig, Wiener–Hopf) identifying the precise obstructions in each; and (v) a characterization of the meta-obstruction underlying all approaches: the prime-zero duality.
The paper establishes that Li's criterion for the Riemann Hypothesis, when transposed into the frequency domain, places the conjecture at exactly the Shannon–Nyquist critical sampling rate. The frequencies φk = arctan(2γk) of the derived almost-periodic function F all lie strictly inside (0, π/2) and accumulate at π/2 — and the sampling spacing Δt = 2 at odd integers gives ωs = π/2, matching the bandwidth supremum exactly.
Theorems 1.1 and 1.2 together suggest that the Riemann zeros encode the prime distribution at exactly the Shannon capacity of the explicit formula, viewed as a communication channel. RH then asserts that this encoding is lossless — all signal energy resides at the Nyquist boundary, with zero redundancy.
We give an unconditional proof that δn = λn+1 − λn > 0 for all n ≤ 43 via a manifestly positive sine decomposition requiring no cancellation, and bootstrap this to n ≤ 511 using tail-dominance arguments. Numerical verification with 300 zeros at 30-digit precision confirms monotonicity and convexity of λn for n ≤ 100.
We exhaustively evaluate the applicable harmonic-analysis machinery (Beurling–Malliavin, de Branges, Kadec, Bochner, Toeplitz–Fisher–Hartwig, Wiener–Hopf, Tauberian theorems) and identify the precise obstruction in each. All obstructions reduce to a single meta-obstruction we term the prime-zero duality: every functional or operator built from the primes is itself controlled by the zeros it seeks to constrain.
We propose the information theory of arithmetic functions, in which RH appears as an optimality statement about the encoding capacity of the explicit formula, viewed as a communication channel — connecting number theory, information theory, and random matrix theory through a single structural identity.
11M26, 42A75, 94A20, 46L87, 11M06, 58B34
Salehi, S. (2026). The Riemann Hypothesis at the Nyquist Limit: A Bridge Between Prime Number Theory, Information Theory, and Harmonic Analysis. Claude Shannon International University.