A new bridge between prime number theory, information theory, and harmonic analysis — placing the deepest unsolved problem in mathematics at the exact boundary of signal reconstruction.
In 1859, Bernhard Riemann conjectured a pattern in the distribution of prime numbers that has resisted proof for 167 years. In 1949, Claude Shannon proved that every signal has a minimum sampling rate — the Nyquist limit — below which perfect reconstruction is impossible.
These two ideas were never supposed to meet.
This paper shows they are the same statement.
Starting from Li's criterion (1997) — which reformulates the Riemann Hypothesis as the non-negativity of a specific sequence — we derive a new representation of the problem 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.
This is not a numerical coincidence. It is an algebraic identity, emerging from a phase relation in the Li coefficients that had not been previously observed.
The implication is striking: the Riemann zeros encode the prime distribution at the theoretical maximum information rate, with zero redundancy. The Riemann Hypothesis is the assertion that this encoding is lossless — that no signal energy leaks outside the critical line.
RH holds if and only if F(t) alternates sign at every odd positive integer — a condition on the zero-crossing pattern of a single continuous function.
The sampling rate and bandwidth match exactly, placing RH at the boundary between signal determination and underdetermination.
The Li coefficient differences δ_n are positive for n ≤ 511, using a manifestly positive sine decomposition that requires no cancellation arguments for n ≤ 43.
Every component of Weil's 1948 proof of RH for curves over finite fields is mapped to a noncommutative geometric counterpart over ℚ, with the missing piece — the Hodge index theorem — identified precisely as the Weil positivity condition.
All applicable harmonic analysis tools — Beurling–Malliavin, de Branges, Kadec, Toeplitz–Fisher–Hartwig, Wiener–Hopf, and Tauberian methods — identifying the precise technical obstruction in each.
The prime-zero duality, in which every tool built from primes is controlled by zeros and vice versa, is identified as the meta-obstruction underlying all approaches to RH.
This work does not prove the Riemann Hypothesis.
What it does is open a door between number theory and information theory that was not known to exist. The Nyquist observation reframes RH as a question about optimal signal encoding — a language spoken by electrical engineers, information theorists, and physicists who may never have engaged with the zeta function. Sometimes progress on an impossible problem comes not from pushing harder, but from showing it to different eyes.
We propose a new research program — the information theory of arithmetic functions — in which the explicit formula is treated as a communication channel, the zeros as an encoded signal, and RH as a channel-capacity optimality statement.
The prime numbers have been speaking in a language we built our entire telecommunications infrastructure on. We just didn't recognize the accent.