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| Möbius randomness law | |
|---|---|
| Name | Möbius randomness law |
| Field | Number theory |
| Introduced | Late 20th century |
| Main contributors | Hugh L. Montgomery; Peter Sarnak; Andrew Odlyzko; Harold Davenport; H. L. Montgomery; P. D. T. A. Elliott |
| Related | Riemann hypothesis; Sarnak conjecture; Möbius function; Liouville function |
Möbius randomness law The Möbius randomness law asserts that the Möbius function behaves like a sequence with strong pseudorandom properties, exhibiting cancellation against structured deterministic sequences. Formulated in modern terms by comparisons to conjectures in analytic number theory, the law connects deep questions about the distribution of primes and zeros of L-functions with dynamics, ergodic theory, and spectral theory.
The Möbius randomness law emerges from interactions among research traditions associated with Bernhard Riemann, G. H. Hardy, John Edensor Littlewood, Atle Selberg, Harald Cramér, Paul Erdős, Alfréd Rényi, and Wacław Sierpiński. It ties concepts developed in the work of Émile Borel, Alan Turing, Andrey Kolmogorov, Norbert Wiener, Salvatore Pincherle, and George Pólya to later formulations by Hugh L. Montgomery, Peter Sarnak, Andrew Odlyzko, and Harold Davenport. The law is motivated by classical results like the Prime Number Theorem and conjectures influenced by Bernhard Riemann’s 1859 memoir, and it has been framed alongside hypotheses such as the Riemann hypothesis and the Generalized Riemann Hypothesis studied by Atle Selberg and Alan Baker.
The principle is often stated informally as orthogonality between the Möbius function and any "low-complexity" deterministic sequence arising in settings studied by Marian von Smoluchowski, Yakov Sinai, Furstenberg, and Jean Bourgain. One canonical presentation relates sums of the Möbius function to correlations studied in the tradition of G. H. Hardy and J. E. Littlewood, while another formulation uses the language of dynamical systems as in work by Peter Sarnak and Mikhael Gromov. Precise formulations invoke objects from Iwaniec and Kowalski’s toolkit, including Dirichlet characters from Adrien-Marie Legendre’s lineage, automorphic forms studied by Robert Langlands, and spectral measures à la David Hilbert and John von Neumann.
The law is tightly connected to the Riemann hypothesis and the Generalized Riemann Hypothesis, to conjectures about L-functions developed by Atle Selberg and Robert Langlands, and to the Sarnak conjecture formulated by Peter Sarnak. It relates to the Prime Number Theorem and to pair-correlation conjectures investigated by Hugh L. Montgomery and computational explorations by Andrew Odlyzko. Links extend to the Mertens conjecture and to results in the style of Littlewood and J. E. Littlewood on sign changes, as well as to probabilistic heuristics used by Paul Erdős and Mark Kac.
Partial progress includes orthogonality results for special classes of sequences proved using tools from Terence Tao and Ben Green's work on linear equations in primes and correlations, extensions by Jean Bourgain in ergodic theory contexts, and results on nilsequence orthogonality developed by Terence Tao, Ben Green, and Terry Tao's collaborators. Computational verifications by Andrew Odlyzko and empirical data from H. L. Montgomery support aspects of the law in ranges accessible to numerical exploration; rigorous theorems include results conditional on the Generalized Riemann Hypothesis and unconditional results for restricted families inspired by techniques of Atle Selberg and Henryk Iwaniec.
Approaches draw on analytic methods from Atle Selberg’s trace formula, spectral techniques from Robert Langlands and H. P. McKean, ergodic methods from Hillel Furstenberg and Jean Bourgain, and combinatorial innovations associated with Paul Erdős, Ronald Graham, and Endre Szemerédi. Tools include exponential sum estimates inspired by I. M. Vinogradov and L. K. Hua, sieve methods from Atle Selberg and Brun, bilinear forms developed in the style of Heath-Brown and K. Soundararajan, and automorphic techniques influenced by Peter Sarnak and Henryk Iwaniec. Recent advances employ additive-combinatorics machinery from Ben Green and Terence Tao and ergodic-theoretic machinery from Eli Glasner and Michail Gromov.
If the Möbius randomness law holds in full generality, consequences would include refined error terms in the Prime Number Theorem as assessed by Bernhard Riemann’s framework, implications for zero-distribution of L-functions investigated by Hugh L. Montgomery and Andrew Odlyzko, and structural results for multiplicative functions akin to theorems studied by P. D. T. A. Elliott and Andrew Granville. The law informs rigidity results in dynamics explored by Peter Sarnak and Jean Bourgain, has potential bearings on pseudorandomness constructions used by Claude Shannon and John von Neumann, and could impact cryptographic assumptions alongside research by Whitfield Diffie and Ronald Rivest.
Central open problems include proving orthogonality to all deterministic sequences as conjectured by Peter Sarnak, establishing unconditional links to the Riemann hypothesis advocated by G. H. Hardy and J. E. Littlewood, extending nilsequence orthogonality results pioneered by Ben Green and Terence Tao, and developing new spectral or trace methods inspired by Robert Langlands and Atle Selberg. Active directions involve blending insights from Jean Bourgain’s harmonic analysis, Terence Tao’s probabilistic methods, computational verification by Andrew Odlyzko, and structural classification questions reminiscent of work by Paul Erdős and Pál Erdős.