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| Mertens function | |
|---|---|
| Name | Mertens function |
| Notation | M(x) |
| Domain | Integers |
| Codomain | Integers |
| Introduced | 1897 |
| Named after | Franz Mertens |
Mertens function The Mertens function is an arithmetic summatory function introduced by Franz Mertens in the late 19th century, defined by a cumulative sum over the Möbius function. It plays a role in analytic number theory, connects to hypotheses about the Riemann zeta function and prime distribution, and has been studied by computational projects and mathematicians across Europe and North America. Results about it relate to work by Bernhard Riemann, G. H. Hardy, John Littlewood, Atle Selberg, Enrico Bombieri, and others.
For a positive integer n, the function is given by the finite sum of values of the Möbius function up to n. The definition was proposed during a period that included contributions from Paul Lévy, Émile Borel, Srinivasa Ramanujan, and contemporaries in Berlin and Paris. The summatory form links classical sequences studied by Carl Friedrich Gauss, Adrien-Marie Legendre, and later by Chebyshev and Dirichlet.
The function takes integer values and alternates irregularly based on prime power factorization, a topic also central to the work of Leonhard Euler, Joseph-Louis Lagrange, and Adrien-Marie Legendre. It is bounded in trivial senses on finite intervals, and parity patterns echo investigations by Sophie Germain and Évariste Galois about multiplicative functions. Elementary identities connect it to sums and convolutions used by Dirichlet and in classical papers by Niels Henrik Abel.
By definition the function is the summatory function of the Möbius function, which itself is multiplicative and tied to inversion formulae from Augustin-Louis Cauchy and Peter Gustav Lejeune Dirichlet. Its generating Dirichlet series relates to the reciprocal of the Riemann zeta function studied by Riemann and later by Hadamard and de la Vallée-Poussin. Analytic continuations and Mellin transforms used by Hendrik Lorentz and G.H. Hardy appear in analyses that involve contour integrals familiar from Bernhard Riemann and J. E. Littlewood.
Questions about growth rates feature conjectures and disproofs historically analogous to disputes involving Poincaré and Hilbert problems. The now-refuted bound inspired by early commentators paralleled discussions by Mertens and critics including Odlyzko and Te Riele. Connections to zero distributions of Riemann zeta function zeros brought attention from Andrew Odlyzko, Alan Turing, Enrico Bombieri, and Hugh Montgomery. Applications of explicit formulae echo methods used by Atle Selberg and Grosswald.
Extensive computations have been carried out by researchers and projects at institutions like University of Illinois, Princeton University, Bell Labs, and national laboratories where staff such as Andrew Odlyzko and teams influenced by Alan Turing and John von Neumann contributed algorithms. Large-scale integer sequence tables maintained by groups including those around Neil Sloane and databases at Mathematical Sciences Research Institute catalog numerical values. Implementations used fast transforms and sieves related to techniques developed by Sieve of Eratosthenes refinements and computational advances by Donald Knuth and Peter L. Montgomery.
Explicit formulae connect summatory arithmetic functions to zeros of the Riemann zeta function, an approach rooted in the work of Riemann and elaborated by Guinand and Rudin. Relationships between zero-free regions, density hypotheses advanced by Selberg and Montgomery, and moment conjectures discussed by Keating and Snaith influence bounds on the function. Numerical verification of zero distributions by Odlyzko and analytic techniques from Ingham and Titchmarsh are central.
Variants include weighted summatory functions and higher-dimensional analogues studied in contexts connected to Dedekind zeta functions for number fields such as those investigated by Hecke and Weil, as well as general Möbius inversions in combinatorial settings traced to Rota and Stanley. Multivariable and automorphic generalizations relate to research by Langlands and Gelbart, while probabilistic models touch on ideas from Erdős and Kac. Recent work by authors affiliated with Princeton University, University of Cambridge, ETH Zurich, and research groups citing Zagier explore extensions in algebraic and spectral frameworks.
Category:Arithmetic functions