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| von Mangoldt function | |
|---|---|
| Name | Von Mangoldt function |
| Notation | Λ(n) |
| Field | Number theory |
| Introduced | 1897 |
| Introduced by | Hans von Mangoldt |
von Mangoldt function is an arithmetic function introduced by Hans von Mangoldt used to encode the distribution of prime powers in multiplicative number theory. It plays a central role in the study of the Prime number theorem, the Riemann zeta function, and the distribution of prime numbers via explicit formulas connecting zeros of the zeta function to counting functions. Named for a German mathematician associated with the proof of explicit formulas, the function appears in classical results by Bernhard Riemann, G. H. Hardy, and J. E. Littlewood.
The von Mangoldt function is defined for positive integers n by Λ(n)=log p if n equals p^k for some prime p and integer k≥1, and Λ(n)=0 otherwise; this definition was formalized by Hans von Mangoldt in his work extending ideas of Bernhard Riemann and Pafnuty Chebyshev. It is supported on prime powers, multiplicative on coprime arguments in the trivial sense, and satisfies summatory identities linking Λ to the Möbius function μ and the logarithm via Dirichlet convolution: log n = sum_{d|n} Λ(d), a relation used by Dirichlet and later exploited by Leonhard Euler and Adrien-Marie Legendre. Under Dirichlet generating series, Λ appears in the identity −ζ′(s)/ζ(s)=sum_{n≥1} Λ(n)n^{−s}, connecting to the analytic properties studied by Riemann, Hadamard, and de la Vallée Poussin.
Summatory functions of Λ link directly to prime-counting functions such as π(x) and Chebyshev’s functions ψ(x) and θ(x), central objects in the proofs by Hadamard and de la Vallée Poussin of the Prime number theorem. The identity ψ(x)=sum_{n≤x} Λ(n) gives an alternative route to prime distribution results used by Paul Erdős and Atle Selberg in elementary proofs, and it appears in comparisons with π(x) studied by Jacques Hadamard and Charles Jean de la Vallée Poussin. In conditional results like the Riemann hypothesis, strong bounds on sums of Λ yield explicit error terms in estimates by John von Neumann and influence conjectures of G. H. Hardy and Littlewood regarding the sign changes and irregularities of π(x).
The von Mangoldt function is the kernel in explicit formulas connecting zeros of the Riemann zeta function and related L-functions to prime distributions, as originally developed by Bernhard Riemann and later refined by Hans von Mangoldt and Atle Selberg. Through −ζ′/ζ one derives explicit formulas expressing ψ(x) in terms of a main term, sums over nontrivial zeros of ζ(s), and contributions from trivial zeros and poles; these techniques were used by G. H. Hardy, John Littlewood, and Alan Turing in zero calculations. The function appears in complex-analytic estimates such as zero-density results by Atle Selberg, zero-free regions by de la Vallée Poussin, and Weil-style trace formulas employed by André Weil and in modern contexts by Pierre Deligne and Alexander Grothendieck in arithmetic geometry analogies.
Generalizations include weighted von Mangoldt-type functions appearing in the study of Dirichlet L-series for characters introduced by Johann Peter Gustav Lejeune Dirichlet and in prime distribution in arithmetic progressions investigated by John von Neumann and Dirichlet’s followers. Modified versions arise in the Selberg trace formula of Atle Selberg and in automorphic settings connected to the Langlands program of Robert Langlands. Variants Λχ(n) twisted by a Dirichlet character χ or by coefficients of modular forms occur in explicit formulas for L-functions studied by Hecke, Hecke, and Goro Shimura. In algebraic number theory, analogous functions appear in the study of ideal norms in number fields treated by Richard Dedekind and Emil Artin.
The von Mangoldt function is pivotal in proofs of the Prime number theorem via complex-analytic and elementary methods by Hadamard, de la Vallée Poussin, Atle Selberg, and Paul Erdős, serving as the summand whose control implies asymptotics for π(x). It is employed in sieve-theoretic arguments by Brun and modern sieve theorists such as Enrico Bombieri and John Friedlander to detect primes and prime powers, and it appears in zero-density and zero-free-region proofs used by Vinogradov and I. M. Vinogradov’s circle method adaptations. In additive problems, Λ is used to weight primes in the Goldbach conjecture investigations by Harald Helfgott and earlier work by Vinogradov, and in equidistribution results concerning primes in arithmetic progressions by Linnik and Bombieri–Vinogradov-type theorems by Enrico Bombieri and A. I. Vinogradov.