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| explicit formula (number theory) | |
|---|---|
| Name | Explicit formula (number theory) |
| Caption | Distribution of primes and zeros of L-functions |
| Field | Number theory |
| Introduced | 19th century |
| Key figures | Bernhard Riemann, Hugh Montgomery, Atle Selberg, John von Neumann, G. H. Hardy |
explicit formula (number theory)
The explicit formula in number theory connects spectral data of analytic objects with arithmetic data, relating zeros of zeta and L-functions to primes and prime powers. It provides precise equalities or trace-type relations used in the study of Riemann zeta function, Dirichlet L-function, Dedekind zeta function, Hasse–Weil L-function and other analytic continuations. The machinery underpins major results such as the Prime Number Theorem, links to the Riemann hypothesis, and informs approaches in analytic number theory, spectral theory, and automorphic forms.
The explicit formula arises from contour integration and Fourier transform techniques applied to complex-analytic objects like the Riemann zeta function, the Dirichlet eta function, and families of Hecke L-series. Its role is to express sums over prime powers in terms of sums over nontrivial zeros and poles, bringing together contributions of Bernhard Riemann, Hermann Weyl, Atle Selberg, John von Neumann, and later contributors such as Hugh Montgomery and Andrew Odlyzko. Variants appear in the contexts of Dedekind zeta function, Artin L-series, and the Selberg trace formula for modular forms and Maass forms.
The idea dates to Bernhard Riemann's 1859 memoir on primes where he sketched a relation between zeros of the Riemann zeta function and prime counting; later rigorous forms were developed by Edmond Landau, Godfrey Hardy, John Littlewood, and G. H. Hardy. In the 20th century, Atle Selberg formulated a trace-style explicit formula linking lengths of closed geodesics and eigenvalues on modular surfaces, inspiring cross-pollination with Harish-Chandra's harmonic analysis on Lie groups. The mid-20th century work of Alan Turing, Hugh Montgomery, Enrico Bombieri, and Andrew Weil refined explicit formulas for general L-functions and highlighted connections to the Riemann hypothesis and the Generalized Riemann Hypothesis.
A typical derivation begins with an analytic continuation and functional equation for an L-function such as the Riemann zeta function or a Dirichlet L-function, together with a Mellin or Fourier transform of a compactly supported test function. Contour integration around zeros and poles invokes residues at nontrivial zeros and trivial zeros, producing identities that relate sums over prime powers (via logarithmic derivatives) to sums over zeros. Key technical inputs trace to methods used by Bernhard Riemann, Edmond Landau, Harish-Chandra, and Atle Selberg and leverage distributions and test functions as in Laurent Schwartz's theory.
Concrete instances include Riemann's original formula expressing the prime counting function π(x) in terms of the zeros of the Riemann zeta function, explicit class number formulas for Dedekind zeta functions of number fields as developed by Hecke and Emil Artin, and prime distribution results in arithmetic progressions using Dirichlet L-function explicit formulas as in work by Peter Dirichlet and Hadamard. Selberg-type trace formulas yield spectral identities for compact quotients of SL(2,R), applied by Atle Selberg and Harish-Chandra to the theory of modular forms and equidistribution problems studied by S. Lang and H. Iwaniec.
The explicit formula requires analytic continuation and a functional equation for the underlying L-function as proved for many cases by Riemann, Hecke, Godfrey Hardy, Erich Hecke, and André Weil. Convergence and error-term control use test functions with prescribed decay and support, invoking estimates from Vinogradov, I. M. Vinogradov, H. L. Montgomery, and results about zero-free regions from Charles de la Vallée Poussin and Hadamard. When automorphic methods apply, input from Langlands program conjectures and Harish-Chandra's Plancherel theorem also appears.
Generalized explicit formulas have been established for Dirichlet L-functions (Dirichlet characters, primes in arithmetic progressions), Dedekind zeta functions (number fields and class numbers), Artin L-series (Galois representations and Frobenius elements), and Hasse–Weil L-functions attached to elliptic curves and higher-dimensional algebraic varieties. Work by Emil Artin, A. Selberg, Andrew Weil, Pierre Deligne, and Robert Langlands provided frameworks tying monodromy, Frobenius conjugacy classes, and automorphic spectra to prime-counting terms appearing in explicit formulas.
Explicit formulas yield quantitative primes results: proofs and refinements of the Prime Number Theorem by Hadamard and Charles de la Vallée Poussin, zero-density estimates due to H. L. Montgomery and Atle Selberg, distribution of primes in arithmetic progressions by Dirichlet and L. J. L. Brower-style developments, and links between zero statistics and random matrix theory as investigated by Hugh Montgomery, Freeman Dyson, and Michael Berry. They underpin conditional results dependent on the Riemann hypothesis and the Generalized Riemann Hypothesis, affecting bounds in areas studied by Enrico Bombieri, Daniel Goldston, and János Pintz.