This article was accepted into the corpus but its outbound wikilinks were never NER-processed — typical at the deepest BFS hop or when the run's entity cap was reached. No expansion funnel to show.
| Johnston series | |
|---|---|
| Name | Johnston series |
| Field | Mathematics |
| Introduced | 19th century |
| Notable | Examples include arithmetic, geometric, and harmonic analogues |
Johnston series
The Johnston series is a class of mathematical sequences and series studied in connection with analytic number theory, functional analysis, and special functions. It intersects research on the Riemann zeta function, Euler–Maclaurin summation, and Fourier analysis, and has been cited in work involving the Hardy–Littlewood circle method and Ramanujan's mock theta functions.
The Johnston series refers to sequences defined by specified generating functions, Dirichlet series, or q-series whose coefficients obey recursive or multiplicative constraints studied by researchers following the traditions of Bernhard Riemann, Leonhard Euler, Srinivasa Ramanujan, Godfrey Hardy, and John Edensor Littlewood. Early treatments relate the Johnston series to the formalism of Dirichlet series, Laurent series, Taylor series, and q-series in the context of expansions used by Niels Henrik Abel and Augustin-Louis Cauchy. Modern formulations situate them alongside objects from the theories of Modular forms, L-functions, Bessel functions, and Gamma function techniques developed by Atle Selberg and Harold Davenport.
Studied initially in correspondence between contemporaries of Carl Friedrich Gauss and Adrien-Marie Legendre, the concept matured through work influenced by Bernhard Riemann's 1859 memoir, Leonhard Euler's zeta investigations, and later extensions by G. H. Hardy and J. E. Littlewood. In the 20th century, contributions by Atle Selberg, Hans Rademacher, Tom M. Apostol, and Hecke linked Johnston-type constructs to multiplicative number theory and the theory of Hecke operators. Research in the late 20th and early 21st centuries by groups around Andrew Wiles, Roger Heath-Brown, Ken Ono, and Don Zagier connected these series to modularity, mock modularity, and trace formulas inspired by the Selberg trace formula and the Atiyah–Singer index theorem.
Johnston series admit formulations as Dirichlet series A(s) = Σ a_n n^{-s}, as ordinary generating functions F(z)=Σ a_n z^n, or as q-series Σ a_n q^n whose coefficients a_n satisfy arithmetic or analytic constraints encountered in the work of Euler, Gauss, and Ramanujan. Key properties include analytic continuation akin to the Riemann zeta function, functional equations analogous to those for Dedekind eta function and Modular forms, and growth estimates studied with Phragmén–Lindelöf principle techniques used by Edmund Landau and Rudolf Lipschitz. Spectral interpretations employ operators in the spirit of Hecke operators and the Selberg trace formula, while asymptotic behavior is addressed via methods from Tauberian theorems and the Hardy–Littlewood circle method.
Important invariants include special values related to Bernoulli numbers, connections with the Euler–Maclaurin summation formula, and moments linked to central values of L-functions appearing in the work of Iwaniec and Kowalski. Convolution identities mirror those studied by Dirichlet and Ramanujan, and orthogonality relations reflect harmonic analysis on adelic groups and GL(2) automorphic representations developed by Jacquet and Langlands.
Classic examples echo familiar series studied by Euler and Dirichlet: arithmetic Johnston-type series produce analogues of the Dirichlet L-series and specializations yield analogues of the Riemann zeta function and Hurwitz zeta function. q-series special cases exhibit relationships to Ramanujan's mock theta functions and to the Dedekind eta function, while polynomially-weighted examples recover expansions similar to those for Bernoulli polynomials and Euler polynomials. Modular examples arise from Hecke eigenforms and theta series associated with quadratic forms studied by Carl Gustav Jacobi and Srinivasa Ramanujan. Spectral series analogues are comparable to Eisenstein series appearing in Selberg's work and Maass wave forms investigated by Hans Maass.
Johnston series are applied in analytic investigations of prime distribution via analogues of Chebyshev functions and in zero-density estimates for L-functions relevant to the Generalized Riemann Hypothesis research of Andrew Odlyzko and Hugh Montgomery. They surface in combinatorial partition theory in the lineage of Freeman Dyson and George Andrews, and in quantum modularity contexts explored by Zagier and Witten. Relations to representation theory involve Langlands program concepts and automorphic representations for GL(n), while computational aspects draw on algorithms influenced by Atkin–Lehner theory and explicit formulae akin to those used by Titchmarsh and Ivić.
See also works by Tom M. Apostol, G. H. Hardy, Srinivasa Ramanujan, and surveys connecting q-series, modular forms, and L-functions.
Category:Mathematical series