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| P-series | |
|---|---|
| Name | P-series |
| Field | Mathematics |
| Subtype | Infinite series |
| Notable | Leonhard Euler; Augustin-Louis Cauchy; Niels Henrik Abel |
P-series are infinite numerical series of the form ∑_{n=1}^∞ n^{-p} with real parameter p. They provide a fundamental family in analysis linking work by Leonhard Euler, Bernhard Riemann, Augustin-Louis Cauchy, and Niels Henrik Abel to questions of convergence, zeta functions, and analytic continuation. P-series serve as prototypes for comparison tests used in proofs by Joseph Fourier, Karl Weierstrass, Srinivasa Ramanujan, and Évariste Galois-era developments in series theory.
A P-series is typically denoted ∑_{n=1}^∞ n^{-p}, where p∈ℝ; common notation links to the Riemann zeta function ζ(p) when Re(p)>1 via work of Bernhard Riemann. Standard symbols include n^{-p}, p>0, and special values like ζ(2) evaluated by Leonhard Euler in correspondence with Christian Goldbach and results later used by Joseph-Louis Lagrange. Notation choices appear in texts by Augustin-Louis Cauchy, Karl Weierstrass, Richard Dedekind, and David Hilbert.
The basic convergence criterion: the series converges for p>1 and diverges for p≤1, a result established by comparison techniques employed by Jean le Rond d'Alembert and formalized by Augustin-Louis Cauchy. The integral test, attributed to Brook Taylor-era calculus and refined by Bernhard Riemann and Niels Henrik Abel, compares ∑ n^{-p} to ∫ x^{-p} dx used in analyses by George Boole and Oliver Heaviside. For p=1 the harmonic series diverges, a fact proved in demonstrations by Nicole Oresme and revisited by Leonhard Euler and Carl Friedrich Gauss. Conditional convergence phenomena and absolute convergence concepts are discussed in treatises by Srinivasa Ramanujan, Gustav Lejeune Dirichlet, and Karl Weierstrass when extending to complex p and linking to analytic continuation in the work of Bernhard Riemann and Hermann Weyl.
Classic special cases include p=1 (the harmonic series), p=2 where Euler computed ζ(2)=π^2/6 connecting to Johann Bernoulli-family investigations, and p=3 studied in correspondence of Leonhard Euler with Christian Goldbach. Other values tie to notable constants investigated by Srinivasa Ramanujan, Leonard Euler-type series, and the Apéry's theorem proof by Roger Apéry showing ζ(3) is irrational, discussed among Enrico Bombieri and Alan Baker. Alternating analogues lead to the Dirichlet eta function explored by Peter Gustav Lejeune Dirichlet and applied by Bernhard Riemann; p as complex variable ties directly to the Riemann zeta function and to conjectures examined by Andrew Wiles and Alan Turing in computational verifications. Finite truncations feature in numerical work by John von Neumann and Norbert Wiener and appear in approximations in texts by Isaac Newton and Brook Taylor.
Origins trace to medieval and early modern writings by Nicole Oresme and later systematic study in the 17th century by Johannes Kepler and Isaac Newton. The 18th century saw major advances by Leonhard Euler who evaluated ζ(2) and developed summation techniques communicated to Christian Goldbach and Daniel Bernoulli. 19th-century formalization by Augustin-Louis Cauchy and Bernhard Riemann introduced rigorous convergence tests and complex-variable extensions; subsequent contributions by Karl Weierstrass, Eduard Heine, and Georg Cantor influenced function-theoretic perspectives. 20th-century work by Roger Apéry, Atle Selberg, Harold Davenport, and Enrico Bombieri expanded arithmetic and analytic properties, with computational checks by Alan Turing and D. H. Lehmer.
P-series underpin comparison tests in analysis used by Augustin-Louis Cauchy and appear in spectral theory contexts studied by David Hilbert and John von Neumann. Values of ζ(p) relate to Fourier series work of Joseph Fourier and to partition function studies connected to Srinivasa Ramanujan and G. H. Hardy. Physics applications include regularization methods in quantum field theory examined by Paul Dirac and Richard Feynman, while analytic continuations link to scattering problems treated by Werner Heisenberg and Hermann Weyl. Generalizations include Dirichlet series of Peter Gustav Lejeune Dirichlet, L-functions central to Andrew Wiles's work on the Taniyama–Shimura conjecture and the Modularity theorem, and multivariate zeta functions appearing in research by Bernard Julia and Alexander Grothendieck. Related summation methods and transforms feature in the works of Joseph Fourier, Bernhard Riemann, Hjalmar Mellin, and John von Neumann.
Category:Mathematical series