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A Summation Formula Involving σk(n), k > 1
- Source :
- Canadian Journal of Mathematics. 21:951-964
- Publication Year :
- 1969
- Publisher :
- Canadian Mathematical Society, 1969.
-
Abstract
- The existence of certain formulae analogous to Poisson's summation formula (9, pp. 60-64),where αβ = 2π, α > 0, and Fc(x) is the Fourier cosine transform of f(x), but involving number-theoretic functions as coefficients, was first demonstrated by Voronoï (10) in 1904. He proved thatwhere r(n) is an arithmetic function,/(x) is continuous in (a, b) and a(x) and i?(x) are analytic functions dependent on τ(n). Later, numerous papers were published by various authors giving formulae of this type involving d(n), the number of divisors of n (3), and rp(n), the number of ways of expressing n as the sum of p squares of integers (8).In 1937, Ferrar (4) developed a general theory of summation formulae, using complex analysis. Around that time, Guinand (5) also published papers where he developed the general theory from a different point of view. He applied the theory of mean convergence for the transforms of class L2(0, ∞ ). Later in 1950, Bochner (1) gave a general summation formula.
- Subjects :
- Class (set theory)
General Mathematics
010102 general mathematics
Divisor function
Type (model theory)
Poisson distribution
01 natural sciences
Combinatorics
symbols.namesake
0103 physical sciences
symbols
Arithmetic function
010307 mathematical physics
0101 mathematics
Voronoi diagram
Sine and cosine transforms
Mathematics
Analytic function
Subjects
Details
- ISSN :
- 14964279 and 0008414X
- Volume :
- 21
- Database :
- OpenAIRE
- Journal :
- Canadian Journal of Mathematics
- Accession number :
- edsair.doi...........50b048a7a29bb950cda4973187450e39