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Arithmetical Inversion Formulas

Published online by Cambridge University Press:  20 November 2018

Eckford Cohen
Eckford Cohen*
Affiliation:
University of Tennessee
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Let n and r be integers, r positive, and define the coreγ(r) of r to be the product of the distinct prime factors of r (γ(1) = 1). Let f(n,r) be a complex-valued, arithmetical function of n and r. If for all n,f﹛n,r) = f((n,r), r) then f(n, r) is called an even function (mod r), and if f(n,r) = f(γ(n, r), r) for all n, γ(n, r) = γ((n, r)), then f(n, r) is said to be a primitive function (mod r). Clearly, both classes of functions are subclasses of the periodic functions (mod r), while the primitive functions form a subclass of the even functions (mod r).

In a series of three papers (3; 5; 6) the author developed parallel, though interrelated, trigonometric and arithmetical theories of the even and primitive functions (mod r).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 399 - 409
Copyright
Copyright © Canadian Mathematical Society 1960

References

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3. Cohen, Eckford, A class of arithmetical functions, Proc. Nat. Acad. Sci., 41 (1955), 939–944.Google Scholar
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5. Cohen, Eckford, Representations of even functions (mod r), I. Arithmetical identities, Duke Math. J. 25 (1958), 401–421.Google Scholar
6. Cohen, Eckford, Representations of even functions (mod r), II. Cauchy products, Duke Math. J., 26 (1959), 165–182.Google Scholar
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