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Dedekind Function

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The Dedekind psi-function is defined by the divisor product

psi(n)=nproduct_(p|n)(1+1/p),
(1)

where the product is over the distinct prime factors of n, with the special case psi(1)=1. The first few values are

psi(1)=1(1+1/1)
(2)
psi(2)=2(1+1/2)
(3)
psi(3)=3(1+1/3)
(4)
psi(4)=4(1+1/2)
(5)
psi(5)=5(1+1/5)
(6)
psi(6)=6(1+1/2)(1+1/3)
(7)
psi(7)=7(1+1/7)
(8)
psi(8)=8(1+1/2)
(9)
psi(9)=9(1+1/3)
(10)
psi(10)=10(1+1/2)(1+1/5),
(11)

giving 1, 3, 4, 6, 6, 12, 8, 12, 12, 18, ... (OEIS A001615).

Sums for psi(n) include

psi(n)=sum_(d|n)n([mu(d)]^2)/d
(12)
=sum_(d|n)dmu(n/d)^2,
(13)

where mu(n) is the Möbius function.

The Dirichlet generating function is given by

sum_(n=1)^(infty)(psi(n))/(n^s)=1/(1^s)+3/(2^s)+4/(3^s)+...
(14)
=(zeta(s)zeta(s-1))/(zeta(2s)),
(15)

where zeta(z) is the Riemann zeta function.

See also

Dedekind Eta Function, Distinct Prime Factors, Euler Product, Totient Function

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References

Cox, D. A. Primes of the Form x2+ny2: Fermat, Class Field Theory and Complex Multiplication. New York: Wiley, p. 228, 1997.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 96, 1994.Sloane, N. J. A. Sequence A001615/M2315 in "The On-Line Encyclopedia of Integer Sequences."

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Dedekind Function

Cite this as:

Weisstein, Eric W. "Dedekind Function." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DedekindFunction.html

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