

A106398


Binomial transform of denominators in a Zeta function.


0



1, 1, 6, 19, 39, 66, 98, 129, 172, 330, 908, 2502, 5955, 12107
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OFFSET

1,3


COMMENTS

The formula 1/Zeta(s) = 1  1/2^s  1/3^s  1/5^s + 1/6^s is shown on p. 249 of Derbyshire and relies upon strategies pioneered by Euler.


REFERENCES

John Derbyshire, "Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics", Joseph Henry Press, 2003, p. 249.


LINKS



FORMULA

Given 1/Zeta(s) = 1  1/2^s  1/3^s  1/5^s + 1/6^s  1/7^s + 1/10^s  1/11^s ..., we apply the binomial transform to the terms [1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, ...], which is the set of squarefree numbers (A005117), along with the Mobius function of each term.


EXAMPLE

The terms 1, 2, 3, 5, 6, 7, ... = A005117, squarefree numbers. Applying the Mobius function rules to each of these, we get 1, 2, 3, 5, 6, .... The Mobius function rules are:
Given the domain N, the natural numbers 1,2,3,...,
Mu(1) = 1; Mu(n) of N = 0 if n has a square factor; Mu(n) = 1 if n is a prime or the product of an odd number of different primes; Mu(n) = 1 if n is the product of an even numbers of different primes.


CROSSREFS



KEYWORD

sign,more


AUTHOR



STATUS

approved



