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A106398
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Binomial transform of denominators in a Zeta function.
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0
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1, -1, -6, -19, -39, -66, -98, -129, -172, -330, -908, -2502, -5955, -12107
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OFFSET
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1,3
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COMMENTS
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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.
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REFERENCES
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John Derbyshire, "Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics", Joseph Henry Press, 2003, p. 249.
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LINKS
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Table of n, a(n) for n=1..14.
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FORMULA
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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 deficient numbers (A087246), along with the Mobius function of each term.
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EXAMPLE
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The terms 1, 2, 3, 5, 6, 7... = A087246, 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.
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CROSSREFS
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Cf. A087246, A008683.
Sequence in context: A031014 A010899 A090381 * A179986 A054567 A096957
Adjacent sequences: A106395 A106396 A106397 * A106399 A106400 A106401
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KEYWORD
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sign
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AUTHOR
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Gary W. Adamson, May 01 2005
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STATUS
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approved
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