1,3

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.

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

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 square-free deficient numbers (A087246), along with the Mobius function of each term.

The terms 1, 2, 3, 5, 6, 7... = A087246, square-free numbers. Applying the Mobius function rules to each of these, we get 1, -2, -3, -5, 6.... the Modius 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.

Cf. A087246, A008683.

Sequence in context: A031014 A010899 A090381 this_sequence A054567 A096957 A035495

Adjacent sequences: A106395 A106396 A106397 this_sequence A106399 A106400 A106401

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Gary W. Adamson (qntmpkt(AT)yahoo.com), May 01 2005