%I #22 Sep 21 2024 19:16:37
%S 1,-1,-6,-19,-39,-66,-98,-129,-172,-330,-908,-2502,-5955,-12107
%N Binomial transform of denominators in a zeta function.
%C 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.
%D John Derbyshire, "Prime Obsession, Bernhard Riemann and the Greatest Unsolved Problem in Mathematics", Joseph Henry Press, 2003, p. 249.
%H Les Mathematiques.net, <a href="http://www.les-mathematiques.net/phorum/read.php?5,1667406">Formule 1/zeta(s)</a> (in French)
%F 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.
%e 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:
%e Given the domain N, the natural numbers 1,2,3,...,
%e 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.
%Y Cf. A005117, A008683.
%K sign,more
%O 1,3
%A _Gary W. Adamson_, May 01 2005