A106398 Binomial transform of denominators in a zeta function.

1, -1, -6, -19, -39, -66, -98, -129, -172, -330, -908, -2502, -5955, -12107, -21447, -34063, -50415, -72754, -104236, -136279, -103796, 222458, 1424920, 4630232, 11295484, 21347890, 26638757, -6669206, -178424687, -735476377, -2213929254, -5647949871, -12781614932, -26019627953

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.

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.

References

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

Links

Table of n, a(n) for n=1..34.

Les Mathematiques.net, Formule 1/zeta(s) (in French)

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 Moebius function rules are: Given the domain N, the natural numbers 1, 2, 3, ..., mu(1) = 1; mu(n) = 0 if n has a square factor > 1; 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.

Mathematica

With[{s = Select[Table[MoebiusMu[n]*n, {n, 1, 55}], # != 0 &]}, Table[Sum[Binomial[n, k]*s[[k+1]], {k, 0, n}], {n, 0, Length[s]-1}]] (* Amiram Eldar, May 31 2025 *)

Crossrefs

Cf. A005117, A008683.

Sequence in context: A090381 A354343 A395007 * A179986 A054567 A096957

Adjacent sequences: A106395 A106396 A106397 * A106399 A106400 A106401

Keyword

sign

Author

Gary W. Adamson, May 01 2005

Extensions

More terms from Amiram Eldar, May 31 2025

Status

approved