A276048 Sequence associated with the functional equation of the Riemann zeta zero spectrum (see formulas).

0, 2, 9, 2, 625, 1, 117649, 2, 9, 1, 25937424601, 1, 23298085122481, 1, 1, 2, 48661191875666868481, 1, 104127350297911241532841, 1, 1, 1, 907846434775996175406740561329, 1, 625, 1, 9, 1, 88540901833145211536614766025207452637361, 1

Offset

1,2

Comments

The functional equation formula in the answer by Peter Humphries is for the Dirichlet eta function and corresponds to the second term in this sequence. This sequence corresponds to zeta function products over all the divisors.

Links

Table of n, a(n) for n=1..30.

Peter Humphries, What completes the Dirichlet generating function ζ(s+c-1) where c is a constant?, Math StackExchange.

Peter Humphries, What is the symmetric functional equation of the Dirichlet eta function?, Math StackExchange.

Formula

a(n) = exp(lim_{s->1} zeta(s)*Sum_{d|n} mu(d)*d^(1 - s)*Sum_{d|n} mu(d)*d^(s)).

a(n) = A014963(n)^(A014963(n)-1), n > 1.

a(n) = A014963(n)^(-A120112(n)), n > 1.

a(prime(n)) = A000169(prime(n)).

Mathematica

Clear[s]; -Table[Limit[Zeta[s]*Total[MoebiusMu[Divisors[n]]*Divisors[n]^(1 - (s))]*Total[MoebiusMu[Divisors[n]]*Divisors[n]^(s)], s -> 1], {n, 1, 30}]; Exp[%]

Crossrefs

Cf. A000169, A014963, A120112, A230283, A230284.

Sequence in context: A091943 A318511 A345299 * A339203 A179451 A124918

Adjacent sequences: A276045 A276046 A276047 * A276049 A276050 A276051

Keyword

nonn

Author

Mats Granvik, Aug 17 2016

Status

approved