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

Author

Mats Granvik, Aug 17 2016

Keywords

• nonn

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

• 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.

Crossrefs

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

Programs

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[%]

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)).