A276048 - OEIS

%I #21 Sep 12 2016 17:15:40

%S 0,2,9,2,625,1,117649,2,9,1,25937424601,1,23298085122481,1,1,2,

%T 48661191875666868481,1,104127350297911241532841,1,1,1,

%U 907846434775996175406740561329,1,625,1,9,1,88540901833145211536614766025207452637361,1

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

%C 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.

%H Peter Humphries, <a href="http://math.stackexchange.com/a/1901492/8530">What completes the Dirichlet generating function ζ(s+c-1) where c is a constant?</a>, Math StackExchange.

%H Peter Humphries, <a href="http://math.stackexchange.com/a/1895154/8530">What is the symmetric functional equation of the Dirichlet eta function?</a>, Math StackExchange.

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

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

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

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

%t 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[%]

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

%K nonn

%O 1,2

%A _Mats Granvik_, Aug 17 2016