[seqfan] Taniguchi's constant, perhaps worth submission

[Richard Mathar]

Taniguchi's constant is an Euler product mentioned in

S. Finch, <a href="http://www.inria.fr/csolve/clss.pdf">Class Number Theory</a>
Anonymous, <a href="http://en.wikipedia.org/wiki/Euler_product">Euler Product</a>, Wikipedia

product_{p=prime} (1+3/p^3+2/p^4+1/p^5-1/p^6)

0.678234491917391978035...

6, 7, 8, 2, 3, 4, 4, 9, 1, 9, 1, 7, 3, 9, 1, 9, 7, 8, 0, 3, 5, 5, 3, 8, 2, 7,
    9, 4, 8, 2, 9

apparently not in the OEIS. Perhaps someone can add more digits (?).
The calculation is simple in principle: take 1/(1-3*x^3+2*x^4+x^5-x^6), that
is the inverse Euler product above with p -> 1/x, then the Taylor series of
this (offset 0),  1,0, 0, 3, -2, -1, 10, -12, -2, 37, -57, 7, 137, -255,...
then the inverse Euler  transform of this sequence, which is
1, 0, 0, 3, -2, -1, 4, -6, 0, 13, -19, 2, 37, -67, 26, 112, -235,..
and build the infinite product of Riemann zeta functions with these exponents,
without the leading 1, product_{i>0} zeta(i)^coeff(i):
1/ [ zeta(3)^3 *zeta(4)^(-2) *zeta(5)^(-1) *zeta(6)^4 *zeta(7)^(-6)*... ]

For the likewise elusive Barban's constant I get 
2.5965362904505420736327406566
2, 5, 9, 6, 5, 3, 6, 2, 9, 0, 4, 5, 0, 5, 4, 2, 0, 7, 3, 6, 3, 2, 7, 4, 0, 6, 5, 6
which might also be worth an independent calculation.


Richard Mathar

# Maple of Taniguchi's constant
read("transforms") ; # http://www.oeis.org/classic/transforms.txt

b2 := 1-3/p^3+2/p^4+1/p^5-1/p^6 ;  

Digits := 130 ;
tm := 310 ;
subs (p=1/x,1/b2) ;
taylor(%,x=0,tm) :
L := [seq(coeftayl(%,x=0,i),i=1..tm-1)] ;
Le := EULERi(L) ;
x := 1.0 ;

for i from 2 to nops(Le) do
        x := x/evalf(Zeta(i))^op(i,Le) ;
        x := evalf(x) ;
        print(x) ;
end do: