A159282 Denominator of the rational coefficient in the main term in the dynamical analog of Mertens's theorem for a full n-dimensional shift, n >= 2.

6, 12, 1620, 2160, 2551500, 3061800, 33756345000, 38578680000, 4060381958325000, 4511535509250000, 3168740859543387253125000, 3456808210410967912500000, 34159303730702924635072148437500

Offset: 2

Author

Thomas Ward, Apr 08 2009

Keywords

• easy
• frac
• nonn

Comments

a(n) for n >= 2 may be defined as follows. For a full n-dimensional shift, let M(N) = Sum_{L} O(L)/exp(h[L]), where the sum is over subgroups L of finite index in Z^n, O(L) is the number of points with stabilizer L, and exp(h) is the number of symbols.

Then M(N) is asymptotic to a rational times a power of Pi times a product of values of the zeta function at odd integers, and a(n) is the denominator of that rational.

Examples

			For n = 3, using the formula in terms of residues, we have residue(zeta(z-1) * zeta(z) * zeta(z+1) * N^z/z, z=2) = (1/12) * zeta(3) * Pi^2 * N^2, so a(3) = 12 (and A159283(3) = 1). [Because A159283(n) = 1 for n = 2..11, these ten values are not listed in the OEIS.]
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 2..63
• R. Miles and T. Ward, Orbit-counting for nilpotent group shifts, Proc. Amer. Math. Soc. 137 (2009), 1499-1507.

Crossrefs

This is the denominator of a rational sequence whose numerator is A159283.

Programs

Maple

# The following program generates an expression from which denominator a(n) can be read off:
f:=n->residue(product(Zeta(z-j),j=-1..(n-2))*N^z/z,z=n-1):
seq(f(n), n=2..30);

Mathematica

Denominator[Table[Residue[Product[Zeta[z - j], {j, -1, n-2}]/z, {z, n-1}], {n, 2, 14}]] (* Vaclav Kotesovec, Sep 05 2019 *)

Formula

By Perron's formula, M(N) = residue(zeta(z+1) * ... * zeta(z-n+2) * N^z, z=n-1) = (b(n)/a(n)) * N^(d-1) * Pi^(floor(n/2)*(floor(n/2)+1)) * Product_{j=1..floor((n-1)/2)} zeta(2*j+1), where b(n) = A159283(n).

Extensions

Various sections edited by Petros Hadjicostas, Feb 20 2021