A159283 Numerator of the rational coefficient in the main term in the dynamical analog of Mertens's theorem for a full n-dimensional shift, n >= 12 (it is 1 for 2 <= n <= 11).

691, 691, 691, 691, 2499347, 2499347, 109638854849, 109638854849, 19144150084038739, 19144150084038739, 1487175010978381361737, 1487175010978381361737, 351514769627820131218308186067

Offset: 12

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 numerator of that rational.

Examples

			For n = 12, using the formula in terms of residues, we have residue(zeta(z+1) * ... * zeta(z-10) * N^z/z, z=11) = (691/3168740859543387253125000) * zeta(3) * zeta(5) * zeta(7) * zeta(9) * zeta(11) * Pi^42 * N^11, so a(12) = 691 and A159282(12) = 3168740859543387253125000.
		

Links

• R. Miles and T. Ward, Orbit-counting for nilpotent group shifts, Proc. Amer. Math. Soc. 137 (2009), 1499-1507.

Crossrefs

This is the numerator of a rational sequence whose denominator is A159282.

Programs

Maple

# The following program generates an expression from which numerator 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

Numerator[Table[Residue[Product[Zeta[z - j], {j, -1, n-2}]/z, {z, n-1}][[1]], {n, 12, 24}]] (* Vaclav Kotesovec, Sep 05 2019 *)

Formula

M(N) = residue(zeta(z+1) * ... * zeta(z-n+2) * N^z, z=n-1) = (a(n)/b(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) = A159282(n).

Extensions

Various sections edited by Petros Hadjicostas, Feb 20 2021