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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. 2

%I

%S 6,12,1620,2160,2551500,3061800,33756345000,38578680000,

%T 4060381958325000,4511535509250000,3168740859543387253125000,

%U 3456808210410967912500000,34159303730702924635072148437500

%N 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.

%C 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.

%C 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.

%H Vaclav Kotesovec, <a href="/A159282/b159282.txt">Table of n, a(n) for n = 2..63</a>

%H R. Miles and T. Ward, <a href="https://doi.org/10.1090/S0002-9939-08-09649-4">Orbit-counting for nilpotent group shifts</a>, Proc. Amer. Math. Soc. 137 (2009), 1499-1507.

%F By Perron's formula, M(N) = residue(z=n-1, zeta(z+1)...zeta(z-n+2)N^z) = (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).

%e 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^2N^2, so a(3)=12.

%p residue(product(Zeta(z-j),j=-1..(n-2))*N^z/z,z=n-1) # generates an expression from which a(n) can be read off

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

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

%K easy,frac,nonn

%O 2,1

%A Thomas Ward (t.ward(AT)uea.ac.uk), Apr 08 2009

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Last modified June 4 10:50 EDT 2020. Contains 334825 sequences. (Running on oeis4.)