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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). 1

%I

%S 691,691,691,691,2499347,2499347,109638854849,109638854849,

%T 19144150084038739,19144150084038739,1487175010978381361737,

%U 1487175010978381361737,351514769627820131218308186067

%N 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).

%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 numerator of that rational.

%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 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).

%e 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.

%p # The following program generates an expression from which numerator a(n) can be read off:

%p f:=n->residue(product(Zeta(z-j),j=-1..(n-2))*N^z/z,z=n-1):

%p seq(f(n), n=2..30);

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

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

%K easy,frac,nonn

%O 12,1

%A _Thomas Ward_, Apr 08 2009

%E Various sections edited by _Petros Hadjicostas_, Feb 20 2021

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Last modified September 20 22:20 EDT 2021. Contains 347596 sequences. (Running on oeis4.)