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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
691, 691, 691, 691, 2499347, 2499347, 109638854849, 109638854849, 19144150084038739, 19144150084038739, 1487175010978381361737, 1487175010978381361737, 351514769627820131218308186067 (list; graph; refs; listen; history; text; internal format)
OFFSET

12,1

COMMENTS

b(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 b(n) is the numerator of that rational.

LINKS

Table of n, a(n) for n=12..24.

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

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).

EXAMPLE

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^42N^11, so b(12)=691.

MAPLE

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

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 *)

CROSSREFS

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

Sequence in context: A189683 A029825 A180320 * A106281 A127341 A135316

Adjacent sequences:  A159280 A159281 A159282 * A159284 A159285 A159286

KEYWORD

easy,frac,nonn

AUTHOR

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

STATUS

approved

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Last modified May 29 04:17 EDT 2020. Contains 334696 sequences. (Running on oeis4.)