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A159283
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Numerator of the rational coefficient in the main term in the dynamical analog of Mertens' theorem for a full n-dimensional shift, n >= 12 (it is 1 for 2 <= n <= 11).
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1
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691, 691, 691, 691, 2499347, 2499347, 109638854849, 109638854849, 19144150084038739, 19144150084038739, 1487175010978381361737, 1487175010978381361737, 351514769627820131218308186067
(list; graph; refs; listen; history; internal format)
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OFFSET
| 12,1
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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.
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REFERENCES
| R. Miles and T. Ward, Orbit-counting for nilpotent group shifts, Proc. Amer. Math. Soc. 137 (2009), 1499-1507.
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FORMULA
| M(N)=residue((\zeta(z+1)...\zeta(z-n+2)N^z)/z,z=n-1)=(a(n)/b(n))N^{d-1}\pi^{\lfloor\frac{n}{2}\rfloor(\lfloor\frac{n}{2}\rfloor+1)}\prod_{j=1}^{\lfloor(n-1)/2\rfloor}\zeta(2j+1)
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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.
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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
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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
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KEYWORD
| easy,frac,nonn
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AUTHOR
| Thomas Ward (t.ward(AT)uea.ac.uk), Apr 08 2009
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