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
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
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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);
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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
Comments