A370412 Square array T(n, k) = numerator( zeta_r(2*n) * sqrt(A003658(k + 2)) / Pi^(4*n) ), read by antidiagonals, where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).
0, 2, 0, 4, 1, 0, 536, 11, 1, 0, 2888, 361, 23, 2, 0, 3302008, 24611, 1681, 116, 4, 0, 12724582576, 2873041, 257543, 267704, 328, 4, 0, 18194938976, 27233033477, 67637281, 3741352, 92656, 88, 1, 0, 875222833138832, 11779156811, 18752521534133, 1156377368, 479214352, 287536, 29, 2, 0
Offset: 0
Examples
The array begins:
0, 0, 0, 0, 0
2, 1, 1, 2, 4
4, 11, 23, 116, 328
536, 361, 1681, 267704, 92656
2888, 24611, 257543, 3741352, 479214352
3302008, 2873041, 67637281, 1156377368, 14816172016
12724582576, 27233033477, 18752521534133, 753075777246704, 16476431095568992
Crossrefs
Programs
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PARI
\p 700 row(n) = {v=[]; for(k=2, 50, if(isfundamental(k), v=concat(v, numerator(bestappr(sqrt(k)*lfun(x^2-(k%2)*x-floor(k/4), 2*n)/Pi^(4*n)))))); v} z(n,d) = if(n == 0, 0,(1/(-2*n))*bernfrac(2*n)*d^(2*n-1)*sum(k=1,d-1, kronecker(d, k)*subst(bernpol(2*n),x,k/d)*(1/(-2*n)))) row(n) = {v=[]; for(k=2, 100, if(isfundamental(k), v=concat(v, numerator((2^(n*4)*n^2*z(n,k))/((2*n)!^2 * (k^(2*n-1))))))); v} \\ more accuracy here -
Sage
# Only suitable for small n and k def T(n, k): discs = [fundamental_discriminant(i) for i in range(1, 4*k+10)] D = sorted(list(set(discs)))[k+1] zetaK = QuadraticField(D).zeta_function(1000) val = (zetaK(2*n)*sqrt(D)/(pi^(4*n))).n(1000).nearby_rational(2^-900) return val.numerator() # Robin Visser, Mar 19 2024
Formula
T(n, k) = numerator( 2^(n*4) * n^2 * zeta_r(1 - 2*n) /((2*n)!^2 * A003658(k + 2)^(2*n - 1)) ), where zeta_r is the Dedekind zeta-function over r and r is the real quadratic field with discriminant A003658(k + 2).
T(n, 0) = numerator((5^(-2*n)*(zeta(2*n, 1/5) - zeta(2*n, 2/5) - zeta(2*n, 3/5) + zeta(2*n, 4/5) ))*zeta(2*n)*sqrt(5)*Pi^(-4*n)). A sum of Hurwitz zeta functions with signs according A080891.
T(n, 1) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000464(n+1) /((2*n)!^2 * 8^(2*n - 1)) ).
T(n, 2) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000191(n+1) /((2*n)!^2 * 12^(2*n - 1)) ).
T(n, 3) = numerator((13^(-2*n)*(zeta(2*n, 1/13) - zeta(2*n, 2/13) + zeta(2*n, 3/13) + zeta(2*n, 4/13) - zeta(2*n, 5/13) - zeta(2*n, 6/13) - zeta(2*n, 7/13) - zeta(2*n, 8/13) + zeta(2*n, 9/13) + zeta(2*n, 10/13) - zeta(2*n, 11/13) + zeta(2*n, 12/13) ))*zeta(2*n)*sqrt(13)*Pi^(-4*n)). A sum of Hurwitz zeta functions with signs according the Dirichlet character X13(12,.).
T(n, 6) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000411(n+1) /((2*n)!^2 * 24^(2*n - 1)) ).
T(n, 7) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064072(n+1) /((2*n)!^2 * 28^(2*n - 1)) ).
T(n, 11) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064075(n+1) /((2*n)!^2 * 40^(2*n - 1)) ).
T(n, k) = numerator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * d(A003658(k+2)/4, n+1) /((2*n)!^2 * 40^(2*n - 1)) ), for all k where A003658(k+2) is a multiple of four (The discriminant of the quadratic field is from 4*A230375). d() are the generalized tangent numbers.
T(0, k) = 0, because for a real quadratic number field the discriminant D is positive, hence the Kronecker symbol (D/-1) = 1. This means the associated Dirichlet L-function will be zero at s = 0 inside the expression zeta_r(s) = zeta(s)*L(s, x).
Comments