cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A097916 Numerator of 2*zeta_K(-1) where K is the totally real field Q(sqrt(n)), as n runs through the squarefree numbers.

Original entry on oeis.org

1, 1, 1, 1, 4, 7, 7, 1, 10, 4, 2, 19, 2, 23, 20, 25, 1, 34, 40, 2, 46, 38, 5, 41, 52, 8, 18, 21, 74, 56, 26, 7, 92, 14, 33, 85, 11, 28, 16, 112, 41, 4, 134, 116, 22, 41, 4, 46, 56, 54, 43, 6, 155, 52, 26, 206, 6, 212, 172, 34, 19, 206, 76, 12, 87, 197, 9, 206, 244, 12, 88, 278, 277, 248
Offset: 1

Views

Author

N. J. A. Sloane, Sep 04 2004

Keywords

Examples

			1/6, 1/3, 1/15, 1, 4/3, 7/3, 7/3, 1/3, 10/3, 4, ...

References

  • F. Hirzebruch, Hilbert modular surfaces, Ges. Abh. II, 225-323.

Links

Crossrefs

Cf. A097917.

Programs

  • PARI
    z(d) = -(1/2)*bernfrac(2)*d*sum(k=1, d-1, kronecker(d, k)*subst(bernpol(2), x, k/d)*(-1/2))
{v=[]; for(k=2, 100, if(issquarefree(k), my(d=k); if(k%4 <> 1, d = 4*k); v=concat(v, numerator(2*z(d)) ))); v} \\ Thomas Scheuerle, Feb 28 2024
  • Sage
    [(round(60*QuadraticField(d).zeta_function(100)(-1).real())/30).numerator() for d in range(2, 100) if Integer(d).is_squarefree()]  # Robin Visser, Feb 28 2024

Extensions

More terms from Robin Visser, Feb 28 2024

A097914 Number of points in a certain complex manifold associated with the Hilbert modular group of the totally real field Q(sqrt(n)), n squarefree, that arise from isotropy groups of order 2.

Original entry on oeis.org

1, 2, 3, 2, 6, 4, 6, 10, 2, 12, 8, 4, 10, 4, 6, 12, 18, 6, 12, 12, 4, 12, 20, 2, 18, 16, 8
Offset: 1

Views

Author

N. J. A. Sloane, Sep 04 2004

Keywords

Links

Crossrefs

Cf. A097915, A097916, A097917, A097918.

A097915 Number of points in a certain complex manifold associated with the Hilbert modular group of the totally real field Q(sqrt(n)), n squarefree, that arise from isotropy groups of order 3.

Original entry on oeis.org

1, 2, 2, 2, 3, 4, 4, 4, 4, 4, 6, 2, 4, 5, 8, 8, 4, 6, 10, 4, 3, 4, 8, 8, 8, 10, 2
Offset: 1

Views

Author

N. J. A. Sloane, Sep 04 2004

Keywords

Links

Crossrefs

Cf. A097914, A097916, A097917, A097918.

A370411 Square array T(n, k) = denominator( 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).

Original entry on oeis.org

1, 75, 1, 16875, 24, 1, 221484375, 34560, 18, 1, 116279296875, 116121600, 58320, 39, 1, 12950606689453125, 780337152000, 440899200, 296595, 51, 1, 4861333986053466796875, 8899589151129600, 6666395904000, 68420017575, 663255, 63, 1, 677114376628875732421875
Offset: 0

Views

Author

Thomas Scheuerle, Feb 22 2024

Keywords

Examples

			The array begins:
           1,            1,             1,              1,                 1
          75,           24,            18,             39,                51
       16875,        34560,         58320,         296595,            663255
   221484375,    116121600,     440899200,    68420017575,       20126472975
116279296875, 780337152000, 6666395904000, 93393323989875, 10382542981248375

Links

Crossrefs

Cf. A370412 (numerators).
Cf. A003658, A080891, A097916, A097917, A230375, A370413.
Cf. A000191, A000411, A000464, A064072, A064075.
Cf. A002432 (denominators zeta(2*n)/Pi^(2*n)).
Cf. A046988 (numerators zeta(2*n)/Pi^(2*n)).
Coefficients of Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

Programs

  • PARI
    \p 700
row(n) = {v=[]; for(k=2, 30, if(isfundamental(k), v=concat(v, denominator(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, denominator((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.denominator() # Robin Visser, Mar 19 2024

Formula

T(n, k) = denominator( 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) = denominator((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) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000464(n+1) /((2*n)!^2 * 8^(2*n - 1)) ).
T(n, 2) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000191(n+1) /((2*n)!^2 * 12^(2*n - 1)) ).
T(n, 3) = denominator((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) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A000411(n+1) /((2*n)!^2 * 24^(2*n - 1)) ).
T(n, 7) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064072(n+1) /((2*n)!^2 * 28^(2*n - 1)) ).
T(n, 11) = denominator( 2^(n*4) * n^2 * zeta(1 - 2*n) * (-1)^n * A064075(n+1) /((2*n)!^2 * 40^(2*n - 1)) ).
T(n, k) = denominator( 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) = 1, 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).

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).

Original entry on oeis.org

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

Views

Author

Thomas Scheuerle, Feb 22 2024

Keywords

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

Links

Crossrefs

Cf. A370411 (denominators).
Cf. A003658, A080891, A097916, A097917, A230375, A370413.
Cf. A000191, A000411, A000464, A064072, A064075.
Cf. A002432 (denominators zeta(2*n)/Pi^(2*n)).
Cf. A046988 (numerators zeta(2*n)/Pi^(2*n)).
Coefficients of Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

Programs

  • 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).
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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