A345267 Conjectural order of the torsion subgroup of the group K_n(Z) (the algebraic K-theory groups of the integers).

1, 2, 2, 48, 1, 1, 1, 240, 1, 2, 2, 1008, 1, 1, 1, 480, 1, 2, 2, 528, 1, 1, 691, 65520, 1, 2, 2, 48, 1, 1, 3617, 16320, 1, 2, 87734, 57456, 1, 1, 174611, 13200, 1, 2, 155366, 1104, 1, 1, 236364091, 131040, 1, 2, 1315862, 48, 1, 1, 3392780147, 6960, 1, 2

Offset: 0

Author

Tom Harris, Jun 12 2021

Keywords

• nonn
• easy

Comments

a(n) is known for n != 0 mod 4 and is related to the Bernoulli numbers via the Riemann zeta function. See Section VI.9 of Weibel's K-book.

K_0(Z) = Z is classical, so a(0) = 1. Rognes proved that K_4(Z) = 0 in 2000 so a(4) = 1.

Otherwise the value of a(4i) = 1 is conjectural. Kurihara observed that this follows from the Kummer-Vandiver conjecture (and in fact is equivalent with it). The Kummer-Vandiver conjecture has been verified for primes up to 163 million, from which it follows that a(4i) must be at least that large if it is not 1.

References

• C. Weibel, The K-book: An Introduction to Algebraic K-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. ISBN: 978-0-8218-91322.

Links

• M. Kurihara, Some remarks on conjectures about cyclotomic fields and K-groups of Z, Compositio Math. 81 (1992), 223-236.
• J. Rognes, K_4(Z) is the trivial group, Topology 30 (2000), 267-281.
• C. Weibel, The K-book: An Introduction to Algebraic K-theory. Graduate Studies in Mathematics, 145. American Mathematical Society, Providence, RI, 2013. ISBN: 978-0-8218-91322.

Crossrefs

Cf. A345225 (the order of the 2-primary subgroup, divides a(n)).

A000367 / A006863 (numerator / denominator of B_2n/4n, where B_m are the Bernoulli numbers).

Programs

Python

from sympy import bernoulli
def a(n):
    n_ = n % 8
    k = n//4 + 1
    b = bernoulli(2*k)/(4*k)
    d = {0:1, 1:2, 4:1, 5:1}
    if n_ == 2:
        return 2 * b.numerator
    elif n_ == 3:
        return 2 * b.denominator
    elif n_ == 6:
        return -1 * b.numerator
    elif n_ == 7:
        return b.denominator
    else:
        return d[n_]

Formula

Let k be the integer part of 1 + n/4.

a(n) is:

1 if n = 0 mod 8

2 if n = 1 mod 8

2*A000367(k) if n = 2 mod 8

2*A006863(k) if n = 3 mod 8

1 if n = 4 mod 8

1 if n = 5 mod 8

-1*A000367(k) if n = 6 mod 8

A006863(k) if n = 7 mod 8.