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

0,2

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

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

Table of n, a(n) for n=0..57.

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.

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.

Prog

(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_]

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).

Sequence in context: A015167 A086204 A318085 * A316090 A345463 A322106

Adjacent sequences: A345264 A345265 A345266 * A345268 A345269 A345270

Keyword

nonn,easy

Author

Tom Harris, Jun 12 2021

Status

approved