Multiple zeta

The Riemann zeta function, introduced by Leonard Euler, is defined by

\zeta(k) = \sum n^{-k}

where the sum is over all positive integers n.

Euler also introduced a multivariate generalization of the zeta function

\zeta(k_1, \ldots, k_r) = \sum n_1^{-k_1}\cdots n_r^{-k_r}

where the sum is over all decreasing k-tuples of positive integers. This generalized zeta function satisfies the following beautiful identity:

 \zeta(a)\,\zeta(b) = \zeta(a, b) + \zeta(b, a) + \zeta(a+b)

The multivariate zeta function and identities such as the one above are important in number theory and are the subject of open conjectures.


3 thoughts on “Multiple zeta

  1. Marcial Fonseca

    Hi there, about this, could you produce an article easy enough to understand how this is possible:
    S = 1+2+3+4+5+… = -1/12
    z(-1) = 1+2+3+4+…. = -1/12.

  2. The definition of the Riemann zeta function given in this post only makes sense for arguments with real part greater than 1. However, it defines an analytic function of a complex argument that has a unique analytic extension to the rest of the complex plane (except 1). That extended function, when evaluated at -1, has the value -1/12.

  3. I think that only the result for 2-ary generalization of the zeta function is pretty. If I’ve worked it out right then the
    equivalent formula involving 3-ary zeta function generalization is:

    z(a)z(b)z(c) = z(a,b,c) + z(a,c,b) + z(b,a,c) + z(b,c,a) + z(c,a,b) + z(c,b,a) +
    z(a, b+c) +z (b+c, a) + z(b, a+c) + z(a+c, b) + z(c, a+b) + z(a+b, c) +

    Where I’ve written z() instead of \zeta(). For the four-ary formula, it gets much worse…

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