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.

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3 thoughts on “Multiple zeta

  1. 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
    or
    z(-1) = 1+2+3+4+…. = -1/12.
    Regards

  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) +
    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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