Barnes zeta function

In mathematics, a Barnes zeta function is a generalization of the Riemann zeta function introduced by E. W. Barnes (1901). It is further generalized by the Shintani zeta function

Definition

The Barnes zeta function is defined by

$\zeta _{N}(s,w|a_{1},...,a_{N})=\sum _{n_{1},\dots ,n_{N}\geq 0}{\frac {1}{(w+n_{1}a_{1}+\cdots +n_{N}a_{N})^{s}}}$

where w and a_j have positive real part and s has real part greater than N.

It has a meromorphic continuation to all complex s, whose only singularities are simple poles at s = 1, 2, ..., N. For N = w = a₁ = 1 it is the Riemann zeta function.

References

• Barnes, E. W. (1899), "The Theory of the Double Gamma Function. [Abstract]", Proceedings of the Royal Society of London, 66, The Royal Society: 265–268, ISSN 0370-1662
• Barnes, E. W. (1901), "The Theory of the Double Gamma Function", Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 196, The Royal Society: 265–387, ISSN 0264-3952
• Barnes, E. W. (1904), "On the theory of the multiple gamma function", Trans. Cambridge Philos. Soc., 19: 374–425
• Friedman, Eduardo; Ruijsenaars, Simon (2004), "Shintani–Barnes zeta and gamma functions", Advances in Mathematics, 187 (2): 362–395, ISSN 0001-8708, MR2078341
• Ruijsenaars, S. N. M. (2000), "On Barnes' multiple zeta and gamma functions", Advances in Mathematics, 156 (1): 107–132, ISSN 0001-8708, MR1800255