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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

\( \zeta_N(s,w|a_1,...,a_N)=\sum_{n_1,\dots,n_N\ge 0}\frac{1}{(w+n_1a_1+\cdots+n_Na_N)^s} \)

where w and aj 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 (The Royal Society) 66: 265–268, ISSN 0370-1662, JSTOR 116064
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 (The Royal Society) 196: 265–387, ISSN 0264-3952, JSTOR 90809
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, doi:10.1016/j.aim.2003.07.020, ISSN 0001-8708, MR2078341
Ruijsenaars, S. N. M. (2000), "On Barnes' multiple zeta and gamma functions", Advances in Mathematics 156 (1): 107–132, doi:10.1006/aima.2000.1946, ISSN 0001-8708, MR1800255

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