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
ζ N ( s , w ∣ a 1 , … , a N ) = ∑ n 1 , … , n N ≥ 0 1 ( w + n 1 a 1 + ⋯ + n N a N ) s {\displaystyle \zeta _{N}(s,w\mid a_{1},\ldots ,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 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, 66, The Royal Society: 265–268, doi:10.1098/rspl.1899.0101, ISSN 0370-1662, JSTOR 116064, S2CID 186213903 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 (274–286), The Royal Society: 265–387, Bibcode:1901RSPTA.196..265B, doi:10.1098/rsta.1901.0006, ISSN 0264-3952, JSTOR 90809 Barnes, E. W. (1904), "On the theory of the multiple gamma function", Trans. Camb. 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, MR 2078341 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, MR 1800255