Barnes zeta function

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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[edit]

The Barnes zeta function is defined by

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[edit]

  • 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, doi:10.1098/rspl.1899.0101 
  • 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, doi:10.1098/rsta.1901.0006 
  • 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, MR 2078341, doi:10.1016/j.aim.2003.07.020 
  • Ruijsenaars, S. N. M. (2000), "On Barnes' multiple zeta and gamma functions", Advances in Mathematics, 156 (1): 107–132, ISSN 0001-8708, MR 1800255, doi:10.1006/aima.2000.1946