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Barnes zeta function

In mathematics, a Barnes zeta function is a generalization of the Riemann zeta function introduced by {{harvs|txt|authorlink=E. W. Barnes|first=E. W. |last=Barnes|year=1901}}. It is further generalized by the Shintani zeta function.

Definition

The Barnes zeta function is defined by

: \zeta_N(s,w\mid a_1,\ldots,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

  • {{Citation | last1=Barnes | first1=E. W. | title=The Theory of the Double Gamma Function. [Abstract] | jstor=116064 | publisher=The Royal Society | year=1899 | journal=Proceedings of the Royal Society of London | issn=0370-1662 | volume=66 | pages=265–268 | doi=10.1098/rspl.1899.0101| s2cid=186213903 }}
  • {{Citation | last1=Barnes | first1=E. W. | title=The Theory of the Double Gamma Function | jstor=90809 | publisher=The Royal Society | year=1901 | journal=Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character | issn=0264-3952 | volume=196 | issue=274–286 | pages=265–387 | doi=10.1098/rsta.1901.0006| bibcode=1901RSPTA.196..265B | doi-access=free }}
  • {{citation|first=E. W. |last=Barnes|title=On the theory of the multiple gamma function|journal= Trans. Camb. Philos. Soc. |volume=19 |year=1904|pages=374–425}}
  • {{Citation | last1=Friedman | first1=Eduardo | last2=Ruijsenaars | first2=Simon | title=Shintani–Barnes zeta and gamma functions | doi=10.1016/j.aim.2003.07.020 | mr=2078341 | year=2004 | journal=Advances in Mathematics | issn=0001-8708 | volume=187 | issue=2 | pages=362–395| doi-access=free }}
  • {{Citation | last1=Ruijsenaars | first1=S. N. M. | title=On Barnes' multiple zeta and gamma functions | doi=10.1006/aima.2000.1946 | mr=1800255 | year=2000 | journal=Advances in Mathematics | issn=0001-8708 | volume=156 | issue=1 | pages=107–132| url=https://ir.cwi.nl/pub/2100 | doi-access=free }}

Category:Zeta and L-functions

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