Russian Journal of Mathematical Physics

, Volume 20, Issue 2, pp 129–137 | Cite as

Values of twisted Barnes zeta functions at negative integers

  • A. BayadEmail author
  • Y. Simsek


In this paper, we study analytical and arithmetical properties of the twisted zeta function \(\Gamma (s)^{ - 1} \int_0^\infty {e^{ - xt} t^{s - 1} } \prod\nolimits_{j = 1}^N {\frac{{a_j t - \log (w^a j)}} {{1 - w^{a_j } e^{a_j t} }}dt} \), where ℜ(s) > N, ℜ(x) > 0, w ∈ ℂ\{0}, N ∈ ℤ, and a 1, …, a N have positive real parts. These functions have many interesting properties. We prove a collection of fundamental identities satisfied by zeta functions of this kind. For instance, special values of these zeta functions are related to twisted Barnes numbers and polynomials. This gives us a new elementary approach to new and known results concerning the Barnes zeta functions. In particular, we derive some well-known results on the Hurwitz zeta functions.


Zeta Function Negative Integer Positive Real Part Arithmetical Property Euler Polynomial 
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© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Département de mathématiquesUniversité d’Evry Val d’Essonne, Bâtiment I.B.G.B.I.Evry CedexFrance
  2. 2.Faculty of Arts and Science, Department of MathematicsUniversity of AkdenizAntalyaTurkey

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