Integral Transforms and Special Functions
Vol. 20, No. 9, September 2009, 673–686
The discrete universality of the periodic Hurwitz zeta function
A. Laurinˇ cikas
a
* and R. Macaitien˙ e
b
a
Vilnius University, Naugarduko 24 03225,Vilnius, Lithuania;
b
Šiauliai University, Višinskio 19 77156,
Šiauliai, Lithuania
(Received 11 November 2007 )
The periodic Hurwitz zeta function ζ(s,α; A), s = σ + it,0 <α ≤ 1, is defined, for σ> 1, by ζ(s,α; A) =
∑
∞
m=0
a
m
/(m + α)
s
and by analytic continuation elsewhere. Here {a
m
} is a periodic sequence of complex
numbers. In this paper, a discrete universality theorem for the function ζ(s,α; A) with a transcendental
parameter α is proved. Roughly speaking, this means that every analytic function can be approximated
uniformly on compact sets by shifts ζ(s + imh, α; A), where m is a non-negative integer and h is a fixed
positive number such that exp{2π/h} is rational.
Keywords: periodic Hurwitz zeta function; limit theorem; universality
2000 Mathematics Subject Classification: 11M41; 60F17; 33E20
1. Introduction
Let A ={a
m
: m ∈ N
0
}, where N
0
denotes a set of all non-negative integers, be a periodic sequence
of complex numbers with minimal period k ∈ N. The periodic Hurwitz zeta function ζ(s,α; A),
s = σ + it, with parameter α,0 <α ≤ 1, is defined, for σ> 1, by
ζ(s,α; A) =
∞
m=0
a
m
(m + α)
s
.
The periodicity of the sequence A implies, for σ> 1, the equality
ζ(s,α; A) =
1
k
s
k−1
l =0
a
l
ζ
s,
k + α
k
, (1)
*Corresponding author. Email: antanas.laurincikas@maf.vu.lt
ISSN 1065-2469 print/ISSN 1476-8291 online
© 2009 Taylor & Francis
DOI: 10.1080/10652460902742788
http://www.informaworld.com