THE PERIODIC ZETA-FUNCTION 1. Introduction

Fizikos ir matematikos fakulteto
Seminaro darbai,
’iauliu universitetas,
7, 2004, 5769
THE PERIODIC ZETA-FUNCTION
Algimantas Ambraziejus LAURUTIS, Darius ’IAUƒI UNAS
’iauliai University, P. Vi²inskio 19, 77156 ’iauliai, Lithuania;
e-mail: darius.siauciunas@centras.lt
Abstract. A survey of results on the periodic zeta-function is given. The
main attention is devoted to author's results: approximate functional equation,
mean square, fourth moment and joint limit theorems in functional
spaces.
Key words and phrases: functional equation, fourth moment, periodic zetafunction.
Mathematics Subject Classication: 11M41.
1. Introduction
The functions of complex variable in some half-plane dened by Dirichlet
series with coecients having a certain arithmetical sense are called the
zeta-functions or L-functions. They play an important role in the analytic
number theory. For example, the properties of the classical Riemman zeta
and Dirichlet L-functions are directly connected to the distribution of prime
numbers.
Now we dene the periodic zeta-function. Let A = {a m : m ∈ Z} be
a periodic sequence of complex numbers with period k > 0. So we have
a m+k = a m , m ∈ Z. The periodic zeta-function ζ(s; A) is dened, for σ > 1,
by
ζ(s; A) =
∞∑
m=1
a m
m s .
If A = {1} and k = 1, then the function ζ(s; A) reduces to the Riemann
zeta-function ζ(s). For σ > 1 we have that
ζ(s; A) =
∞∑
m=1
a m
m s = 1 k s
k∑
a q ζ ( s, q )
,
k
q=1

58 On the periodic zeta-function series
where ζ(s, α) is the Hurwitz zeta-function
ζ(s, α) =
∞∑
m=0
1
, σ > 1, 0 < α 1.
(m + α) s
Since ζ(s; α) is regular everywhere, except for a simple pole at s = 1 with
residue 1, the last equality gives the analytic continuation of ζ(s; A) to the
whole complex plane except, possibly, a simple pole at s = 1 with residue
a = 1 k
k∑
a m .
m=1
If a m = χ(m), where χ(m) is a Dirichlet character modulo k, then ζ(s; A)
becomes the Dirichlet L-function
L(s, χ) =
∞∑
m=1
χ(m)
m s , σ > 1.
So, the periodic zeta function is a generalization of the classical Riemann
zeta and Dirichlet L-functions.
The function ζ(s; A) was studied by many authors. We will mention some
their results. W. Schnee obtained [14] the functional equation for ζ(s; A). Let
B = {b m : m ∈ Z} with
b m = 1 ∑k−1
a j e −2πij m k ,
k
j=0
and let B ∗ = {b ∗ m, m ∈ Z} with b ∗ m = b −m . Then the function ζ(s; A)
satises the following functional equation [14]
ζ(1 − s; A) = ( k
2π
) sΓ(s) (
e
πi s 2 ζ(s; B) + e
−πi s 2 ζ(s; B ∗ ) ) .
As a special case of the periodic Lerch zeta-function, the function ζ(s; A) also
appears in [2], where its functional equation by another method is derived.
The papers [4] and [12] are devoted to Hamburger-type theorems for
ζ(s; A). We recall that Hamburger's theorem states that any Dirichlet series
which satises the same functional equation as the Riemann zeta-function
ζ(s), and some additional conditions, is a constant multiple of ζ(s). In [4]
it is proved that a special type of functional equations for Dirichlet series
is equivalent to periodicity of the coecients of these series. The theorem

A. A. Laurutis, D. ’iau£i unas 59
of [12] gives a complete characterization of Dirichlet series with periodic
multiplicative coecients which satisfy certain additional conditions.
The paper [3] of T. Funakura is devoted to Kronecker's limit formula for
ζ(s; A). Let
â r = 1 √
k
k∑
a m e −2πim r k
m=1
denote the Fourier transform of a m . Then it is proved that
(
)
â r
lim ζ(s; A) − √
s→1
k(s − 1)
= −√ 1 ∑k−1
k
+ π 2k
m=1
k−1
â r log ( sin πm k
∑
a m ctg πm k
m=1
+ ( â k
√
k
− a k
)
log 2,
)
+
âk √
k
γ 0
where γ 0 is Euler's constant.
The paper [5] is very rich by results on the function ζ(s; A). The statements
of them are complicated, therefore we will mention only some problems.
Let, for σ > 1,
Then it is proved [5] that
F (s, r k ) = ∞ ∑
m=1
e 2πim r q
m s .
ζ(s; A) = √ âk ζ(s) + 1 ∑k−1
√ â r F (s, r
k k k ).
Also, the Laurent expansion at s = 1 for ζ(s; A) is given with explicit form
for the rst three coecients. Similar expressions are obtained also for derivatives
ζ (l) (s; A).
Further, the powers
are considered. Here
ζ l (s; A) =
d l (m) =
∞∑
m=1
∑
d 1 ...d l =m
r=1
d l (m)
m s , σ > 1,
a d1 . . . a dl .

60 On the periodic zeta-function series
Using the properties of ζ(s; A) the Voronoï type formulae for d l (m) are obtained.
J. Steuding in [15] investigated the zeros distribution of ζ(s; A). To state
his results on zeros of ζ(s; A) we need some notations and denitions. Denote
the zeros of ζ(s; A) by ϱ = β + iγ. Moreover, let c A = max{|a m | : 1 m
k}, m A = min {1 m k : a m ≠ 0}, and
A(A) = c A m A
|a m
A | .
Then in [15] it was established that ζ(s; A) ≠ 0 for σ > 1 + A(A).
Dene the numbers â ± m, where â − m coincides with â m , and â + m is obtained
from â − m taking "+" in place "-" in the exponent. Let A ± = {â ± m} and
B(A) = max{A(A ± )}. Then in [15] it was noted that for σ < −B(A) the
function ζ(s; A) can only have zeros close to the negative real axis, if m A
+ =
m A
−, and close to the line
σ = 1 +
πt
log m A −
m A
+
if m A
+ ≠ m A
−. The zeros ϱ of ζ(s; A) with β < −B(A) are called trivial, and
other zeros of ζ(s; A) are nontrivial. So, nontrivial zeros lie in the vertical
strip
−B(A) σ 1 + A(A).
Denote by N(T ; A) the number of nontrivial zeros ϱ of ζ(s; A) with |γ| T
counted with multiplicity. Then in [15] it was proved that
N(T ; A) = T π log kT
√ + B log T.
2πlm A mA −m A
+
Moreover, from the asymptotic formulae for
∑
ϱ nontrivial
|γ|T
( 1)
β −
2
it follows that the most of nontrivial zeros are approximately symmetrically
distributed or lie close to the critical line σ = 1 2. If
lim
T →∞
∑
ϱ nontrivial
|γ|T
( 1) β − ≠ 0,
2

A. A. Laurutis, D. ’iau£i unas 61
then ζ(s; A) has innitely many zeros o the critical line.
Now dene for the nontrivial zeros ϱ = β + iγ of ζ(s; A) the functions
N + (c, T, A) = #{ϱ : β c, |γ| T },
N − (c, T, A) = #{ϱ : β c, |γ| T }.
Then the last result of [15] states that, for δ > 0,
N +( 1
2 + δ, T, A) + N −( 1
2 − δ, T, A) B log log T
= N(T, A).
δ log T
This means that the nontrivial zeros of ζ(s; A) are clustered around the critical
line σ = 2.
1
We else will mention an important result of J. Steuding [16] on the universality
of ζ(s; A). Let meas{A} denote the Lebesgue measure of the set
A ⊂ R, and let, for T > 0,
ν T (. . . ) = 1 meas{τ ∈ [0; T] : . . . },
T
where in place of dots a condition satised by τ is to be written. Suppose
that k is an odd prime, a m is not a multiple of a character modulo k, and
1
a k = 0. Let K be a compact subset of the strip {s ∈ C :
2
< σ < 1} with
connected complement, and let f(s) be a continuous function on K which is
analytic in the interior of K. Then, for any ε > 0,
lim inf
T →∞
ν T
Let, for r 0 and σ 1 2
(
max
s∈K
I r (T, σ) = I r (T, σ; A) =
∣
∣ζ(s + iτ; A) − f(s) ∣ < ε ) > 0.
∫ T
0
∣ ζ(σ + it; A)
∣ ∣
2r dt.
In [7] the quantity I 1 (T, σ) was begun to study using a simple approximation
by a Dirichlet polynomial for ζ(s; A). The length of this polynomial was too
large to obtain a precise asymptotics for I 1 (T, σ). Also, in [7] a limit theorem
for ζ(s; A) in the space of analytic functions was proved, i. e. it was obtained
that the probability measure
ν T
(
ζ(s + iτ; A) ∈ A
)
,
A ∈ B(H( ˆD)),
converges weakly to some probability measure on ( H( ˆD), B(H( ˆD)) ) as T →
∞. Here ˆD = {s ∈ C : 1 2
< σ < 1}, B(S) stands for the class of Borel sets

62 On the periodic zeta-function series
of the space S, and H(G) is the space of analytic on G functions equipped
with the topology of uniform convergence on compacta.
We recall the denition of the weak convergence of probability measures.
Let P n and P be probability measures on ( S, B(S) ) . Then P n converges to
P as n → ∞ if
∫
S
fdP n
∫
−→
n→∞
S
fdP
for any real bounded continuous function f on S.
A more general limit theorem related to the residue a was proved in [8].
2. An approximate functional equation
Now we discuss our results. The rst theorem is devoted to an approximate
functional equation for the function ζ(s; A). In some sense this theorem
has an auxiliary character because an approximate equation is the principal
tool for the investigation of the mean square of ζ(s; A).
Let
and
ψ(a) = cos π( a 2
2 − a − 1 )
8
cos πa
f(α, t) = t log ( 2π ) t +
t 2 − 7π 8 + πα2
2 + πl + πn − παl + 2πy(l − α),
2
and let, as usual, [u] denote the integer part of u.
Theorem 1 ([10]). Let t 1, y = ( t
2π ) 1 2, n = [y], r = [y − q k
Then, for 1 2 σ 1,
ζ(s; A) = k −s k∑
where
q=1
a q
∑
0mr
1
(m + q k )s
+k −s( 2π ) s−
1
k∑
2
e i(t+ π 4 )
t
+k −s( 2π ) σ
2
t
q=1
a q
∑
1mn
e −2πim q k
m 1−s
], l = n − r.
k∑
a q e if( q k ,t) ψ(2y − 2n + l − q k ) + k−s R(s, k),
q=1
R(s, k) = Bt σ 2 −1
k∑
|a q |.
q=1

A. A. Laurutis, D. ’iau£i unas 63
Theorem 1 is derived from an approximate functional equation for the
Hurwitz zeta function ζ(s, α), which gives an approximation for ζ(s, α) in
the critical strip {s ∈ C : 1 2 σ 1}. A similar result for σ = 1 2
only was
also obtained in [13]. As it was observed by A. Balanzario [1], the latter
theorem is a special case of the analogue of the Riemann-Siegel formula for
the Hurwitz zeta-function.
3. The mean square
Now we discuss the asymptotics for the mean square of the function
ζ(s; A) in the critical strip is obtained. For this Theorem 1 is applied. The
cases of a xed σ, 1 2 < σ < 1, of σ = 1 2
and of σ = σ T → 1 2
are considered. Let
+ 0 as T → ∞
K(k) =
k∑
|a q | 2 ,
q=1
and, for 1 2 < σ < 1, C(σ) = max((2σ − 1) −1 , (1 − σ) −1 ).
Moreover, let M q (T ) = [ ( T 2π )1/2 − q k ] .
Theorem 2 ([10]). Let σ, 1 2
< σ < 1, be xed and T → ∞. Then
∫T
( )
I 1 T, σ =
1
= k −2σ T
|ζ(σ + it; A)| 2 dt
k∑ ∑
|a q | 2
q=1
1mM q (T )
1
(m + q k )2σ
+ (2π)2σ−1
2 − 2σ k−2σ K(k)T 2−2σ ∑
− 2πk −2σ
−
k∑ ∑
|a q | 2
q=1
(2π)
2 − 2σ k−2σ K(k)
1mM q(T )
∑
1mM 0 (T )
1mM 0 (T )
1
1
m 2−2σ
(m + q k )2σ−2
1
m 2σ−2
+ Bk 1−2σ K(k)C(σ)T σ log T + Bk 1−σ K(k)C(σ)T 1 2 .

64 On the periodic zeta-function series
Note that all estimates in Theorem 2 are uniform with respect to σ and
k. Theorem 2 improves a similar result of [8]. This is inuenced by the
application of Theorem 1 instead of a simple approximation of ζ(s; A) by a
long nite sum in [8].
Theorem 3 ([17]). Let T → ∞. Then
Here
∫T
( 1) I 1 T, = |ζ( 1 2
2 + it; A)|2 dt
0
= k −1 K(k)T log T
+k −1 K(k)T (2γ 0 − log π − 1) − k −1 T ( K 1 (k) − K 2 (k) )
+Bk 1 2 K(k)T
1
2 log T + BkK(k).
K 1 (k) =
K 2 (k) = k
k∑
q|a q | 2
q=1
k∑
q=1
|a q | 2
q .
∞ ∑
m=1
1
m(mk + q) ,
Note that in [8] the remainder term in a similar theorem has the form
BK(k)T.
Now we consider the mean square of ζ(s; A) near the critical line. Let
σ T = 1 2 + 1
l T
, where l T > 0 and l T → ∞ as T → ∞. Three cases of l T are
l
considered: l T = o(log T), log T = o(l T ) as T → ∞ and lim T
T →∞
log T = κ ≠ 0.
The results are stated in the following theorems.
Theorem 4 ([11]). Let l T = o(log T ) as T → ∞. Then
I 1 (T, σ T ) = k −2σ T
K(k)T l T + T
+Bk 1−2σ T
K(k)T l T exp
k∑ |a q | 2
q 2σ T
q=1
+Bk 1−σ T
K(k)T 1 2 log T.
Theorem 5 ([11]). Let log T = o(l T ) as T → ∞. Then
I 1 (T, σ T ) = k −2σ T
K(k)T log T + T
{
− log T }
+ Bk −2σ T
K(k)T
l T
k∑
q=1
|a q | 2
q 2σ T
+ Bk1−2σ T
K(k)T

A. A. Laurutis, D. ’iau£i unas 65
Theorem 6 ([11]). Let lim
T →∞
+Bk −2σ T
K(k) T log2 T
l T
+ Bk 1−σ T
K(k)T 1 2 log T.
l T
log T
= κ ≠ 0. Then, as T → ∞,
I 1 (T, σ T ) = k −2σ T
K(k) κ ( )
k∑
1 − e
− 2 |a q | 2
κ T log T + T
2 q 2σ T
q=1
+o(k −2σ T
K(k)T log T ) + Bk 1−2σ T
K(k)T
+Bk 1−σ T
K(k)T 1 2 log T.
In Theorems 46 all estimates again are uniform with respect to k.
4. The fourth moment
The investigations of the fourth power moment is a very dicult problem
even for the Riemann zeta-function and Dirichlet L-functions. Therefore, we
obtain only estimate for I 2 (T, 1 2
), moreover, we consider a simple case of
prime k. In this case the function ζ(s; A) can be written in the form
ζ(s; A) = 1 ∑k−1
ϕ(k)
∑
q=1 χ( mod k)
χ(q)a q L(s, χ) + a k
ζ(s). (1)
ks Here ϕ(k) denotes the Euler function. From this, using the well-known
approximate functional equations for L(s, χ), L(s, χ 1 )L(s, χ 2 ) and for ζ(s),
ζ 2 (s), an approximate functional equation for ζ 2 (s; A) can be obtained. However,
the later functional equation is too complicated to use it for the investigation
of the asymptotics for I 2 (T, σ). Therefore, we use directly (1) and
known estimates for the fourth power moment of L(s, χ) and ζ(s) to obtain
the following result.
Theorem 7 ([18]). Let k be a prime number, and T → ∞. Then
I 2 (T, 1 2 ) = Bk3 K 2 (k)T log 4 T.
The estimate of Theorem 7 with respect to T has a precise order, because
the same estimate is valid in the case a m = χ(m).

66 On the periodic zeta-function series
5. Joint limit theorem
Now we discuss the functional limit theorems for periodic zeta-function.
Let D = {s ∈ C : σ > 1 2
}, and let H(D) and M(D) denote the spaces of
analytic and meromorphic on D functions, respectively, equipped with the
topology of uniform convergence on compacta. Let γ be the unit circle on
C, and let
Ω = ∏ p
where γ p = γ for each prime p. The innite-dimensional torus Ω is a compact
topological Abelian group, therefore on (Ω, B(Ω)) the probability Haar
measure m H exists. Thus we obtain a probability space (Ω, B(Ω), m H ). Denote
by ω(p) the projection of ω ∈ Ω to the coordinate space γ p , and let, for
m ∈ N,
ω(m) = ∏
p α ‖m
γ p ,
ω α (p),
where p α ‖m means that p α | m but p α+1 ∤ m.
Let
ˆζ(s; A) = (1 − 2 1−s )ζ(s; A).
Theorem 8 ([9]). The probability measure
ν T (ˆζ(s + iτ; A) ∈ A),
A ∈ B(H(D)),
weakly converges to the distribution of the H(D)-valued random element
(
1 − 2ω(2)
2 s ) ∞
∑
m=1
a m ω(m)
m s
as T → ∞.
Theorem 8 is used to obtain joint limit theorems for periodic zeta-functions.
Let A j = {a jm : m ∈ Z}, j = 1, . . . , n, be a sequence of complex
numbers with periods k j > 0. Dene, for σ > 1,
Let
ζ(s; A j ) =
∞∑
m=1
a(j) = 1 k j
a jm
, j = 1, . . . , n.
ms k j
∑
m=1
a jm .

We also use the notation
A. A. Laurutis, D. ’iau£i unas 67
H r (D) = H(D) × · · · × H(D) ,
} {{ }
M r (D) = M(D) × · · · × M(D) .
} {{ }
r
We suppose that a(1) = · · · = a(r) = 0 and a(r + 1), . . . , a(k) ≠ 0, where
r = 0, 1, . . . , n. On the probability space (Ω, B(Ω), m H ), dene the H n (D)-
value random element F (s, ω; A 1 , . . . , A n ) by the formula
where
F (s, ω; A 1 , . . . , A n ) = (ζ(s, ω; A 1 ), . . . , ζ(s, ω; A n )),
ζ(s, ω; A j ) =
∞∑
m=1
r
a jm ω(m)
m s , ω ∈ Ω, j = 1, . . . , n.
Denote H r M n−r = H r (D) × M n−r (D). Our main result is contained in the
following theorem.
Theorem 9 ([9]). The probability measure
ν T ((ζ(s + iτ; A 1 ), . . . , ζ(s + iτ; A n )) ∈ A), A ∈ B(H r M n−r ),
weakly converges to the distribution of the random element F (s, ω; A 1 , . . . , A n )
as T → ∞.
Theorem 9 is derived from the following multidimensional result. Let
and
ˆζ(s, ω; A j ) =
Moreover, we put
p(s) = 1 − 2 2 s ,
(
1 − 2ω(2)
2 s ) ∞
∑
2ω(2)
p(s, ω) = 1 −
2 s ,
m=1
a m ω(m)
m s , j = 1, . . . , n.
H r,2(n−r) = H r (D) × H 2(n−r) (D),
and dene a probability measure
P T (A) = ν T ((ζ(s + iτ; A 1 ), . . . , ζ(s + iτ; A r ), p(s + iτ),
ˆζ(s + iτ; A r+1 ), . . . , p(s + iτ)ˆζ(s + iτ; A n )) ∈ A),
A ∈ B(H r,2(n−r) ).
Theorem 10 ([9]). The probability measure P T converges weakly to the distribution
of the random element
(ζ(s, ω; A 1 ), . . . , ζ(s, ω; A r ), p(s, ω), ˆζ(s, ω; A r+1 ), . . . , p(s, ω), ˆζ(s, ω; A n )).
as T → ∞.

68 On the periodic zeta-function series
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A. A. Laurutis, D. ’iau£i unas 69
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Periodine dzeta funkcija
A. A. Laurutis, D. ’iau£i unas
Straipsnyje pateikta periodines dzeta funkcijos rezutatu apºvalga. Ypatingas
demesys skiriamas rezultatams, gautiems autoriaus: artutinei funkcinei lyg£iai,
vidurkiui, ketvirtajam momentui bei daugiamatems ribinems teoremoms funkcinese
erdvese.
Rankra²tis gautas
2004 10 20