on some aspects in the theory of the estermann zeta-function

115
Proc. Sci. Seminar
Faculty of Physics and Mathematics, Šiauliai University,
5, 2002, 115–130
ON SOME ASPECTS IN THE THEORY
OF THE ESTERMANN ZETA-FUNCTION
Rasa ŠLEŽEVIČIENĖ
Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, 2600
Vilnius, Lithuania
e-mail: rasa.slezeviciene@maf.vu.lt
Abstract. This paper gives an overview on some results on the mean-square, universality
and zero-distribution of the Estermann zeta-function.
Key words: Estermann zeta-function, zero-distribution, universality, mean-square.
1. Introduction
As usual, let s = σ + it be a complex variable. Further, let k and l be coprime
integers and define for a fixed complex number α the generalized divisor function
by
σ α (n) = ∑ d α .
d|n
Then, for σ > max{1 + Re α, 1}, the Estermann zeta-function with parameters k l
and α is defined by
E
(s; k )
l , α =
One can find the representation
E
(s; k )
l , α
= l α−s l∑
f=1
∞∑
n=1
σ α (n)
n s
{
exp 2πin k }
. (1)
l
{
exp 2πi fk }
L
(1, f ) ( ) fk
l l , s − α L
l , 1, s , (2)
where L(λ, α, s) stands for the Lerch zeta-function, given by
L(λ, α, s) =
∞∑
n=0
exp{2πinλ}
(n + α) s
for σ > 1, 0 < α 1 and real λ. Thus, E(s; k l
, α) can be analytically continued to a
meromorphic function in s, which is regular in the whole complex plane up to two

116 On some aspects in the theory of the Estermann zeta-function
simple poles at s = 1 and s = 1 + α if α ≠ 0, resp. one double pole at s = 1 if
α = 0. With the functional equation of the Lerch zeta-function one gets
E
(s; k )
l , α
= 1 ( 2π
π l
{
×
− cos
) 2s−1−α
Γ(1 − s)Γ(1 + α − s)
( )
)
πα
cos E
(1 + α − s; k∗
2
l , α
(
πs − πα 2
)
E
(1 + α − s; − k∗
l , α )}
,
(3)
where k ∗ is defined by kk ∗ ≡ 1 mod l. Since σ α (n) = n α σ −α (n) we have further
E
(s; k )
l , α = E
(s − α; k )
l , −α . (4)
Therefore, throughout the paper we may assume that a := Re α 0.
The Estermann zeta-function was introduced by Estermann in 1930 for α = 0
[2]. He investigated the number D(n) of solutions of
xy + zt = n
and showed that
D(n) = n
2∑
(log n) r
r=0
2∑ ∑
c rs d −1 (log d) s + E(n),
s=0
∑
with certain constants c rs and an error term E(n) ≪ n 7 8 (log n) 23 4
d|n d− 3 4 .
In 1987 Kiuchi in his work on exponential sum involving the generalized divisor
function [11] defined E(s; k l , α) for α ∈ (−1, 0]. Since σ α(n) is an analytic function
in α the results above hold by analytic continuation for any complex α.
The Estermann zeta-function was studied in papers of Jutila and Furuya with
regard to the divisor problem (see [3], [9]), and by Ivić [7] for an analytic continuation
of the error term in the mean square formula of the Riemann zeta-function.
The values at the integers were investigated by Ishibashi [8].
In this paper we will present some mean-value theorems for the Estermann
zeta-function. Also we will discuss the universality property of the Estermann zetafunction
with certain parameters and we will conclude with some results on the
zero-distribution.
d|n

R. Šleževičienė 117
2. Mean-square
It is very important to have knowledge on the moments of the Estermann zetafunction
in the investigations on the zero-distribution and the universality of this
function. Kamiya [10] obtained an approximate functional equation for E(s; k l , 0)
and deduced, for A > 49,
l∑
∫
k=1
(k,l)=1[−T,T ]\[−A,A]
( 1
∣ E 2 + it; k )∣ ∣∣∣
2
l , 0 dt ≪ lT (log(lT )) 4 , (5)
as T → ∞. In this section we present further mean-square estimates and formulae
for the Estermann zeta-function, which were obtained by Steuding and the author
in [20].
Theorem 1. We have uniformly for l T , as T → ∞,
1
ϕ(l)
l∑
k=1
(k,l)=1
the implicit constants do not depend on l.
∫ T
1
( 1
∣ E 2 + it; k )∣ ∣∣∣
2
l , 0 dt ≍ T (log T ) 4 ;
This improves slightly Kamiya’s result (5) for highly composite l T . The
proof even allows to give a (complicated) mean-square formula. For simplicity we
consider only prime l.
Theorem 2. Let l be a prime number. Then for sufficiently large T
l−1 ∫ T
∑
k=1 1
( 1
∣ E 2 + it; k )∣ ∣∣∣
2
l , 0
dt = l5 − l 4 + 7l 3 − 11l 2 + 5l + 1
2π 2 (l − 1)l 2 T (log T ) 4
(l + 1)
+ O(T (log T ) 3 ).
The proofs of these results use a representation of E(s; k l
, α) by Dirichlet L-functions.
First, we recall some basics on characters and their associated Dirichlet series.
The Dirichlet L-function to a character mod q, q ∈ N, is for σ > 1 given by
L(s, χ) =
∞∑
n=1
χ(n)
n s
= ∏ p
(
1 − χ(p)
p s ) −1
,
where the product is taken over all prime numbers p. We define the Gauss sum
associated to a Dirichlet character χ mod q by
τ(χ) =
∑ (
χ(f) exp 2πi f )
.
q
f mod q

118 On some aspects in the theory of the Estermann zeta-function
It is known that for coprime integers f, q and for a character χ mod q the following
relation holds
(
exp 2πi f )
= 1 ∑
τ(χ)χ(f), (6)
q ϕ(q)
χ mod q
where ϕ is Euler’s totient. In view of the orthogonality relation for characters this
identity is equivalent to
χ(f)τ(χ) =
∑ (
χ(j) exp 2πi jf )
.
q
j mod q
Note, that the latter formula holds without any restrictions (not necessarily f coprime
to q) if χ is primitive. Each character is induced by a primitive character.
Using (6) one can show that
(
exp 2πi f )
= 1
q ϕ(q)
∑
m|q
(m,q/m)=1
µ(q/m)
∑
χ mod m
primitive
τ(χ)χ(f(q/m) ∗ ).
This gives, for arbitrary s,
E
(s; k )
l , α = ∑ d|l
= ∑ d|l
×
∞∑
n=1
σ α (n)
n s
1
d s ϕ(l/d)
∞∑
m=1
(
exp 2πi kn/d )
l/d
∑
b|l/d
(b,l/db)=1
σ α (dm)
m s χ(m).
µ(l/db)
∑
χ mod b
primitive
τ(χ)χ(kd(l/b) ∗ )
(7)
Using the Euler product representation of ∑ ∞
m=1
where
∞∑
m=1
λ(s; d, χ, α) := ∏ p|d
σ α (dm)
m
χ(m), we obtain
s
σ α (dm)
m s χ(m) = λ(s; d, χ, α)L(s, χ) L(s − α, χ),
1 + χ(p)p α(ν(d;p)+1)−s − χ(p)p α−s − p α(ν(d;p)+1)
1 − p α
if α ≠ 2πiv
log p
, v ∈ Z, and
(
λ s; d, χ, 2πiv )
:= ∏ (
ν(d; p) + 1 − ν(d; p)χ(p)p
−s ) ;
log p
p|d

R. Šleževičienė 119
here ν(d; p) is the exponent of the prime p in the unique prime factorization of d.
By the orthogonality relation for characters we have
l∑
(
∣ E s; k )∣ ∣∣∣
2
l , α =ϕ(l) ∑ 1 ∑
d 2σ ϕ(l/d) 2 µ(l/db) 2 b
d|l
k=1
(k,l)=1
This gives
l∑
k=1
(k,l)=1
∫2T
T
× ∑
χ mod b
primitive
( 1
∣ E 2 + it; k )∣ ∣∣∣
2
l , 0 dt =ϕ(l) ∑ d|l
b|l/d
(b,l/db)=1
|λ(s; d, χ, α)| 2 |L(s, χ) L(s − α, χ)| 2 .
× ∑
χ mod b
primitive
1
dϕ(l/d) 2
∫2T
T
∑
b|l/d
(b,l/db)=1
( )∣ ×
1 ∣∣∣
4
∣ L 2 + it, χ dt.
µ(l/db) 2 b
( )∣ 1 ∣∣∣
2
∣ λ 2 + it; d, χ, 0
Let η(b) denote the number of primitive characters χ mod b. Taking into account
Wang’s asymptotic formula for the fourth power moment of Dirichlet L-functions
[24]
∑
χ mod b
primitive
∫2T
T
for 1 < b T , we get
where
( )∣ 1 ∣∣∣
4
∣ L 2 + it, χ
l∑
k=1
(k,l)=1
Σ := ∑ d|l
∫2T
T
dt = η(b)
2π 2 ∏
p|b
(
1 − 1 ) 4 (
1 − 1 ) −1
p p 2 T (log(bT )) 4
+ O((bT ) ε min{b 9 8 T
7
8 , bT
11
12 }),
( 1
∣ E 2 + it; k )∣ ∣∣∣
2
l , 0 dt ≍ ϕ(l)T (log T ) 4 · Σ,
σ 0 (d) 2
dϕ(l/d) 2
∑
b|l/d
(b,l/db)=1
µ(l/db) 2 η(b)b ∏ p|b
(
1 − 1 p) 4
;
note that Wang’s formula holds also in the case b = 1, the well-known case of the
fourth moment of ζ(s), first proved by Ingham [6]. As a Dirichlet convolution of
multiplicative functions Σ is multiplicative itself. This leads to the estimate
Σ ≍ ∏ (1 + 4 ( )) ( 1
p + O p 2 1 − 1 4
≍ 1,
p)
p|l
(8)

120 On some aspects in the theory of the Estermann zeta-function
which proves Theorem 1.
To obtain Theorem 2 note that for prime l
( )∣ 1 ∣∣∣
2
∣ λ 2 + it; l, χ, 0 = 4 − 4Re {χ(l)l − 1 2 −it } + |χ(l)| 2 l −1 .
Therefore, we can rewrite (8) as
2T
l∑
∫
( 1
∣ E 2 + it; k )∣ ∣∣∣
2
l , 0 dt
or
k=1
T
(k,l)=1
⎧
( ⎪⎨ ∫2T
( )∣
1
= ϕ(l)
1 ∣∣∣
4
ϕ(l) 2 ⎪ ∣ ζ ⎩ 2 + it dt + l
l∑
k=1
(k,l)=1
+ 1 l
∫2T
T
∫2T
T
T
∑
χ mod l
primitive
∫2T
(
4 + 1 ∣ ( )∣ ∣∣∣
l − 4Re {l− 1 1 ∣∣∣
4
2 }) −it ζ
2 + it dt
( 1
∣ E 2 + it; k )∣ ∣∣∣
2
l , 0
dt = 4l3 − 6l 2 + 2l + 1
l 2 (l − 1)
+ l
l − 1
∑
χ mod l
primitive
T
∫2T
T
⎫
( )∣ 1 ∣∣∣
4 ⎪⎬
∣ L 2 + it, χ dt
⎪⎭
∫2T
T
)
,
( )∣ 1 ∣∣∣
4
∣ ζ 2 + it dt
( )∣ 1 ∣∣∣
4
∣ L 2 + it, χ dt
plus an error term which does not exceed the modulus of the real part of the integral
∫2T
( )∣ I := exp(−it log l)
1 ∣∣∣
4
∣ ζ 2 + it dt
T
( )∣ ≪ max
1 ∣∣∣
4 (
∣) T t2T
∣ ζ 2 + it min
d
∣∣∣ −1
T t2T
∣dt exp(−it log l) .
A simple calculation gives the mean-square formula of the theorem.
To the right we have
Theorem 3. For σ > 1 2 ,
∫
1
T
(
lim
T →∞ T ∣ E σ + it; k )∣ ∣∣∣
2
l , α
1
Further, if a < 0, then for T > 2
⎧
∫ T
(
∣ E σ + it; k )∣ ∣∣∣
2 ⎨
l , α dt ≪
1
dt = ζ(2σ − 2a)ζ(2σ − a)2 ζ(2σ)
.
ζ(4σ − 2a)
T (log T ) 2 if σ = 1 2 ,
T
⎩
2(1−σ) if a + 1 2 < σ < 1 2 ,
T 1−2a (log T ) 2 if σ = a + 1 2 ;
the implicit constants in the estimates depend on l, σ and α.

R. Šleževičienė 121
In the formulation of the theorem, ζ(s) is the famous Riemann zeta-function,
for σ > 1 defined by
ζ(s) =
∞∑ 1
n s = ∏ p
n=1
(
1 − 1 p s ) −1
.
This theorem can be proved by classical arguments, i.e. the mean-square theorems
for Dirichlet series and Carlson’s theorem.
3. Universality
Voronin [22] proved a remarkable universality property of the Riemann zetafunction,
namely that any non-vanishing analytic function g(s), defined on the disc
|s| < 1 4
, and continuous on the boundary, can be approximated to any degree of
accuracy by ζ(s) somewhere in the strip 1 2
< σ < 1. Later, it was even proved that
the set of translations of ζ(s) which approximate a given function g(s) has a positive
lower (Lebesgue-) density; see [12]. This result was extended, for example, to
joint universality for Dirichlet L-functions by Voronin himself [23], to universality
for Dedekind zeta-functions by Reich [17], for Hurwitz zeta-functions by Bagchi [1],
for Dirichlet series with multiplicative coefficients by Laurinčikas [12], Laurinčikas
and the author [14], and for more general Dirichlet series by Gonek [5], Laurinčikas,
Schwarz and Steuding [13]. The Linnik-Ibragimov conjecture states that all functions
given by Dirichlet series, analytically continuable to the left of the half-plane
of absolute convergence, and which satisfy some natural growth conditions, are
universal.
We add to the known examples of Dirichlet series with the universality property
certain Estermann zeta-functions. For this purpose we exchange, as in Section
2, in formula (1) the additive character exp(2πin k l
) by a multiplicative character
χ mod l. Therefore we define, for σ > 1,
E(s; χ, α) =
∞∑
n=1
σ α (n)
n s χ(n). (9)
The identity (6) leads to a representation of Estermann zeta-functions by the Dirichlet
series E(s, χ, α) given by (9). For the sake of simplicity we may assume that
α ≠ 2πiv
log l
, v ∈ Z, throughout this section.
For prime l we have
E
(s; k )
l , α = 1 ∑
τ(χ)χ(k)E(s; χ, α) + Λ(s; α)E(s; χ 0 , α), (10)
ϕ(l)
χ mod l
χ≠χ 0
where
Λ(s; α) := l − l1+α−s − l 1+2α + l 1+2α−s − l s + l α+s
l s (l − 1)(1 − l α )(1 − l −s )(1 − l α−s .
)

122 On some aspects in the theory of the Estermann zeta-function
This formula should be compared with (7).
As a Dirichlet series with multiplicative coefficients E(s; χ, α) possesses an Euler
product. Note that for prime l and a character χ mod l the following relations hold:
if χ ≠ χ 0 , and otherwise
E(s; χ, α) = 1
τ(χ)
∑
j mod l
χ(j)E
(s; j )
l , α
E(s; χ 0 , α) = ls − l s+α − 1 + l α−s + l 2α − l 2α−s
l s (1 − l α E(s; 1, α).
)
In any case,
E(s; χ, α) = ∏ p
(
1 − χ(p)
p s ) −1 (
1 − χ(p)
p s−α ) −1
= L(s, χ) L(s − α, χ);
the Euler product representation is valid for σ > max{1 + Re α, 1} while the latter
identity holds throughout the whole complex plane. In particular, E(s; χ, α) is an
entire function if χ is non-principal, whereas E(s; χ 0 , α) is analytic in C except for
poles at s = 1, 1 + α.
Note that the functions E(s; χ, α) have better analytic properties than Estermann
zeta-functions (which seems to be caused by the Euler product). We know this
phenomenon from multiplicative twists (Dirichlet L-functions) and additive twists
(Lerch zeta-functions) of the Riemann zeta-function.
Now we can add to the known examples of Dirichlet series with the universality
property certain Estermann zeta-functions. Garunkštis, Laurinčikas, Steuding and
the author proved in [4]
Theorem 4. Suppose that a < −1, l 5 is prime, and that
∞∑
j=1
|σ α (p j )|
p jβ < 1 (11)
holds for p = 2, 3 and some β ∈ ( 1 2
, 1). Let K be a compact subset of the vertical
strip β < σ < 1 with connected complement, and let g(s) be a continuous function
on K which is holomorphic in the interior of K. Then, for any ɛ > 0,
lim inf
T →∞
{
1
T meas τ ∈ [0, T ] : max
s∈K
(
∣ E s + iτ; k ) }
l , α − g(s)
∣ < ε > 0.
The proof of this theorem is based on a joint universality theorem for E(s; χ, α)
proved by the author in [18].
We say that a multiplicative function f belongs to the class M χ = M χ (c 1 , β,
θ, c 2 ) if the following axioms are satisfied:

R. Šleževičienė 123
1) there exists a constant c 1 such that |f(n)| c 1 for all n ∈ N;
2) the associated Dirichlet series
Z(s, f) :=
∞∑
n=1
f(n)
n s
has an analytic continuation of finite order to the half plane σ β with some
β ∈ ( 1 2
, 1), except for at most one simple pole at s = 1;
3) for σ β the estimate
∫ T
0
|Z(σ + it, f)| 2 dt ≪ T
holds, as T → ∞;
4) for 1 h m, coprime to m, exist positive constants θ h such that
lim
x→∞
1
π(x; h, m)
∑
px
p≡h mod m
|f(p)| 2 = θ h ,
where the sum is taken over all prime numbers p x, and where π(x; h, m)
counts the primes p x in the residue class h mod m;
5) there exists a constant c 2 such that the estimate
∞∑
j=1
|f(p j )|
p jβ c 2 < 1
holds for all primes p.
Let χ be a character. We consider the associated twists of the Dirichlet series
with multiplicative coefficients f(n), given by
Then
Z(s, fχ) =
∞∑
n=1
f(n)χ(n)
n s
= ∏ p
⎛
⎝1 +
∞∑
j=1
⎞
f(p j )χ(p j )
⎠
p js .
Theorem 5. Suppose that χ 1 , . . . , χ N are pairwise non-equivalent caharcters, and
that fχ j ∈ M χ for 1 j N. For 1 j N let K j be a compact subset of
D := {s ∈ C : β < σ < 1} with connected complement, and g j (s) be a continuous
non-vanishing function on K j which is analytic in the interior of K j . Then, for
any ε > 0,
lim inf
T →∞
{
}
1
T meas max max |Z(s + iτ, fχ j ) − g j (s)| < ε > 0.
1jn s∈K j
From this joint universality we will deduce

124 On some aspects in the theory of the Estermann zeta-function
Corollary 6. Suppose that Re α = a < −1, l 5 is prime, and that axiom (11)
is fulfilled for p = 2, 3 with some β ∈ ( 1 2 , 1). For 1 j n let χ j be pairwise nonequivalent
characters mod l, and let K j be a compact subset of D with connected
complement, and let g j (s) be a continuous non-vanishing function on K j which is
analytic in the interior of K j . Then, for any ε > 0,
lim inf
T →∞
{
}
1
T meas max max |E(s + iτ, χ j , α) − g j (s)| < ε > 0.
1jn s∈K j
Proof. Since σ α (n) is multiplicative, and, for a < −1,
∑
|σ α (n)χ(n)| σ a (n) d a < ζ(−a) = c 1 .
In view of well-known properties of Dirichlet L-functions axiom 2 on analytic continuation
and, by Theorem 3, axiom 3 on the mean-square are fulfilled for any
β ∈ ( 1 2
, 1). Since
⎛
∑
px
p≡h mod l
|σ α (p)χ(p)| 2 = ∑ px
p≡h mod l
dn
⎜
1 + O ⎝
∑
px
p≡h mod l
p a ⎞
⎟
⎠ = π(x; h, l) + O(1),
axiom 4 is also fulfilled. It remains to check axiom 5. Since |σ α (p j )χ(p j )| σ a (p j ),
a short computation shows
∞∑
j=1
|σ α (p j )χ(p j )|
p jβ
1 + p a+β − p a − p β−a
(p β − 1)(p β−a − 1)(p a − 1) .
Therefore, if a < −1, then inequality (11) holds for all primes p 5 with any β > 1 2 .
Hence, the assertion of the corollary follows from Theorem 5.
Sketch of the proof of Theorem 4. Let
then c χ ≠ 0. Then, by (10),
E
(s; k )
l , α = ∑
c χ := τ(χ)χ(k) ,
ϕ(l)
c χ E(s; χ, α) + Λ(s; α)E(s; χ 0 , α). (12)
χ mod l
Put F := 1 + sup s∈K |g(s)|. Then g(s) + F ≠ 0 for s ∈ K. Now define
g χ1 (s) = g(s) + F
c χ1
, g χ2 = − F
c χ2
, and g χ0 (s) = g χj (s) = η

R. Šleževičienė 125
for j = 3, . . . , l − 2, where η is a small positive parameter, chosen later (here we
need l 5). Hence, using (12), for any τ,
E
(s + iτ, k )
l , α − g(s) = ∑
c χ (E(s + iτ; χ, α) − g χ (s))
χ≠χ 0
⎛
⎞
∑l−1
+ η ⎝ c χj + Λ(s; α) ⎠ + Λ(s + iτ; α)E(s + iτ; χ 0 , α)
j=3
− Λ(s; α)g χ0 (s).
Taking the maximum over all s ∈ K, using joint universality for E(s; χ, α) and
taking into account that Λ(s; α) ≪ l, we obtain
max
s∈K
(
∣ E s + iτ, k ) l , α − g(s)
∣ Cl(δ + η),
where the constant C is absolute. With δ and η sufficiently small we obtain the
assertion of the theorem.
It remains to show that the estimate (11) holds for p = 2, 3 with certain values
of α in the half plane a < −1 and some β ∈ ( 1 2 , 1). Therefore, take β = 4 5 , then
(11) is true for p = 3. Further, α with
Im α ≡
π mod 2π,
log 2
leads to
σ α (2 j ) = 1 − 2 a ± . . . + (−1) j 2 a = 1 − (−2a ) j+1
1 + 2 a ,
and we can bound the left hand side of (11) by
∞∑ |σ α (2 j )|
j=1
2 j 4 5
= 1 ( 1
1 + 2 a 2 4 5 − 1 − 2 a )
< 1
2 4 5 −a + 1
for −1.7 a < −1. Hence, for example E(s; 1 5 , − 3 2 + i π
log 2
) has the universality
property in the strip 4 5
< σ < 1. Obviously, a choice of β close to 1 would imply
universality for E(s; k l
, α) with less restricted α. We have failed to prove universality
for arbitrary Estermann zeta-functions E(s; k l
, α) or functions E(s; χ, α). However,
we believe that all in this paper appearing zeta-functions have the universality
property.

126 On some aspects in the theory of the Estermann zeta-function
4. Zero-distribution
Denote the zeros of E(s; k l
, α) by ρ = β + iγ. The Estermann zeta-function
is presented by a Dirichlet series not identically equal to 0. Hence, there exists a
zero-free region on the right. It is not difficult to show that
E
(s; k )
l , α ≠ 0 for σ > 3.
Using the functional equation (3) and taking into account the non-vanishing of the
Gamma-function and the zero-free region on the right, we get that for σ < −2 + a
the function E(s; k l
, α) can only have zeros close to the negative real axis. We call
zeros ρ of E(s; k l
, α) with β < −2 + a trivial. One can easily locate those zeros and
prove that the number of trivial zeros ρ with |ρ| T is ≍ T . We call other zeros of
E(s; k l
, α) nontrivial. Therefore, the nontrivial zeros lie in the vertical strip
−2 + a σ 3.
Applying ideas of Littlewood [16] and Levinson and Montgomery [15] Steuding
and the author proved in [19]
Theorem 7. Let B > 3 − a be a constant. Then, as T tends to infinity,
∑
β>−B
|γ|T
(B + β) = (2B + a + 1) T π log T l + O(log T ).
2πe
Sketch of the proof of Theorem 7. The proof relies mainly on
Lemma 8 (Littlewood). Let f(s) be regular in and upon the boundary of the rectangle
R with vertices b, b + iT, a + iT, a, and not zero on σ = b. Denote by ν(σ, T ) the
number of zeros ρ = β + iγ of f(s) inside the rectangle with β > σ including those
with γ = T but not γ = 0. Then
∫
R
∫ a
log f(s)ds = −2πi ν(σ, T )dσ.
b
This is an integrated version of the principle of the argument (for a proof see
[21], §9.9 or [16]).
By the condition on B, all nontrivial zeros of E(s; k l
, α) have real parts in (−B, 4).
Write N(σ, T ; k l , α) for the number of nontrivial zeros ρ of E(s; k l
, α) with β > σ

R. Šleževičienė 127
and |γ| T . Let R be a rectangle with vertices 4 ± iT, −B ± iT Then Littlewood’s
lemma states
∫
R
log E
(s, k )
∫ 4
l , α ds + O(1) = −2πi
−B
N
(σ, T ; k )
l , α dσ,
here the error term occurs from the poles at s = 1 and s = 1 + α. Hence
2π ∑
(B + β) + O(1) =
β>−B
|γ|T
∫ T
−T
∫ T
−
−
+
( log
∣ E − B + it; k )∣ ∣∣∣
l , α dt
−T
∫ 4
−B
∫ 4
−B
log
∣
(4 E + it; k )∣ ∣∣∣
l , α dt
arg E
(σ − iT ; k )
l , α dσ
arg E
(σ + iT ; k )
l , α dσ =
4∑
I j ,
say. To define log E(s; k l
, α) we choose the principal branch of the logarithm on the
real axis, as σ → ∞; for other points s the value of the logarithm is obtained by
analytic continuation.
The first integral gives the main term in the formula of Theorem 7. For its
computation we use the functional equation for the Estermann zeta-function and
Stirling’s formula. The second integral is bounded and the horizontal integrals are
of oder log T by a standard argument, using Jensen’s formula.
By Theorem 7 we can obtain a formula for the number of nontrivial zeros. Denote
by N(T ; k l , α) the number of nontrivial zeros ρ of E(s; k l
, α) with |γ| T (according
multiplicities). Using the formula of Theorem 7 with B + 1 instead of B, we get
after subtracting the resulting formula from the one above
Corollary 9.
As T tends to infinity,
N
(T ; k )
l , α = 2 T lT
log + O(log T ).
π 2πe
j=1
Note that the main term in the asymptotic formula does not depend on k and
α.
Multiplying the formula of Corollary 9 with B and subtracting it from the formula
of Theorem 7 gives

128 On some aspects in the theory of the Estermann zeta-function
Corollary 10.
We have, as T tends to infinity,
1 ∑
N(T ; k l , α) β = a + 1 + O(T −1 ).
2
ρ nontrivial
|γ|T
One may interpret the last formula in the sense that the mean value of the real
parts of the nontrivial zeros of E(s; k a+1
l
, α) is
2
. However, in the example
E(s; 1, α) = ζ(s)ζ(s − α) (13)
we expect no zeros lying on the line σ = a+1
2
unless a ≠ 0.
Moreover, we can prove upper bounds for the number of zeros in half-planes on
the right, namely
Theorem 11. We have uniformly in δ > 0, as T tends to infinity,
( 1
N
2 + δ, T ; k )
(
l , α T log log T log log T
≪ ≪
δ δ log T N T ; k )
l , α ,
and, for fixed σ > 1 2 ,
N
(σ, T ; k )
l , α ≪ T.
Sketch of the proof of Theorem 11. With similar arguments as in the proof of
Theorem 7 we obtain for every δ > 0
δN
(σ + δ, T ; k )
l , α =δ ∑ ∑
1 (β − σ) = 1 ∫
log E
(s; k )
2π
l , α ds
= 1
2π
β>σ+δ
∫ T
−T
β>σ
|γ|T
By Jensen’s inequality,
∫
1
T
log
2π ∣
(σ E + it; k )∣ ∣∣∣
l , α dt T { 1
4π log T
−T
log
∣
(σ E + it; k )∣ ∣∣∣
l , α dt + O(log T ).
∫ T
−T
(
∣ E σ + it; k )∣ ∣∣∣
2 }
l , α dt .
Now Theorem 3 yields the estimates of Theorem 11.
By the last theorem the number of zeros on the right of
σ = 1 log t
+ φ(t)log
2 log t
has zero density in the set of all nontrivial zeros, whenever φ(t) is a positive function
which tends with t to infinity. We conjecture that the example (13) is typical, i. e.
that a positive proportion of the nontrivial zeros is clustered around the lines σ = 1 2
and σ = 1 2 + a.
We conclude with some consequence of the universality property. Therefore, we
assume that E(s; k l
, α) satisfies all conditions of Theorem 4. In combination with
the upper estimates of Theorem 11 we obtain now for the number N(σ, T ; k l
, α) the
right order.

R. Šleževičienė 129
Corollary 12.
T → ∞,
Under the assumptions of Theorem 4 we have for σ > 1 2 , as
T ≪ N
(σ, T ; k )
l , α ≪ T.
Therefore, such Estermann zeta-functions have many zeros off the lines σ = 1 2
and σ = 1 2
+ a whereas for E(s; 1, α) = ζ(s)ζ(s − α) it is expected that all nontrivial
zeros lie exactly on these lines.
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