On Zeros of Periodic Zeta Functions
We consider zeta functions ζ(s; \( \mathfrak{a} \) ) given by Dirichlet series with multiplicative periodic coefficients and prove that, for some classes of functions F , the functions F(ζ(s; \( \mathfrak{a} \) )) have infinitely many zeros in the critical strip. For example, this is true for sin(...
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508368168288256 |
|---|---|
| author | Laurinčikas, A. Šiaučiūnas, D. Лаурінчікас, А. Шяучюнас, Д. |
| author_facet | Laurinčikas, A. Šiaučiūnas, D. Лаурінчікас, А. Шяучюнас, Д. |
| author_sort | Laurinčikas, A. |
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| datestamp_date | 2020-03-18T19:16:10Z |
| description | We consider zeta functions ζ(s; \( \mathfrak{a} \) ) given by Dirichlet series with multiplicative periodic coefficients and prove that, for some classes of functions F , the functions F(ζ(s; \( \mathfrak{a} \) )) have infinitely many zeros in the critical strip. For example, this is true for sin(ζ(s; \( \mathfrak{a} \) )). |
| first_indexed | 2026-03-24T02:24:06Z |
| format | Article |
| fulltext |
UDC 511.3
A. Laurinčikas (Vilnius Univ., Lithuania),
D. Šiaučiūnas (Šiauliai Univ., Lithuania)
ON ZEROS OF PERIODIC ZETA FUNCTIONS
ПРО НУЛI ПЕРIОДИЧНИХ ДЗЕТА-ФУНКЦIЙ
We consider the zeta functions ζ(s; a) given by Dirichlet series with multiplicative periodic coefficients and prove that, for
some classes of functions F , the functions F (ζ(s; a)) have infinitely many zeros in the critical strip. For example, this is
true for sin(ζ(s; a)).
Розглянуто дзета-функцiї ζ(s; a), що заданi рядами Дiрiхле з мультиплiкативними перiодичними коефiцiєнтами, та
доведено, що для деяких класiв функцiй F функцiї F (ζ(s; a)) мають нескiнченну кiлькiсть нулiв у критичнiй смузi.
Наприклад, це виконується для sin(ζ(s; a)).
1. Introduction. The zero distribution of zeta functions is of particular interest in analytic number
theory, and, in general, in mathematics. The most important problems are related to the Riemann zeta
function ζ(s), s = σ + it, which is defined, for σ > 1, by Dirichlet series
ζ(s) =
∞∑
m=1
1
ms
,
and is analytically continued to the whole complex plane, except for a simple pole at the point s = 1
with residue 1. It is well known that s = −2m, m ∈ N, are so called trivial zeros of ζ(s). Moreover,
ζ(s) 6= 0, for σ ≥ 1, and for σ ≤ 0, t 6= 0, however, the function ζ(s) has infinitely many complex
(nontrivial) zeros in the critical strip {s ∈ C : 0 < σ < 1}. The famous Riemann hypothesis (RH)
says that all nontrivial zeros of ζ(s) lie on the critical line σ =
1
2
, and this is equivalent to the non-
vanishing of ζ(s) in the half-plane
{
s ∈ C : σ >
1
2
}
. The last known result on zero-free regions for
ζ(s) is of the form: there exists an absolute constant c > 0 such that ζ(s) 6= 0 in the region{
s ∈ C : σ ≥ 1− c
(log(|t|+ 2))2/3(log log(|t|+ 2))1/3
}
.
G. H. Hardy proved [1] that infinitely many nontrivial zeros lie on the critical line. This result
was improved by A. Selberg, N. Levinson, B. Conrey. The last result in this direction says [2] that
at least 41 percent of all nontrivial zeros of ζ(s) in the sense of of density are on the critical line.
Numerical calculations also support RH: the first 1013 nontrivial zeros of ζ(s) lie on the critical line
σ =
1
2
[3].
A natural generalization of the function ζ(s) is the periodic zeta function. Let a = {am : m ∈ N}
be a periodic sequence of complex numbers with minimal period k ∈ N. The periodic zeta function
ζ(s; a) is defined, for σ > 1, by the series
ζ(s; a) =
∞∑
m=1
am
ms
.
c© A. LAURINČIKAS, D. ŠIAUČIŪNAS, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6 857
858 A. LAURINČIKAS, D. ŠIAUČIŪNAS
Moreover, the function ζ(s; a) is analytically continuable to the whole complex plane. Really, let
ζ(s, α) denote the Hurwitz zeta function with parameter α, 0 < α ≤ 1, given, for σ > 1, by the
series
ζ(s, α) =
∞∑
m=0
1
(m+ α)s
,
and by analytic continuation elsewhere, except for a simple pole at s = 1 with residue 1. Then the
periodicity of the sequence a implies, for σ > 1, the equality
ζ(s; a) =
1
ks
k∑
l=1
alζ
(
s,
l
k
)
.
Therefore, in virtue of the above remarks, the later equality gives analytic continuation for ζ(s; a) to
the whole complex plane. If
a
df
=
1
k
k∑
l=1
al 6= 0,
then the function ζ(s; a) has a simple pole at s = 1 with residue a, otherwise, the function ζ(s; a) is
an entire function.
Obviously, if a1 = 1 and k = 1, then ζ(s; a) = ζ(s).
We use the notation
a±m =
1
k
k∑
l=1
al exp
{
±2πil
m
k
}
and a± = {a±m : m ∈ N}. Then the sequences of complex numbers a± are also periodic with period
k. In [4], it was proved that the function ζ(s; a) satisfies the functional equation
ζ(1− s; a) =
(
k
2π
)s
Γ(s)
(
exp
{
πis
2
}
ζ(s; a−) + exp
{
−πis
2
}
ζ(s; a+)
)
,
where Γ(s), as usual, stands for the Euler gamma function.
In [5], J. Steuding began to study the zero distribution of the function ζ(s; a). Denote the zeros of
ζ(s; a) by ρ = β + iγ. Moreover, let ca = max(|am| : 1 ≤ m ≤ k), ma = min{1 ≤ m ≤ k : am 6=
6= 0}, and
A(a) =
maca
|ama |
.
Then it was established in [5] that ζ(s; a) 6= 0 for σ > 1 +A(a).
Now let
â±m =
1√
k
k∑
l=1
al exp
{
±2πil
m
k
}
,
â± = {â±m : m ∈ N} and B(a) = max{A(â±)}. Then it was obtained in [5] that the function ζ(s; a),
for σ < −B(a), can have only zeros close to the negative real axis if mâ+ = mâ− , and close to the
line
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
ON ZEROS OF PERIODIC ZETA FUNCTIONS 859
σ = 1 +
πt
log
mâ−
mâ+
if mâ+ 6= mâ− . 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 strip −B(a) ≤ σ ≤ 1 +A(a).
In [5], an asymptotic formula for the number of nontrivial zeros ρ of ζ(s; a) with |γ| ≤ T also
was obtained, and proved that the nontrivial zeros of ζ(s; a) are clustered around the critical line.
Suppose that k > 2, am is not a multiple of a Dirichlet character modk, and am = 0 for
(m, k) > 1. Then it was observed in [6, p. 223] that ζ(s; a) has infinitely many zeros in the strip
D =
{
s ∈ C :
1
2
< σ < 1
}
. Note that, in this case, the sequence a is non multiplicative (we recall
that a is multiplicative if a1 = 1 and amn = aman for all m,n ∈ N, (m,n) = 1), and the function
ζ(s; a) has no the Euler product over primes.
Our aim is to consider the case of a multiplicative sequence a, and to prove that the function
F (ζ(s; a)) with certain F has infinitely many zeros in the strip D. In other words, we will construct
composite functions of zeta functions with Euler product for which RH is not true. This is motivated
by a better understanding of the RH problem.
Let G be a region on the complex plane. Denote by H(G) the space of analytic functions
on G equipped with the topology of uniform convergence on compacta. Define some classes of
functions F : H(G) → H(G) for certain regions G. Let V > 0 be an arbitrary fixed number,
DV =
{
s ∈ C :
1
2
< σ < 1, |t| < V
}
, and SV = {g ∈ H(DV ) : g(s) 6= 0 or g(s) ≡ 0} . Denote
by UV the class of continuous functions F : H(DV ) → H(DV ) such that, for each polynomial
p = p(s), the set (F−1{p}) ∩ SV is nonempty.
It is easily seen that the function
F (g) =
r∑
k=1
ckg
(k), g ∈ H(DV ), c1, . . . , cr ∈ C \ {0},
where g(k) stands for the kth derivative of g, is an element of the class UV . Really, for arbitrary
polynomial p(s) of degree k, there exists a polynomial p̂(s) of degree k + 1, p̂(s) 6= 0 for s ∈ DV ,
such that F (p̂) = p.
Let S = {g ∈ H(D) : g(s) 6= 0 or g(s) ≡ 0} . Now we introduce a class of functions F for
which the image F (S) is a certain subset of H(D). For a1, . . . , ar ∈ C, denote by Ua1,...,ar the class
of continuous functions F : H(D)→ H(D) such that F (S) ⊃ Ha1,...,ar;F (0)(D), where
Ha1,...,ar;F (0)(D) = {g ∈ H(D) : g(s) 6= aj , j = 1, . . . , r} ∪ {F (0)}.
For example, the functions F (g) = sin g, F (g) = cos g, F (g) = sinh g and F (g) = cosh g belong
to the class U−1,1. To see this, it suffices to solve the equation F (g) = f in g ∈ S. In the case of
F (g) = cos g, we have that
eig + e−ig
2
= f.
Hence, we find that
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
860 A. LAURINČIKAS, D. ŠIAUČIŪNAS
g± =
1
i
log
(
f ±
√
f2 − 1
)
.
Thus, if f ∈ H−1,1;1(D), then we can choose, say, the solution g+ which belongs to S. Therefore,
F ∈ U−1,1.
Our last class is very simple. We say that a continuous function F : H(D) → H(D) belongs to
the class U, if s− a ∈ F (S) for all a ∈
(
1
2
, 1
)
.
It is easily seen that the function F (g) = gg′, g ∈ H(D), belongs to the class U. Really, solving
the equation gg′ = s − a, we find that g = ±
√
s2 − 2as+ C with arbitrary constant C. We can
choose C such that s2 − 2as+ C 6= 0 for s ∈ D. Thus, there exists g ∈ S satisfying F (g) = s− a.
Now we are ready to state the theorems on zeros of the function F (ζ(s; a)). In the notation used
in Introduction, we suppose that ca <
√
2− 1. Note that this inequality implies, for all primes p, the
inequality
∞∑
α=1
|apα |
pα/2
≤ c < 1. (1)
Theorem 1. Suppose that the sequence a is multiplicative such that the inequality ca <
√
2−1
is satisfied, and that F belongs to at least one of the classes UV and U with sufficiently large V.
Then, for every σ1, σ2,
1
2
< σ1 < σ2 < 1, there exists a constant c = c(σ1, σ2, a, F ) > 0 such that,
for sufficiently large T (in the case of the class UV , we suppose that T < V ), the function F (ζ(s; a))
has more than cT zeros in the rectangle {s ∈ C : σ1 < σ < σ2, 0 < t < T}.
Theorem 2. Suppose that the sequence a is the same as in Theorem 1, and F ∈ Ua1,...,ar ,
where Reaj 6∈
(
−1
2
,
1
2
)
, j = 1, . . . , r. Then the same assertion as in Theorem 1 is true.
For the proof of Theorems 1 and 2, we apply the universality property of the function ζ(s; a).
2. Universality. The universality property of zeta functions was discovered by S. M. Voronin in
1975. In [8], he proved that the Riemann zeta function ζ(s) is universal in the sense that its shifts
ζ(s + iτ), τ ∈ R, approximate a wide class of analytic functions. The last version of the Voronin
theorem is contained in the following theorem, see, for example, [9, p. 225]. Denote by meas{A}
the Lebesgue measure of a measurable set A ⊂ R.
Theorem 3. Let K ⊂ D be a compact set with connected complement, and let f(s) be a
continuous nonvanishing function on K which is analytic in the interior of K. Then, for every ε > 0,
lim inf
T→∞
1
T
meas
{
τ ∈ [0, T ] : sup
s∈K
|ζ(s+ iτ)− f(s)| < ε
}
> 0.
The first result on the universality of the function ζ(s; a) was obtained in [10, p. 145], see also
[11]. We state a more general case given in [6, p. 219].
Theorem 4. Suppose that k > 2, am is not a multiple of a Dirichlet character modk, and
am = 0 for (m, k) > 1. Let K ⊂ D be a compact set with connected complement, and let f(s) be a
continuous function on K which is analytic in the interior of K. Then, for every ε > 0,
lim inf
T→∞
1
T
meas
{
τ ∈ [0, T ] : sup
s∈K
|ζ(s+ iτ ; a)− f(s)| < ε
}
> 0.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
ON ZEROS OF PERIODIC ZETA FUNCTIONS 861
Note that the sequence a in Theorem 4 is not multiplicative. The general case was discussed in
[12] and [13]. We recall an universality theorem for ζ(s; a) with multiplicative sequence a [14].
Theorem 5. Suppose that the sequence a is multiplicative and inequality (1) holds. Let K and
f(s) be the same as in Theorem 3. Then the same assertion as in Theorem 4 is true.
Since f(s) is nonvanishing on K, Theorem 5 does not give any information on zeros of the
function ζ(s; a).
In [7], the first author began to study the universality of F (ζ(s; a)) for some classes of functions
F, and in theorems obtained the shifts F (ζ(s + iτ ; a)) approximate not necessarily nonvanishing
analytic functions. Therefore, the theorems of such a kind provide an information on the zero-
distribution of the function F (ζ(s; a)). For the proof of Theorems 1 and 2, we use the following
universality statements.
Lemma 1. Suppose that the sequence a is multiplicative and inequality (1) holds, K ⊂ D is a
compact set with connected complement, and f(s) is a continuous function on K and is analytic in
the interior of K. Let V > 0 be such that K ⊂ DV , and F ∈ UV . Then, for every ε > 0,
lim inf
T→∞
1
T
meas
{
τ ∈ [0, T ] : sup
s∈K
|F (ζ(s+ iτ ; a))− f(s)| < ε
}
> 0.
Proof of the lemma is given in [7].
Now we state an universality theorem for the functions from the class Ua1,...,ar .
Lemma 2. Suppose that a is the same as in Lemma 1, and F ∈ Ua1,...,ar . For r = 1, let
K ⊂ D be a compact set with connected complement, and f(s) be a continuous and 6= a1 function
on K and analytic in the interior of K. For r ≥ 2, let K ⊂ D be an arbitrary compact set, and
f ∈ Ha1,...,ar;F (0)(D). Then the same assertion as in Lemma 1 is true.
Note that in [7], the universality of F (ζ(s; a)) with F satisfying a stronger condition F (S) =
= Ha1,...,ar;F (0)(D) has been considered.
Lemma 3. Suppose that a is the same as in Lemma 1, K ⊂ D is a compact subset, and
f ∈ F (S). Then the same assertion as in Lemma 1 is true.
Lemmas 2 and 3 are deduced from a limit theorem on the weak convergence of probability
measures in the space H(D) [14] as well as from the Mergelyan theorem on the approximation of
analytic functions by polynomials [15], see also [16, p. 436].
3. Remarks on Theorems 1 and 2. Theorems 1 and 2 are consequences of the classical Rouché
theorem, see, for example, [17, p. 246] and Lemmas 1, 3, and 2, respectively.
1. Hardy G. H. Sur les zéros de la function ζ(s) de Riemann // C. R. Acad. Sci. Paris. – 1914. – 158. – P. 1012 – 1014.
2. Bui H. M., Conrey B., Young M. P. More than 41% of the zeros of the zeta function are on the critical line // Acta
Arith. – 2011. – 150. – P. 35 – 64.
3. Gourdon X. The 1013 first zeros of the Riemann zeta function and zeros computation at very large height /
http://numbers.computation.free.fr (2004).
4. Schnee W. Die Funktionalgleichung der Zetafunction und der Dirichletschen Reichen mit Periodishen Koeffizienten
// Math. Z. – 1930. – 31. – S. 378 – 390.
5. Steuding J. On Dirichlet series with periodic coefficients // Ramanujan J. – 2002. – 6. – P. 295 – 306.
6. Steuding J. Value Distribution of L-Functions // Lect. Notes Math. – 2007. – 1877.
7. Laurinčikas A. Universality of composite functions of periodic zeta functions (in Russian) // Mat. Sb. – 2012. – 203,
№ 11. – P. 105 – 120 (English transl.: Sb. Math. – 2012. – 203, № 11. – P. 1631 – 1646).
8. Voronin S. M. Theorem on the "universality"of the Riemann zeta function in Russian // Izv. Akad. Nauk SSSR, Ser.
Mat. – 1975. – 39. – P. 475 – 486 (English transl.: Math. USSR Izv. – 1975. – 9. – S. 443 – 453).
9. Laurinčikas A. Limit theorems for the Riemann zeta function. – Dordrecht etc.: Kluwer Acad. Publ., 1996.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
862 A. LAURINČIKAS, D. ŠIAUČIŪNAS
10. Bagchi B. The statistical behaviour and universality properties of the Riemann zeta function and other allied Dirichlet
series: Ph. D. Thesis. – Calcuta, 1981.
11. Bagchi B. A joint universality theorem for Dirichlet L-functions // Math. Z. – 1982. – 181. – S. 319 – 334.
12. Sander J., Steuding J. Joint universality for sums and products of Dirichlet L-functions // Analysis. – 2006. – 26. –
P. 285 – 312.
13. Kaczorowski J. Some remarks on the universality of periodic L-functions // New Dirrections in Value Distribution
Theory of Zeta and L-Functions / Eds R. Steuding, J. Steuding. – Aachen: Shaker Verlag, 2009. – P. 113 – 120.
14. Laurinčikas A., Šiaučiūnas D. Remarks on the universality of periodic zeta functions // Mat. Zametki. – 2006. – 80,
No. 4. – S. 561 – 568 (English transl.: Math. Notes. – 2006. – 80, № 3-4. – P. 532 – 538).
15. Mergelyan S. N. Uniform approximations to functions of complex variable // Usp. Mat. Nauk. – 1954. – 7. – S. 31 – 122
(English transl.: Amer. Math. Soc. Trans. – 1954. – 101. – P. 99).
16. Walsh J. L. Interpolation and approximation by rational functions in the complex domain (in Russian). – Moscow:
Izd. Inost. Lit., 1961.
17. Privalov I. I. Introduction to the theory of functions of complex variable (in Russian). – Moscow: Nauka, 1967.
Received 13.04.12,
after revision — 31.10.12
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 6
|
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| last_indexed | 2026-03-24T02:24:06Z |
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| spelling | umjimathkievua-article-24712020-03-18T19:16:10Z On Zeros of Periodic Zeta Functions Про нулі періодичних дзета-функцій Laurinčikas, A. Šiaučiūnas, D. Лаурінчікас, А. Шяучюнас, Д. We consider zeta functions ζ(s; \( \mathfrak{a} \) ) given by Dirichlet series with multiplicative periodic coefficients and prove that, for some classes of functions F , the functions F(ζ(s; \( \mathfrak{a} \) )) have infinitely many zeros in the critical strip. For example, this is true for sin(ζ(s; \( \mathfrak{a} \) )). Розглянуто дзета-функції ζ(s; \( \mathfrak{a} \) ), що задані рядами Діріхлє з мультиплікативними періодичними коефiцiєнтами, та доведено, що для деяких класів функцій F функції F(ζ(s; \( \mathfrak{a} \) )) мають нескінченну кількість нулів у критичній смузі. Наприклад, це виконується для sin(ζ(s; \( \mathfrak{a} \) )). Institute of Mathematics, NAS of Ukraine 2013-06-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2471 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 6 (2013); 857–862 Український математичний журнал; Том 65 № 6 (2013); 857–862 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2471/1708 https://umj.imath.kiev.ua/index.php/umj/article/view/2471/1709 Copyright (c) 2013 Laurinčikas A.; Šiaučiūnas D. |
| spellingShingle | Laurinčikas, A. Šiaučiūnas, D. Лаурінчікас, А. Шяучюнас, Д. On Zeros of Periodic Zeta Functions |
| title | On Zeros of Periodic Zeta Functions |
| title_alt | Про нулі періодичних дзета-функцій |
| title_full | On Zeros of Periodic Zeta Functions |
| title_fullStr | On Zeros of Periodic Zeta Functions |
| title_full_unstemmed | On Zeros of Periodic Zeta Functions |
| title_short | On Zeros of Periodic Zeta Functions |
| title_sort | on zeros of periodic zeta functions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2471 |
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