1. Introduction
We are dealing with normalized series of the form:
(1)
where
is an arbitrary sequence of complex numbers (the coefficients of the series) and
is a sequence of non decreasing real numbers (the exponents of the series).
When
, then the series has the form
and it is
called ordinary Dirichlet series. There is a vast literature on general Dirichlet series, a part of which has been mentioned in [1] [2] where we brought contributions to different aspects of this topic. In this paper we deal only with the problem of the multiplicity of zeros of Dirichlet functions and it is sufficient to study those two papers in order to get the necessary information supporting the development below.
We suppose that the function
is totally multiplicative, i.e.
for every
. Then for every prime decomposition
we have
.
If the series is an ordinary one, then
Assuming that we have also this last property for a general Dirichlet series, then it can be easily shown (see [1], theorem 11) that the series appears also as an Euler product:
(2)
where
is the set of prime numbers. For such a series we have
(3)
The series on the left hand side of this equality have the same half plane of convergence which must coincide with that of the right hand side.
We denote by
the sequence
.
It is known that (see [1]) if
has a finite abscissa of convergence, then the abscissa of convergence of
is zero.
Moreover, if
has a discrete set of singular points on the imaginary axis, then
can be continued as a meromorphic function in the whole complex plane. Dirichlet L-series can be continued in this way.
Suppose that the series
admits such a continuation and satisfies (2). We will call the extended function Euler product Dirichlet function.
Meromorphic continuations are possible also for the series
and for the right hand side term in (3).
Let us denote by
the meromorphic function obtained on the right hand side in (3) by this continuation.
By the uniqueness theorem of analytic functions,
coincides with
; therefore it can have only simple poles.
2. The Geometry of the Mappings by General Dirichlet Series
From (1) it can be easily seen that
. We have shown in [2] that this happens uniformly with respect to t.
This simple fact has important consequences regarding the landscape of the pre-image of the real axis by a Dirichlet function (see Figure 1). A similar landscape is produced by the pre-image of the real axis by the derivative of
(see Figure 2).
Figure 1. The landscape of the pre-image of the real axis by Dirichlet L-functions defined by a complex and respectively a real Dirichlet character.
Figure 2. A sample of intertwining curves of the Riemann Zeta function.
Let us list a few facts valid for any Dirichlet function. The proofs can be found in [1] [2].
Proposition 2.1: The pre-image of the real axis by
contains infinitely many disjoint curves
extending for
from
to
which are mapped each one bijectively by
onto the interval
of the real axis. Consecutive curves
and
form infinite strips
,
, where
contains the real axis.
Proposition 2.2: Every strip
contains a unique component
of the pre-image of the real axis which is mapped bijectively by
onto the interval
and a unique unbounded component of the pre-image of the unit circle. It contains also a finite number of curves
which are mapped bijectively onto the whole real axis. Every curve
contains a unique zero of
.
Proposition 2.3: The curves
are disjoint, except that
can meet
or
into a double zero of
.
Proposition 2.4: If we color, for example, red the pre-image of the negative real half axis and blue the pre-image of the positive real half axis, then the pre-image of any circle centered at the origin will meet alternatively the color red and the color blue. This is the color alternating rule. It is illustrated in Figure 3. The same rule applies also to the pre-image of the real axis by any derivative of
.
Proposition 2.5: When represented in the same plane, the pre-image of the real axis by both
and
, come in couples of curves
and
, respectively
and
(see [2], Theorem 4) which intersect two by two (the intertwining curves).
Figure 3. An illustration of the color alternating rule. The vertical lines and the lines around zeros are pre-images of circles centered at the origin.
Proposition 2.6: The intertwining curves intersect each other in points where the tangent to
, respectively
is horizontal, or in multiple zeros, where the tangents do not exist. An illustration of this fact can be seen in Figure 2 above.
Proposition 2.7: If we denote by a the color of the pre-image of the negative real half axis by
and b that of the positive real half axis and by c and d the colors of the pre-image of the same half axes by
, then color a can meet only color d and color b can meet only color c, except for the case of
and
, where color d meets both color a and b. This is the color matching rule.
We notice that there is no exception to this rule when instead of the couple
and
we take the couple
and
since
does not change color. The same thing can be said for any couple of consecutive derivatives of
.
Proposition 2.8: The color matching rule forbids double zeros at the intersection of
and
when
.
Proposition 2.9: If
are the zeros of
in
, then the pre-image of the segment
from
to
and the curves
and
bound fundamental domains which are mapped conformally by
onto the whole complex plane with slits alongside these segments and the interval
of the real axis. In a similar way fundamental domains are obtained also for
.
3. Local Mapping Properties of Analytic Functions
It is known (see [3], page 133):
Proposition 3.1: If
is a regular point of the analytic function
, then in a neighborhood of
we have:
(4)
where n is a positive integer and
is analytic at
and
Depending on the value of n, the local mapping by
at
has the form seen in Figure 4.
Here the arcs
are mapped by
onto the interval
, where r is the radius of the image disc, such that
.
Figure 4. The local mapping by an analytic function at a regular point.
They are analytic arcs (see [3], page 234) i.e. the derivative
exists on
. Moreover,
exists. The angles at
are doubled, tripled, etc. by
according with
,
, etc. and they remain the same when
.
4. The Multiplicity of the Zeros of
We proved in [2] that linear combinations of linearly independent Dirichlet functions satisfying the same Riemann type of functional equation have double zeros.
If
is such a linear combination, the when
varies from 0 to 1 the zeros of
move continuously to the zeros of
.
When a couple of zeros of
symmetric with respect to the critical line move to a couple of zeros of
on the critical line, there must be a value of
for which the two zeros coincide, and therefore
has a double zero.
An illustration of such a zero can be seen in Figure 5.
These linear combinations cannot be Euler product functions. We will deal next with Euler product Dirichlet functions.
Theorem 4.1 Euler product Dirichlet functions do not have any multiple zero.
Proof: Suppose that
is an Euler product Dirichlet function. Then the formula (3) is valid for
, where the function
is meromorphic in the whole plane and has only simple poles. Indeed, by the section 3 at any point
the function
has the form
, where
and
is analytic at
and
.
When
, the function
is analytic at
and
, therefore
is analytic at
. When
then
, hence
,
where
, thus
has a simple pole at
.
Figure 5. The zeros of
for two close values of
. They must coincide for an intermediate
.
Let us assume that
is a multiple zero of
, i.e.
. Then, when trying to compute
we get an indetermination of the form 0/0. The l'Hospital rule is applicable, and
. If
then by Proposition 2.9
is an
interior point of a fundamental domain
of
. Since
the function
maps any unbounded curve
originating at
and such that
, for
onto a closed curve
passing through the origin. We can take
completely included in the fundamental domain
. Suppose that
is a point interior to
and let
be such that
. If
is an arbitrary point of
we can connect
and
by a curve
not intersecting
. Then the image of
by
cannot intersect
. But if
was outside
, this is a contradiction, hence we cannot have
.
On the other hand, if
then we still get an indetermination of the form 0/0 and the l'Hospital rule can be applied again giving
. By using the same argument as in the previous case we
infer that
cannot be different of zero and the process continues indefinitely. Hence all the coefficients of the Taylor expansion of
at
cancel and then
is identically equal to zero in a neighborhood of
, which is impossible.
The final conclusion is that
cannot have any multiple zero and the only zeros of
are the zeros of
, which as we have seen cannot be at the same time zeros of
.
Remark: We have seen that the Dirichlet functions
defined in [3] have double zeros for some values of
. They are not Euler product functions. One might think that despite of this fact, taking the ratio
we could deal with a function
as the function
above. However, when trying to define
at
we realize that this is impossible since
and all its derivatives cancel at
, which is absurd. Indeed,
should then be identically zero in a neighborhood of
and by the uniqueness theorem of analytic functions throughout its domain and this is obviously not the case. Therefore the condition on
to be an Euler product function is essential.
We emphasize also the fact that the Dirichlet L-functions, and in particular the Riemann Zeta function, are Euler product Dirichlet functions and therefore they cannot have multiple zeros. For a reader willing to aquire more information on this topic we recommend Chapter 11, Dirichlet series and Euler products of the monograph Multiplicative Number Theory by H. Montgomery and R. C. Vaughan.
5. Conclusion
Theorem 4.1 gives the solution of the outstanding problem, whether the Dirichlet functions admit or not multiple zeros. The answer is negative and it concerns a wider class of functions, namely those obtained by analytic continuation across the converging line of general Dirichlet series which can be written as Euler products.
Acknowledgements
We thank Aneta Costin for her support with technical matters.