APM 2017012314223447(2)

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Advances in Pure Mathematics, 2017, 7, 1-20
http://www.scirp.org/journal/apm
ISSN Online: 2160-0384
ISSN Print: 2160-0368
DOI: 10.4236/apm.2017.71001 January 23, 2017
The Geometry of the Mappings by General
Dirichlet Series
Dorin Ghisa
York University, Glendon College, Toronto, Canada
Abstract
We dealt in a series of previous publications with some geometric aspects of
the mappings by functions obtained as analytic continuations to the whole
complex plane of general
Dirichlet series. Pictures illustrating those aspects
contain a lot of other information which has been waiting for a rigorous
proof. Such a task is partially fulfilled in this paper, where we succeeded
among other things, to prove a theorem about general
Dirichlet series having
as corollary the Speiser’
s theorem. We have also proved that those functions
do not possess multiple zeros of order higher than 2
and the double zeros have
very particular locations. Moreover, their derivatives have only simple zeros.
With these results at hand, we revisited GRH for a simplified proof.
Keywords
General Dirichlet Series,
k
S
Strips, Intertwining Curves,
Fundamental Domains, Riemann Hypothesis
1. Introduction
The study of general Dirichlet series has its origins in the works [1]-[18] of E.
Cahen, J. Hadamard, E. Landau, H. Bohr, G. H. Hardi, M. Riesz, T. Kojima, M.
Kuniyeda, G. Valiron, etc. A lot of contemporary mathematicians created a
diversified theory of general Dirichlet series, some insisting on the connection
with the Laplace-Stieltjes transforms [19]-[24] as J. Yu, Y. Kong, S. Daochun, X.
Luo, Y. Yan, C. Singhal, G. S. Srivastava, etc., others as P.K. Kamthan, S. K.
Shing Gautam, L.H. Khoi, [25] [26] endowing them with some topological
structures, extending to them the Nevanlinna theory (too many to be cited) or
dealing with vector valued Dirichlet series [27]-[32] as A. Defant, M. Maestre, D.
Perez-Garcia, J. Bonet, B.L. Srivastava, A. Sharma, G. S. Srivastava, etc.
We will use normalized series defined as follows. To any sequence
How to cite this paper:
Ghisa, D. (2017
)
The Geometry of the Mappings by General
Dirichlet Series
.
Advances in Pure Math
e-
matics
,
7
, 1-20.
http://dx.doi.org/10.4236/apm.2017.71001
Received:
November 22, 2016
Accepted:
January 19, 2017
Published:
January 23, 2017
Copyright © 201
7 by author and
Scientific
Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY
4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access

D. Ghisa
2
( )
12
1 ,,A aa= = 
of complex numbers and any increasing sequence
( )
12
0
λλ
Λ= = < <
such that
lim
nn
λ
−>∞
= ∞
we associate the series
( )
,1
e
n
s
An
n
sa
λ
ζ
∞−
Λ=
=
∑
(1)
Λ
is called the
type
of the series (1) and the series defined by the same
Λ
will be called series of the
same type
. When
ln
n
n
λ
=
, we obtain the ordinary
(
proprement dites
, [18]) Dirichlet series.
Suppose that
A
and
Λ
are such that the
abscissa of convergence
(see [14]
[18]) of the series (1)
1
1
lim log
sup
n
ck
nk
n
a
σλ
−>∞ =
=
∑
is finite. (2)
Then (see [1])
( )
,A
zs
ζ
Λ
=
is an analytic function in the half plane
( )
u
s
σ
ℜ>
,
where
u
σ
is the
abscissa of uniform convergence
of (1), and where
u
σ
is at
most
cD
σ
+
with
log
limsup
nn
n
D
λ
−>∞
=
(3)
For the ordinary Dirichlet series
1D=
and it is known that in the case of
Dirichlet L-series defined by imprimitive Dirichlet characters
0
c
σ
=
and
1
u
σ
=
, while in the case of primitive characters
0
cu
σσ
= =
.
In general, when
0D=
then
uc
σσ
=
, and the series (1) is an analytic
function in the half plane
()
c
s
σ
ℜ>
.
Suppose that this function can be continued analytically to the whole complex
plane, except possible at
1
s=
which is a simple pole. We keep the notation
( )
,A
s
ζ
Λ
for the function obtained by analytic continuation.
With the exception of a discrete set of points from the complex plane, this
function is locally injective,
i.e
. it maps conformally and hence bijectively small
neighbourhoods of every point onto some domains. Enlarging these neigh-
bourhoods, the image domains get bigger. How big can they get? The answer is:
they become the whole complex plane with some slits (see [33] [34] [35] [36]). A
region with this property is called
fundamental domain
of
( )
,A
s
ζ
Λ
.
Our aim is to show that the complex plane can be divided into a countable
number of sets whose interiors are fundamental domains of this function. We
have done this previously for the Riemann Zeta function [37] [38] as well as for
Dirichlet L-functions [35]. To do the same thing for functions obtained by
analytic continuations of general Dirichlet series we made [36] the assumption
that (1) satisfies a Riemann type of functional equation. We will show next that
such a strong assumption is not necessary and we can obtain the same result by
using just elementary properties of the conformal mapping.
It has been proved [36] that for normalized series (1) the limit
( )
,
lim 1
A
it
σ
ζσ
→+∞ Λ
+=
is uniform with respect to
t
. This means that for any
0>
there is
c
σσ
≥

such that
σσ
>
implies
( )
,
1
A
s
ζ
Λ
−<

, therefore
the whole half plane
( )
s
σ
ℜ>

is mapped by
( )
,A
s
ζ
Λ
into a disc centred at
1z=
of radius

.

D. Ghisa
3
An immediate consequence of this fact is that the abscissa of convergence of a
normalized series is less than
+∞
. Moreover, for
1<
there is no zero of
( )
,
A
s
ζ
Λ
in the half plane
( )
s
σ
ℜ>

. Also, if
( )
,A
s
ζ
Λ
satisfies a Riemann
type of functional equation, then there cannot be non trivial zeros of
( )
,A
s
ζ
Λ
in the half plane
()
s
σ
ℜ <− 
neither.
2. Pre-Images of Lines and Circles
Suppose that, for a point
0
s
with
( )
0c
s
σ
ℜ>
the function
( )
,A
s
ζ
Λ
has a real
value
( )
,01
As
ζ
Λ>
. The continuation from
0
s
along the interval
[
)
1, +∞
, is a
curve
k
′
Γ
such that when
( )
,A
zs
ζ
Λ
=
tends to 1, we have that
σ
tends to
+∞
on
k
′
Γ
, or there is a point
,kj
u
such that
( )
,,
1
A kj
u
ζ
Λ
=
and the con-
tinuation can be carried along the whole real axis giving rise to a curve
,kj
Γ
. We
will show later that when
( )
,A
zs
ζ
Λ
=
tends to
+∞
on the real axis then
σ
tends to
−∞
on
k
′
Γ
, or on
,kj
Γ
, in other words these curves cannot remain in
a right half plane.
Let us notice first that
( )
,A
s
ζ
Λ
are transcendental functions and
s= ∞
is
an essential singular point for them. The value
0
z=
cannot be a lacunary
value for
()
,A
s
ζ
Λ
since then
( )
,
1
A
s
ζ
Λ
would be a non polynomial integer
function for which
s= ∞
is an essential singular point and hence
0s=
would be also an essential singular point for
( )
,A
s
ζ
Λ
, which is not true. Then,
by the Big Picard Theorem,
( )
,A
s
ζ
Λ
has infinitely many zeros in every neigh-
bourhood of
s= ∞
.
Given a bounded region of the plane, there is
0r>
such that the pre-image
of the circle
r
C
centred at the origin and of radius
r
has only disjoint com-
ponents, which are closed curves containing each one a unique zero of
( )
,A
s
ζ
Λ
belonging to that region. When
r
increases those curves expand and they can
touch one another at some points
k
v
(see Figure 1(b)). These are branch
points of the function, since in every neighbourhood of
k
v
the function takes at
least twice any value on the image circle. Therefore the derivative of
( )
,A
s
ζ
Λ
cancels at
k
v
. It is obvious that any zero of the derivative, which is not a zero of
the function itself, can be obtained in this way. Indeed, if
v
is such a zero, we
can take
( )
,A
rv
ζ
Λ
=
and necessarily at least two components of the pre-image
of
r
C
will pass through
v
.
What happens with those components of the pre-image of a circle
r
C
when
1r=
? We have proved [36], Theorem 1 that there is at least one unbounded
component of the pre-image of the unit circle. That proof did not use the as-
sumption of
( )
,A
s
ζ
Λ
satisfying a Riemann type of functional equation and
therefore it is true for any function
( )
,A
s
ζ
Λ
.
Let us notice that two curves
k
′
Γ
and
l
′
Γ
cannot intersect each other. Indeed,
if
0
s
would be a common point of these curves, then when
( )
,A
zs
ζ
Λ
=
moves
on the interval
I
between
( )
0 ,0A
zs
ζ
Λ
=
and 1 the point
s
describes an
unbounded curve which bounds a domain mapped by
( )
,A
s
ζ
Λ
onto the
complex plane with a slit alongside the interval
I
. That domain should contain

D. Ghisa
4
Figure 1. Pre-images of circles and the birth of a strip.
a pole of
( )
,A
s
ζ
Λ
which is not true. Therefore an intersection point
0
s
of the
two curves cannot exist and consecutive curves
k
′
Γ
and
1k+
′
Γ
bound infinite
strips
k
S
. We suppose that
0
S
is the strip containing the point
1s=
and for
every integer
k
, the curve
1k+
′
Γ
is situated above
k
′
Γ
. Figure 1(e) illustrates
the birth of a strip
k
S
when the pre-image of a ray making a small angle
α
with the positive real half axis is taken and then we let
0
α
→
.
We have also proved [36], Theorem 2 that every unbounded component of
the pre-image by
( )
,A
s
ζ
Λ
of the unit circle is contained between two con-
secutive curves
k
′
Γ
and
1k+
′
Γ
and vice-versa, if
0k≠
between two con-
secutive curves
k
′
Γ
and
1k+
′
Γ
there is a unique unbounded component of the
pre-image of the unit circle. It has been shown that the respective component
does not intersect any one of these curves. For
0
k≠
every strip
k
S
contains
also a unique curve
,0k
Γ
which is mapped bijectively by
( )
,A
s
ζ
Λ
onto the
interval
( )
,1−∞
of the real axis, as well as a certain number of curves
,kj
Γ
,
0j≠
which are mapped bijectively by
( )
,A
s
ζ
Λ
onto the whole real axis. There
are infinitely many strips
k
S
covering the whole complex plane [36], Theorem
4. Therefore, there are infinitely many unbounded components of the pre-image
by
( )
,A
s
ζ
Λ
of the unit circle. Some strips
k
S
can contain also bounded
components of the pre-image of the unit circle as well as bounded components
of the pre-image of
r
C
with
1r>
. Figure 1(d) portrays the strip
5
S
of
( )
s
ζ
containing two components of the pre-image of the unit circle: the
unbounded one containing two zeros and the other bounded, containing one
zero.
The use of the pre-image of the real axis can be traced back to Speiser’s work

D. Ghisa
5
[39] on the Riemann Zeta function. After that the pre-image of the real axis does
not appear any more in literature as a tool except for the paper of Arias-de-Reina
[40], who revisits Speiser’s theorem. In the same year John Derbyshire uses both:
the pre-image of the real axis and that of the imaginary axis in his popular book
Prime Obsession
and declares that they are
at the heart
of that book. The
classification of the components of the pre-image of the real axis by the Riemann
Zeta function appears for the first time in [37], where the strips
k
S
are also
introduced and a method is devised of partitioning them into fundamental
domains. Later, such a classification has been extended to Dirichlet L-functions
and finally to functions defined by general Dirichlet series.
A different approach, namely that of phase diagram, has been used by Elias
Wegert for visual exploration of complex functions [41] [42]. Applied to the
Riemann Zeta function, his phase plots revealed interesting patterns pertaining
to the universality property of that function. It is known that such a property
extends to more general Dirichlet series and probably it can complement our
fundamental domains approach.
An even better way to visualize the complexity of conformal mappings by
analytic functions of a complex variable is to use an orthogonal mesh in the
z-plane formed with rays issuing from the origin and circles centred at the origin.
Moreover, a spectre of colors can be superposed to the mesh as seen in Figures
2(a)-(c). By taking the pre-image of that mesh we obtain a coloured orthogonal
mesh in the s-plane in which the color of every point coincides with that of its
Figure 2. Colour-visualization of the conformal mapping by
( )
s
ζ
.

D. Ghisa
6
image. In this way we can locate the corresponding points in the two planes and
have also a global view of the mapping. The pre-image of the real axis and the
k
S
-strips are still identifiable. Figures 2(d)-(f) above illustrates the mapping by
the Riemann Zeta function in the rectangles
[ ] [ ]
20,20 30,30
− ×−
,
[ ] [ ]
1,5 1,000;1,020−×
and
[ ]
99
0.4,1.4 10 ,10 20

×+

. By comparing them, one
can notice the increasing number of zeros in the strips with increasing
t
. The
pattern we can see here is proper to any function
( )
,A
s
ζ
Λ
.
Theorem 1
No zero of
( )
,A
s
ζ
Λ
or of
( )
,A
s
ζ
Λ
′
can belong to a curve
k
′
Γ
.
Proof. The affirmation of the theorem is obvious for the zeros of
( )
,A
s
ζ
Λ
since 0 does not belong to the interval
[
)
1, +∞
of the real axis. A more intricate
argument guarantees that the same is true for the zeros of the derivative of
( )
,A
s
ζ
Λ
. Indeed, even if
1r>
, no bounded component of the pre-image of
r
C
can reach
k
′
Γ
, despite of the fact that
r
C
intersects the interval
[
)
1, +∞
.
Indeed, in the contrary case, the respective component should intersect
k
′
Γ
at
lest twice, or it should be tangent to it, fact which requires that
r
C
intersects
the interval
[
)
1, +∞
the same number of times or to be tangent to it, which is
not possible. The fact that
r
C
intersects the interval
[
)
1, +∞
has as effect the
pre-image of
r
C
intersecting the curves
,
kj
Γ
(and not
k
′
Γ
). It results that no
two bounded components of the pre-image of
r
C
can meet on
k
′
Γ
, neither
can one of these components meet on
k
′
Γ
an unbounded component of the
pre-image of
r
C
into a zero of
( )
,A
s
ζ
Λ
′
.
Remark: Theorem 1 does not imply that
( )
,A
s
ζ
Λ
′
cannot have zeros on some
,kj
Γ
. Such zeros appeared as possible for the Dirichlet L-function
( )
5, 2,Ls
as
seen in Figure 3 when
t
has approximately the values 169.2 and 179.2.
However, we suspected that this was due to the poor resolution of the picture
and indeed, when we zoomed on the respective points, we obtained configu-
rations which show clearly that
( )
,A
s
ζ
Λ
′
does not cancel there. However, as we
will see later, the possibility of such zeros cannot be excluded.
We have seen that the unbounded components of the pre-image of the unit
circle do not intersect any
k
′
Γ
. On the other hand the bounded components of
the pre-image of the unit circle intersect curves
,kj
Γ
at points
,kj
u
where
( )
,,
1
A kj
u
ζ
+
Λ−
=
. In the same way the bounded components of the pre-image of
r
C
with
1r>
will intersect
,kj
Γ
at points
u
where
( )
,A
ur
ζ
+
Λ−
=
. However
the story of the unbounded components of the pre-image of
r
C
is a little more
complicated.
When increasing
r
past 1 all the unbounded components of the pre-image
of
1
C
fuse together into a unique unbounded curve
r
γ
intersecting every
curve
k
′
Γ
, hence they do not generate by this fusion zeros of
( )
,As
ζ
Λ
′
. Indeed,
since the mapping of
k
′
Γ
onto the interval
( )
1, +∞
is bijective, there should be
a unique point
k
s
on every
k
′
Γ
such that
( )
,Ak
sr
ζ
Λ
=
. The continuation over
r
C
from each one of these points can be made clockwise and counter clockwise
into
k
S
, respectively
1k
S
−
, for every
k
, giving rise to that unbounded curve.
The final conclusion is that
( )
,A
s
ζ
Λ
′
does not cancel on any
k
′
Γ
. ■

D. Ghisa
7
Figure 3. Possible zeros of
( )
,
A
s
ζ
Λ
′
on
,kj
Γ
which are not double zeros of
( )
,A
s
ζ
Λ
.
Given any bounded domain in the plane
( )
s
, we can take
1r>
close
enough to 1 such that
r
γ
does not touch any bounded component of the
pre-image of
r
C
included in that domain. However, for bigger values of
r
the
curve
r
γ
comes into contact with bounded components which were turning
around one or several zeros
,kj
s
of
( )
,A
s
ζ
Λ
, fusing with them and getting to
the left of those zeros, hence intersecting also the curves
,
kj
Γ
which contain the
respective zeros. The curve
r
γ
is orthogonal to every component of the pre-
image of the real axis if it intersects that component at a point where
( )
,A
s
ζ
Λ
′
does not cancel.
Since a point turning around the origin in the same direction on an arbitrary
circle
r
C
centred at the origin will meet consecutively the positive and the
negative real half axis, the components of the pre-image of
r
C
(including
r
γ
when
1r>
) should meet consecutively the pre-image of the positive and the
negative half axis (coloured differently). This is [33] [35] the so-called
color
alternating rule
.
An immediate consequence of this rule is that in every strip
k
S
the first and
the last curve
,kj
Γ
should be such that the pre-image of the negative real half
axis faces the corresponding
k
′
Γ
. Then, consecutive
,kj
Γ
have the same orien-
tation, as long as they are on the same side of
,0k
Γ
. A zero
0
s
of
( )
,A
s
ζ
Λ
′
being on
,kj
Γ
and not being a zero of
()
,A
s
ζ
Λ
(where the colors are changing)
implies that different components of the pre-image of a circle
r
C
fuse (for a
value
0
r
of
r
) on
,kj
Γ
. The respective components must obey the color
alternating rule for every
0
rr<
and we realize that after fusion the component

D. Ghisa
8
can be such that the rule is still in force. Indeed, if
,kj
Γ
and
,kj
′
Γ
have the
same color at
0
s
then this curve can continue to meet the respective color
without affecting the color alternation by simply switching the branches of
,kj
Γ
and
,kj
′
Γ
which come to
0
s
. However, as we will see next, the position of
0
s
with respect to the two zeros of
( )
,A
s
ζ
Λ
on these curves cannot be arbitrary.
Theorem 2
For
k
different of
0
, there is no zero of the derivative of
( )
,A
s
ζ
Λ
′
at the left of the leftmost zero of
( )
,A
s
ζ
Λ
in
k
S
Proof. We can reproduce the proof of Theorem 1 from [43] for arbitrary
functions
( )
,A
s
ζ
Λ
. Let
,kj
s
be the leftmost zero of
( )
,A
s
ζ
Λ
from
k
S
and
suppose that the simple zero
,kj
v
of
( )
,A
s
ζ
Λ
′
is a progenitor of
,
kj
s
,
i.e
. a
component of the pre-image of the circle
r
C
with
( )
,,
A kj
rv
ζ
Λ
=
contains the
point ,
kj
s
. It is obviously enough to deal with the case where
0j>
,
i.e
. where
,kj
Γ
is above
,0k
Γ
. Then
( )
,,A kj
v
ζ
Λ
(indicated in Figure 4 as
( )
,kj
Lv
, since it
has been partially computer generated by a Dirichlet L-function) belongs to the
upper half plane and
0π
α
≤<
, where
( )
,,
arg
A kj
v
αζ
Λ
=
.
The pre-image of the ray determined by
()
,,A kj
v
ζ
Λ
contains two curves
which are orthogonal at ,
kj
v
. The angles at ,
kj
v
are doubled by
( )
,A
s
ζ
Λ
,
hence the four arcs of the pre-image of that ray make angles of
2
α
,
π22
α
+
,
π2
α
+
and
3π22
α
+
with a horizontal line whose image passes through
( )
,,A kj
v
ζ
Λ
. The angle
π22
α
+
made by the second arc (which ends in
,
kj
s
)
with this line is less than the angle
β
made by the tangent to the pre-image of
the respective ray at any point between ,
kj
s
and ,
kj
v
with the same horizontal
line. If
( ) ( )
,,kj kj
vsℜ ≤ℜ
, then there must be a point on that arc for which
π2
β
=
, therefore
0
α
<
, which is absurd. ■
Corollary: If
( )
,
A
s
ζ
Λ
satisfies RH, then the zeros of
( )
,
A
s
ζ
Λ
′
from every
strip
k
S
are at the right side of the critical line. When
( )
,A
s
ζ
Λ
is the Riemann
Figure 4. The location of the zeros of
( )
,A
s
ζ
Λ
′
.

D. Ghisa
9
Zeta function this corollary represents the Speiser’s theorem [39].
The existence of multiple zeros of functions obtained by analytic con-
tinuations of Dirichlet series has been documented (probably for the first time)
in [44], where double zeros of a linear combination of Dirichlet L-functions have
been found (see Figure 5 below).
We have shown that all those double zeros are located on the critical line. In
this example the function is
( ) ( )
7,2, 0.34375 7,4,Ls Ls+
and the double zero
is obtained for the approximate value of
s
of
0.5 31.6i
+
. The double zeros we
have found for all the functions of this type were located at the intersection of
,0
k
Γ
and
,1k
Γ
or of
,0
k
Γ
and
,1
k−
Γ
. We can make now a much more general
affirmation about the multiple zeros of functions obtained by analytic con-
tinuation of general Dirichlet series.
Theorem 3
In every strip
k
S
of a function
( )
,A
s
ζ
Λ
this function has at
most one double zero. Such a zero is found at the intersection of
,0k
Γ
and
,1k
Γ
or of
,0k
Γ
and
,1k−
Γ
. There is no multiple zero of
( )
,A
s
ζ
Λ
′
in
k
S
and hence
no zero of a higher order than two of
( )
,A
s
ζ
Λ
.
We need to postpone the proof of this theorem for a while.
3. Intertwining Curves
When studying functions
( )
,A
zs
ζ
Λ
=
it is useful to consider besides the planes
( )
s
and
( )
z
also a plane
( )
w
, where
( )
,A
ws
ζ
Λ
′
=
. Sometimes the planes
( )
z
and
( )
w
will be identified in order to make more obvious certain relations
between the configurations defined by the two functions in the respective planes.
The configurations we have in view are pre-images by both
( )
,A
s
ζ
Λ
and by
Figure 5. A double zero of a linear combination of dirichlet L-functions.

D. Ghisa
10
( )
,A
s
ζ
Λ
′
, of some curves or domains.
Regarding the pre-image by
( )
,A
s
ζ
Λ
and by
( )
,A
s
ζ
Λ
′
of the real axis it has
been found [34] [36] that the components of these pre-images are paired in such
a way that only the components of the same pair can intersect each other. The
respective pairs form the so-called intertwining curves.
Three kinds of intertwining curves have been distinguished [33] [34] [35] [36],
namely:
1)
k
′
Γ
and
k
′
ϒ
,
k∈
, which are mapped bijectively by
( )
,A
s
ζ
Λ
and by
( )
,A
s
ζ
Λ
′
onto the interval
( )
1, +∞
, respectively
()
,0
−∞
,
2)
,0k
Γ
and
,0k
ϒ
,
k∈
, which are mapped bijectively by
( )
,A
s
ζ
Λ
and by
()
,A
s
ζ
Λ
′
onto the interval
( )
,1−∞
, respectively
()
0,+∞
3)
,
kj
Γ
and
,kj
ϒ
,
0j≠
,
k∈
,
{ }
\0
k
jJ∈=
(a finite set of integers),
which are mapped bijectively by
( )
,As
ζ
Λ
, respectively by
( )
,A
s
ζ
Λ
′
onto the
whole real axis.
Theorem 4
The intertwining curves touch each other at the points where the
tangent to
,
kj
Γ
,
respectively
k
′
Γ
is horizontal. Vice-versa
,
if at a point of such
a curve the tangent is horizontal
,
then a component of the pre-image of the real
axis by
( )
,A
s
ζ
Λ
′
passes also through that point.
Proof. Indeed, suppose that
( )
s sx=
is the equation of a curve
k
′
Γ
or of a
curve
,kj
Γ
such that
( )
( )
,Asx x
ζ
Λ=
. Then (see Figure 6 below)
()
()
()
,
1
A
sx s x
ζ
Λ
′′
=
. (4)
Figure 6. Intertwining curves of
( )
it
ζσ
+
for
[ ]
0,100t∈
.

D. Ghisa
11
The Equation (4) shows that the argument of
( )
( )
,A
sx
ζ
Λ
′
is opposite to that
of
( )
sx
′
, therefore they cancel simultaneously, and when one is
π
, the other
should be
π−
. Yet these values of the argument of a point mean that the
respective point is on the real axis, therefore
( )
sx
belongs to both pre-images
of the real axis, by
( )
,A
s
ζ
Λ
and
( )
,A
s
ζ
Λ
′
which completely proves the theorem.
■
Remark: The Theorem 4 is a corollary of a much more general property
which says that if
()
z fs=
is an analytic function in a domain
D
of the
complex plane and
γ
is the image by
( )
fs
of a smooth curve
( )
:s szΓ=
,
then denoting by
ϒ
the pre-image by
()
fs
′
of
γ
, at every point
0
s
where
Γ
and
ϒ
intersect each other we have
( ) ( )
( )
00
arg arg 0 mod 2πf s sz
′′
+=
,
where
( )
00
s sz=
.
Indeed, suppose that the planes
( )
z
and
()
w
are identified, where
( )
w fs
′
=
and write
( )
S Sw=
for the curve
ϒ
. Then at an intersection
point
0
s
of
Γ
and
ϒ
we have
( ) ( )
( )
00 0
s sz S f s
′
= =
and
( )
( )
f sz z=
implies
( )
( )
( )
1f sz s z
′′
=
, hence
( ) ( )
00
1f s sz
′′
=
, etc.
We can show now that:
Theorem 5
No strip
k
S
can be included in a right half plane.
Proof. The formula (4) written for
k
′
Γ
tells us that when
( )
sx
′
tends to
∞
then
( )
( )
,A
sx
ζ
Λ
′
must tend to 0. Yet, there is no zero of
( )
( )
,A
sx
ζ
Λ
′
on
any
k
′
Γ
. On the other hand,
( )
( )
,A
sx
ζ
Λ
′
is an unlimited continuation of
( )
,A
s
ζ
Λ
′
alongside
k
′
Γ
, hence when
()
sx
tends to
∞
we have that
( )
( )
,A
sx
ζ
Λ
′
tends to
∞
not to 0 and this is a contradiction. The conclusion is
that the geometry of the pre-image of the real axis is in the whole complex plane
similar to that we can see in a bounded region of the plane in Figure 2, Figure 3,
Figure 6, Figure 7. ■
Summarizing these facts and having in view [36] we can say:
Theorem 6
The variable
σ
takes any real value on every curve
k
′
Γ
. Con-
secutive curves
k
′
Γ
form infinite strips
k
S
which are mapped
(
not necessarily
bijectively
)
onto the whole complex plane with a slit alongside the interval
[
)
1, +∞
of the real axis. If
k
S
contains
k
m
zeros of the function
( )
,A
s
ζ
Λ
,
then it will contain
1
k
m−
zeros of
( )
,A
s
ζ
Λ
′
. For any given
k
,
the function
( )
,A
s
ζ
Λ
either has a finite number of zeros
,kj
s
in
k
S
,
or
,
lim
j kj
s
→∞
= ∞
. If
( )
,A
s
ζ
Λ
satisfies a Riemann type of functional equation
,
every strip
k
S
,
0k≠
contains a finite number of zeros of this function. The strip
0
S
may contain
infinitely many zeros of
( )
,A
s
ζ
Λ
.
Proof. We only need to justify the numbers
k
m
and
1
k
m−
to which the
theorem makes reference. We have seen that every zero of
( )
,As
ζ
Λ
′
is obtained
when two components of the pre-image of some circle
r
C
centred at the origin
and of radius
r
come into contact. If we consider the zeros of
( )
,A
s
ζ
Λ
as the
leafs of a binary tree whose internal nodes are obtained in this way, that tree is a
complete binary tree and it is known that it must have exactly
1
k
m−
internal
nodes. Figure 7 illustrates this situation for ten strips of the Riemann Zeta
function. The location of the zeros of
( )
s
ζ
and those of its first two derivative

D. Ghisa
12
Figure 7. The zeros of
( )
s
ζ
and of its first two derivatives for
[ ]
0,100t∈
.
are also clearly indicated. If there is a double zero of
( )
,A
s
ζ
Λ
in
k
S
, then the
respective leaf is counted twice as a leaf and once as an internal node, etc. When
the number of zeros
,kj
s
in
k
S
is infinite we must have
,
lim
j kj
s
→∞
= ∞
,
otherwise
,kj
s
would have an accumulation point in
k
S
and hence
( )
,A
s
ζ
Λ
would be identically zero. ■
We can prove now also the Theorem 3.
Proof. Exactly 2 curves
,kj
Γ
and
,kj
′
Γ
should meet at a double zero
0
s
of
( )
,A
s
ζ
Λ
belonging to
k
S
. But
0
s
is also a simple zero of
( )
,A
s
ζ
Λ
′
and
therefore a curve, say
,kj
ϒ
should pass through
0
s
. Yet,
,kj
′
Γ
intertwines
with a curve
,kj
′
ϒ
and unless this curve is
,0k
ϒ
, which does not contain any

D. Ghisa
13
zero of
( )
,A
s
ζ
Λ
′
, the point
0
s
would be a double zero of for
( )
,A
s
ζ
Λ
′
, and this
is impossible.
Hence , necessarily one of the curves
,kj
Γ
passing through
0
s
is
,0k
Γ
. This
shows that there can be only one double zero in
k
S
and the other curve passing
through
0
s
in that case is either
,1k
Γ
, or
,1k−
Γ
.
We notice that the curve
,0k
ϒ
does not contain any zero of
( )
,A
s
ζ
Λ
′
, hence
it cannot pass through a multiple zero of order
m
of
( )
,As
ζ
Λ
′
. Since every
curve passing through that zero has an intertwining curve defined by the second
derivative of
( )
,A
s
ζ
Λ
, the respective point should be a multiple zero of order
m
of this second derivative, which is absurd. Therefore
()
,A
s
ζ
Λ
′
has no
multiple zero and then
( )
,A
s
ζ
Λ
cannot have any zero of the higher order than
2. ■
4. Fundamental Domains
For
0j≠
, the curves
,kj
Γ
as well as
,
kj
ϒ
are parabola like curves with
branches extending to infinity as
σ
→ −∞
. Therefore we can distinguish
between the interior and the exterior of such curves. They can be viewed as
oriented curves, with the same orientation as the real axis whose components of
the pre-image they are. By the same rule
k
′
Γ
and
k
′
ϒ
, as well as
,0k
ϒ
are also
oriented, the positive orientation of
k
′
Γ
and of
,0k
ϒ
being from the right to
the left, while that of
k
′
ϒ
, is from the left to the right. Some curves
,kj
Γ
,
0j≠
can contain in interior some other curves of the same type (
embraced curves
)
and by the color alternating rule the orientation of the
embracing curve
and that
of the embraced curves must be different. We did not find any instance where an
embraced curve is in turn embracing, yet there is no reason to believe that such a
situation is impossible.
Theorem 7
If the strip
k
S
contains
k
m
zeros counted with multiplicities
,
then
k
S
can be partitioned into
k
m
sub-strips which are fundamental do-
mains for
( )
,A
s
ζ
Λ
Proof. Suppose that curves
,
kj
Γ
and
,kj
′
Γ
are containing the simple zeros
,
kj
s
and
,kj
s
′
and two components of the pre-image of a circle
r
C
which are
going around each one of these zeros touch at a point
,kj
v
. This is a zero of
( )
,A
s
ζ
Λ
′
. The pre-image of the segment of line connecting
1z=
and
( )
,,A kj
v
ζ
Λ
has as component an arc
,kj
η
connecting the points on the two
curves where
( )
,
1
A
s
ζ
Λ
=
and passing through
,kj
v
. If one of these curves is
,0k
Γ
then the respective point is
∞
and
,kj
η
is an unbounded curve. The
strip
,kj
Ω
bounded by this curve and the branches of
,kj
Γ
and
,kj
′
Γ
corre-
sponding to the interval
[
)
1, +∞
, turns out to be a fundamental domain of
( )
,A
s
ζ
Λ
. Indeed,
,kj
Ω
is mapped conformally by
( )
,A
s
ζ
Λ
onto the whole
complex plane with a slit alongside the interval
[
)
1, +∞
followed by a slit
alongside the segment from
1z=
to
( )
,,A kj
v
ζ
Λ
. If
,kj
Γ
embraces
,kj
′
Γ
then
,kj
Ω
is bounded to the right.
Suppose now that one of the two zeros is a double zero of
( )
,A
s
ζ
Λ
. We know
that
,0k
Γ
must pass through that zero and the part of
,0k
Γ
corresponding to

D. Ghisa
14
the interval
[ ]
0,1
becomes part of the boundaries of adjacent fundamental
domains. One of them will have as image the complex plane with a slit alongside
the positive real half axis and for the other one a slit from
1z=
to
( )
,,A kj
zv
ζ
Λ
=
should be added, where
j
is 1 or −1.
We know that
,kj
s
cannot have a higher order of multiplicity and therefore
the cases analysed exhaust all the possibilities. Having in view Theorem 6, we
conclude that the strip
k
S
can be always divided into
k
m
fundamental
domains. ■
Figure 8 below illustrates this theorem for the case of the Riemann Zeta
(a) (b)
(c)
Figure 8. Two fundamental domains and their images.

D. Ghisa
15
function and the strip
2
S
.
As seen in the case of Dirichlet L-functions (including the Riemann Zeta
function) the strip
0
S
has infinitely many zeros, yet following the same
technique we can divide it into infinitely many fundamental domains.
If four different colors are used, say color 1 and 2 for the pre-images by
( )
,A
s
ζ
Λ
of the positive and of the negative real half axis and 3 and 4 for the
pre-images by
( )
,A
s
ζ
Λ
′
of the same half axes, then two simple topological facts
can be established [33]:
a) The
color alternating rule
, which states that as a point turns indefinitely in
the same direction on a circle centred at the origin, the pre-images of this point
by each one of the functions
( )
,A
s
ζ
Λ
and
( )
,A
s
ζ
Λ
′
will meet alternatively the
colors 1 and 2, respectively 3 and 4.
b) The
color matching rule
, which states that when intertwining curves meet
each other, then if these are not
,0
k
Γ
and
,0
k
ϒ
color 1 will always meet color 4
and color 2 will always meet color 3. Only the curves
,0
k
Γ
and
,0k
ϒ
can
intersect each other at points where color 2 can meet color 4.
The series (1) which are Euler products display special important properties.
5. Euler Products
It is known that the Dirichlet L-functions are meromorphic continuations of
ordinary Dirichlet series defined by Dirichlet characters and these series can be
expressed as Euler products. This property is a corollary of the fact that the
Dirichlet characters are totally multiplicative functions (see for example [33]. Yet
the property of being total multiplicative can be extended to general Dirichlet
series, as done in [35], and therefore some of the general Dirichlet series
( )
,A
s
ζ
Λ
(see the details in [35]) can also be written as Euler products:
( )
( )
1
,p
1
e 1e
p
n
s
s
A np
n
sa a
λ
λ
ζ
−
∞−
−
Λ∈
=
= =Π−
∑

, (5)
where

is the set of prime numbers. This convention will be kept in the
following for all the products and sums involving the subscript
p
. The product
has the same abscissa of convergence as the series itself.
Looking for counterexamples to the Grand Riemann Hypothesis (GRH), some
Dirichlet series satisfying a Riemann type of functional equation have been
found, whose analytic continuation exhibit off critical line non trivial zeros,
namely the Davenport and Heilbronn type of functions and linear combinations
of L-functions satisfying the same functional equation. Although these are not
counterexamples to GRH, their study allowed us to draw interesting conclusions.
We have seen in [36] that if
( )
,A
s
ζ
Λ
does not satisfy the GRH, then for every
two distinct non trivial zeros
1
s it
σ
= +
and
21s it
σ
=−+
there is
0
τ
,
0
01
τ
<<
such that
( )
( )
,0
0
A
s
ζτ
Λ
′=
, where
( ) ( )
12
1s ss
τ ττ
=−+
,
i.e
. the
derivative of
( )
,A
s
ζ
Λ
cancels at a point
( )
00
ss
τ
=
of the interval
I
deter-
mined by
1
s
and
2
s
. Moreover,
( )
0
12
sℜ<
.
Let us rephrase and give a simplified proof to [36], Theorem 3.
Theorem 8
Suppose that the function
(5)
satisfies a Riemann type of func-

D. Ghisa
16
tional equation and the respective series has the abscissa of convergence
12
c
σ
<
. Then for every non trivial zero
it
σ
+
of
( )
,A
s
ζ
Λ
we have
12
σ
=
.
Proof. Suppose that there is a zero
it
σ
+
of
( )
,A
s
ζ
Λ
for which
12
σ
>
.
Then, due to the functional equation,
1it
σ
−+
is also a zero of
( )
,A
s
ζ
Λ
(see
Figure 9 below). There is
0r>
such that in the components
1
D
and
2
D
of
the pre-image of the disc centred at the origin and of radius
r
containing the
respective zeros the function
( )
,A
s
ζ
Λ
is injective. Then we can define the
function
12
:DD
φ
→
as follows:
( ) ( ) ( )
1
,,
2
AA
D
ss s
φζ ζ
−
ΛΛ
=
(6)
The function
φ
can be continued as an analytic involution of the union
′
Ω∪Ω
of the fundamental domains
Ω
and
′
Ω
of
( )
,A
s
ζ
Λ
containing the
respective zeros. The boundaries of the domains
Ω
and
′
Ω
have a common
component
L
and the union
L
′
Ω∪Ω ∪
is a simply connected domain
H
.
The function
φ
can be continued to an analytic involution of
H
having
0
s
as a fixed point. We have
( )
()
ss
φφ
=
in
H
, in particular
( )
1it it
φσ σ
+ =−+
and
( )
1it it
φσ σ
−+ =+
. Moreover,
( )
( )
( )
,,AA
ss
ζφ ζ
ΛΛ
=
in
H
. Let us define the function
Φ
by
( ) ( ) ( )
( )
,,AA
ss s
ζ ζφ
ΛΛ
Φ=
(7)
Since the numerator and the denominator of
Φ
are analytic functions in
H
and the denominator cancels only at
it
σ
+
and
1it
σ
−+
, the function
Φ
is analytic in
H
except at these two points. Since
( ) ( )
( )
,,AA
ss
ζ ζφ
ΛΛ
=
,
we have that
( )
1sΦ=
for
s
not equal to one of these points. Yet they are
removable singularities, and we can set
( )
1sΦ=
in
H
. By the formula (5) we
have
( )
( )
( )
( )
1e 1e
pp
ss
pp p
sa a
λφ λ
−−
∈
Φ=Π − −

(8)
for
s
∈
H
,
( )
s
ℜ
>
σc
. In particular,
Figure 9. Symmetric zeros with respect to the critical line.

D. Ghisa
17
()
( ) ( )
( )
1
1
1
1e 1e
ee
pp
pp pp
it it
pp p
ti ti
pp p
it
aa
ea ea
λ σ λσ
λλσλλσ
σ
− −+ − +
∈
−− −
∈
=Φ+
  
=Π− −
  


=Π− −




The arguments of these ratios represent the angles under which the segment
between
( )
1
e
p
p
a
λσ
−−
and
e
p
p
a
λσ
−
is seen from the point
ep
ti
λ
on the unit
circle. If a Ramanujan type condition is fulfilled, namely
A
and
Λ
are such
that for every
0>
we have
lim e 0
n
nn
a
λε
−
→∞
=
, then the respective angles
tend to zero as
p→∞
. This appears to be a necessary condition for the
convergence of the series.
()
1
argee ee
pp pp
ti ti
pp
p
aa
λλσλλσ
−− −
∈


−−


∑

However, the condition is implicitly satisfied since we know that
( )
sΦ
is
well defined in the domain
H
. The series is a continuous function of
σ
, yet it
can take only integer multiple values of
2π
, which is possible only if it is a
constant. Since
lim e 0,
n
nn
a
λδ
−
→∞
=
the series can remain constant only if there
is
p
0 such that for
p
>
p
0 we have
( )
1
argee ee0.
pp pp
ti ti
pp
aa
λλσλλσ
−− −


− −=


This can happen in two situations: either the three points
e
p
ti
λ
,
e
p
p
a
λσ
−
and
( )
1
ep
p
a
λσ
−−
are collinear, or
( )
1
ee
pp
pp
aa
λ σ λσ
−− −
=
and in this last case
1
σσ
−=
,
i.e
.
12
σ
=
. In the first case, a shift in
t
will not affect the real part of the zeros,
yet it will destroy the collinearity of the three points and therefore this situation
can be ignored. The final conclusion is that
( )
,A
s
ς
Λ
cannot have any non trivi-
al zero
it
σ
+
with
σ
strictly greater than 1/2. The zero
12 it+
can be
either a simple zero and then we have only one fundamental domain
Ω
, or a
double zero and then it is the fixed point of the involution
( )
s
φ
. In both cases it
is located on the critical line, which completely proves the theorem.
■
We need to point out the fact that Figure 9 above illustrates a situation in
which the Equation (5) is not satisfied.
Remark: In all of our publications we understood by trivial zeros of an L-
function those zeros which can be trivially computed. In this respect, the non
trivial zeros of the alternating Zeta function
( )
( )
( )
1
12
s
a
ss
ζζ
−
= −
are the
same as those of
( )
s
ζ
and
( )
a
s
ζ
satisfies the conditions of Theorem 8,
therefore its non trivial zeros are located on the critical line. Thus, RH is also
true for
( )
s
ζ
.
Similarly, the non trivial zeros of a Dirichlet L-function
( )
( )
( )
1
,1, 1
s
Lq s q s
ζ
−
= −
defined by the principal character modulo
q
are the
non trivial zeros of the Riemann Zeta function.
Also, the non trivial zeros of a Dirichlet L-function induced by an imprimitive
character are the non trivial zeros of the function defined by the associated
primitive character. Consequently, the RH for any Dirichlet L-function is
fulfilled. With this understanding of the concept of non trivial zeros, Theorem 8
represents the proof of GRH for a wide class of functions.

D. Ghisa
18
Acknowledgements
The author is very much thankful to Florin Alan Muscutar for his contribution
with computer generated graphics.
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