On the Non-Trivial Zeros of Dirichlet Functions

The purpose of this research is to extend to the functions obtained by meromorphic continuation of general Dirichlet series some properties of the functions in the Selberg class, which are all generated by ordinary Dirichlet series. We wanted to put to work the powerful tool of the geometry of conformal mappings of these functions, which we built in previous research, in order to study the location of their non-trivial zeros. A new approach of the concept of multiplier in Riemann type of functional equation was necessary and we have shown that with this approach the non-trivial zeros of the Dirichlet function satisfying a Reimann type of functional equation are either on the critical line, or they are two by two symmetric with respect to the critical line. The Euler product general Dirichlet series are defined, a wide class of such series is presented, and finally by using geometric and analytic arguments it is proved that for Euler product functions the symmetric zeros with respect to the critical line must coincide.


Introduction
The fundamental domains, as defined by Ahlfors, are for the complex functions of one complex variable the similar of the intervals of monotony for the real ones, in the sense that in both cases the functions are injective there. However, for the analytic functions of one complex variable the concept of fundamental domain plays a much deeper role. Such a domain is mapped conformally by the function onto the whole complex plane with some slits. The rational functions have a finite number of fundamental domains (the same as their degree), while the transcendental ones have infinitely many fundamental domains. Their closure covers the entire complex plane.
The Dirichlet functions are meromorphic functions in the complex plane having an essential singular point at infinity. We have shown in previous works that every neighbourhood of an essential singular point of an analytic function contains infinitely many fundamental domains. Therefore a Dirichlet function has infinitely many fundamental domains outside any bounded region of complex plane. We found that these domains are infinite strips which can be obtained by using the pre-image of the real axis and the zeros of the derivative of the function, necessarily situated on the common boundaries of these domains. These are basic knowledge in the study of the location of non-trivial zeros of Dirichlet functions. The symmetric zeros with respect to the critical line are located in adjacent fundamental domains, which can be mapped conformally one onto the other such that the image of a zero is the other zero. This function can be extended to an involution on the union of the two domains and this is instrumental in using the Euler product in order to prove RH, which is the final result of this paper.
We are dealing with normalized series defined by using an arbitrary sequence of complex numbers ( )  has been studied by Cahen [1] who proved (see also [12]): Proposition 1: If the series (1) has a finite abscissa of convergence, then the series (4) has the abscissa of convergence zero.
Analogously, if the series (1) has a finite abscissa of absolute convergence, then the series (4) has the abscissa of absolute convergence zero.
The sufficient condition found in [12] for the series (1) to admit meromorphic continuation into the whole complex plane can be relaxed as follows: Theorem 1: Suppose that the series (1) has a finite abscissa of absolute convergence a σ and the series (4) has a discrete set of singular points on the imaginary axis in the neighborhood of the origin. Then the series (1) can be continued meromorphically into the whole complex plane.
Proof: We will use the Riemann's technique of contour integration ( [13], page 214). Namely, in the integral formula of Euler Gamma function, true for we replace x by e n x λ . Then (5) implies: If we multiply here the n-th term by n a and add from 1 n = to ∞ we get . Trying to extend this continuation to the whole complex plane by proceeding as Riemann did for ( ) s ζ (see [13], page 214) we are confronted with a difficulty, since the integration contour used by Riemann is inappropriate in our case,

( )
,e A z ζ Λ not being defined for Re 0 z ≤ . We can avoid this inconvenience by choosing a different contour of integration completely included in the convergence domain of the series (4) (see [12]), namely a contour C r formed with the half circle : e , 2 2 i r z r θ γ θ = − ≤ ≤ π π , where r is small enough such that no divergence point of (4) except the origin is located on the diameter of this half circle, and the half lines Im z r = ± , Re 0 z ≥ (see Figure 1 below).
It can be easily checked that   . The terms which contain r under that integral tend all to zero as 0 r → , and therefore their integrals cancel too at the limit. We conclude that: the formula (9) becomes: The limit (8)   The Hadamard's formula shows that the radius of convergence of this series is 1, i.e. the series converges for 1 w < and it diverges for 1 w > . Let us show that the series diverges at every point of the unit circle. Since the set of points e i w θ = with the argument θ rational multiple of 2π is dense in the unit circle, it will be enough to show that the series diverges at every one of these points.
, where p and q are integers.

The Geometry of Mappings by Dirichlet Functions
We dealt in [15] with different classes of Dirichlet series, all admitting meromorphic continuation into the whole complex plane and we found that the respective Dirichlet functions displayed strong similarities regarding the geometry of their conformal mappings. We will list here those properties for a generic Dirichlet function.   Starting with Speiser's work [17] the pre-image of the real axis has been used in order to describe the geometry of mappings by the Riemann Zeta function.
A historic account of this technique can be found in [16], Section 3. In our In what follows we will use the concept of continuation above a curve (see [19],  = and the continuation can be carried along the real axis for values less than 1. When s does not meet any pole in its way, we have that s → ∞ on Γ when x → −∞ , i.e. Γ is unbounded at both ends and it is mapped one-to-one onto the whole real axis. If s meets a pole, the continuation stops there in the sense that when x → −∞ , s approaches that pole. Let us take a ray α η making a small angle 0 α > with the positive real half axis in the z-plane and let z α be the intersection of this ray with the unit circle. We summarize by saying (see [16]) that there can be three types of connected components of the pre-image of the real axis by  from that strip (see [16], Theorem 6), which makes possible the construction of the arcs and curves

The Zeros of Dirichlet Functions
It is obvious that every curve Moreover, no zero can exist outside these curves and if a zero 0 s has the order of multiplicity m then m curves , k j Γ must pass through 0 s . We have shown (see [15], Theorem 3) that 2 m ≤ and every strip k S can contain at most one double zero. When such a zero exists it must be at the intersection of In his axiomatic approach, dealing with ordinary Dirichlet series, another property that those series can have has been taken into account, namely of being expressible as an Euler product. We continue to deal with general Dirichlet series which admit meromorphic extensions into the whole complex plane and look for properties of the respective functions similar to those postulated by Selberg.
Let us notice first that the functional equation Riemann has found for the ζfunction has been a by-product of the continuation process of the ζ -series  [13], page 216). By a happy coincidence, the sum of residues of the integrand has as a factor ( ) . It is obvious that besides the trivial zeros of ( ) and not the fact that they are real zeros. It is just a coincidence that in the particular case of ( ) s ζ they are the same. This is a crucial fact to be settled, since the extension of RH depends essentially on it. We have proved in [16]:  In what follows we will deal with general Dirichlet series whose coefficients and exponents satisfy some special conditions. We consider the coefficients as = for all n ∈  . We say that χ is a Dirichlet character modulo m. The ordinary Dirichlet series defined by such coefficients are called Dirichlet L-series. We notice that if χ is totally multiplicative and n has the prime decomposition converges absolutely for some s and is such that ( )  We say that the series (14) is an Euler product series. When it can be continued as a meromorphic function into the whole complex plane we will say that the respective function is an Euler product function. It can be easily checked that