Mustapha Bekkhoucha

Riemann

RIEMANN HYPOTHESIS PROOF

Keywords: Mathematics Subject Classification. 11M26. Key words and phrases. Riemann zeta function, nontrivial zeros, Riemann hypothesis. 1

Introduction

This work started from question in connection with the following remark : ζ * (∆), for ∆ a half-line of origin ω, 1 2 -< ω < 1, sweeping the first quadrant of apex ω, image of which at the beginning coincide with the segment [ζ * (ω), 1], with ζ * (ω) > 0 and must varies according to the properties of the holomorphic transformations, by which distortions comes infinitely close to the origin, when the half-line tends towards the vertical ? (this was suggested by result of Titchmarsh ([2])). The answer given to this questions necessited the intervention of almost all the known properties of ζ(s), namely :

• • Family of points of the axis s = 1/2 associated with each zero of ζ.

• Similarly, some of these families could be associated by the ordinate of the corresponding ζ-zeros. (This is suggested by a theorem of Ingham, in connection with a conjecture of Mertens, see ( [START_REF] Ellison | Les nombres premiers[END_REF]). In fact, the necessary condition for this, namely the multiplicity of all zeros = 1, can be established easily by using the complements in this paper.)

Position of the problem

In order to locate the complex zeros of the Riemann function ζ(z), which, as we know, are all situated in the strip 0 < (z) < 1, we'll first locate the image of the quadrant of apex ω, ω real neighboring 1 2 at left, by the function ζ(z), or rather by its regular part

ζ * (z) = ζ(z) - 1 z -1
The quadrant is sweeping by a half-straight ∆ m , of origin ω and slope m, m ≥ 0, in the direct way. The image ζ * (∆ m ) that start at the point ζ * (ω) on the real positive half axis with a tangent which has the same direction and the same way as ∆ m (because ζ * (ω) > 0, as we shall prove further) finish at the point 1 with a tangent whose direction is symmetrical to ∆ m direction related to the vertical. It must exists a value of m, for which ζ * (∆ m ) goes for the first time through the vertical of ζ * (ω). We know in fact that the origin 0 belong to the closure image of the quadrant by ζ * : we are basing this statement on a well-know result ( [START_REF] Titchmarsh | The theory of the Riemann zeta function[END_REF]), namely: there is a sequence of points z n which tends to infinity on the vertical (z) = 1 so that ζ(z n ) → 0. Thus, there is, for m enough large, a vertical tangent to ζ * (∆ m ) situated the most on the left.

Main result

Proposition 1. The contact point M (unique or not) of the vertical tangent to ζ * (∆ m ) the most on the left, depends analytically on m.

Proof. M corresponds to a value of the parameter t on ∆ m , of equation σ -ω = t/m, where σ = (z), t = (z), so that (ζ * (ω + s)) has a derivative related to t, equals to zero: we express by this that the tangent on M is vertical. So we have

(3.1) ∂ ∂t (ζ * (t, m)) = 0.
Let us consider now this equation in C(complex field), namely with (t, m) complex variables, (ζ * ) representing the analytical extension of the real part of ζ * . Let us place on m = m 1 , positive real. The equation (3.1) has among its roots, a finite number N 1 of real roots t j (m 1 ) which corresponds to (ζ * (t j (m 1 ), m 1 )) the smallest of the values taken by (ζ * (t, m 1 )) on t real. That is, on one hand the roots are isolated, and on the other hand, (ζ * (t, m 1 )) -→ 1 when t -→ ∞ Some of these roots can be multiple (then, they are counted as their multiplicity order). So, in a neighborhood of m 1 sufficiently small, let us say a disc centered on m 1 in the complex plan, there among the roots of the equation (3.1), N 1 simple complex roots t j (m) which tend to t j (m 1 ) when m → m 1 . That not means, we have N 1 functions on disk. But, if we consider those of roots which have as imaginary part the smallest in absolute value, and next among these, those for which (ζ * (t j (m), m)) (here, it's not that analytical extension of (ζ * (t, m)), but its the real part of the analytical extension of ζ * (t, m)) is the smallest, and finally, among these last, those for which (ζ * (t j (m), m)) (its about the imaginary part of the analytical extension of ζ * (t, m)) is the greatest, we really define a function on the disc by setting,

(ζ * (t h (m), m)) + i (ζ * (t h (m), m))
that is to say ζ * (t h (m), m), where t h (m) is some root carried after the three successive selection. This function is holomorphic in the disc. It is in fact possible to decide on an unique choice of t h (m) among the carried roots, by the two following successive selections : we take the t h (m) which the imaginary part is the smallest, in absolute value and on the whole, and among these last one, those which the real part is the smallest. Then t h (m) is well an holomorphic function of m, in a neighborhood of each point m of the pointed disk ∆(m 1 , r). In fact, the index h will be the same for all the points of such a neighborhood if it is sufficiently small. Let us place in a neighborhood of m 0 , in the pointed disk ∆(m 1 , r).

The different solutions of the equation for each point m, make so much holomorphic function t h (m) in this neighborhood supposed sufficiently small. Let t h 0 (m) one among those which take in m 0 the value associated by the preceding process. We'll prove that h 0 is still the value of h associated to all the points of a neighborhood of m 0 if it is sufficiently small. Let any index h = h 0 : Whether ζ * (t h (m 0 )) = ζ * (t h 0 (m 0 )), and then, by continuity, it's impossible that h being hold again by the first selections, if m is sufficiently approaching by m 0 . For Instance, if, at first, we have

| (ζ * (t h (m 0 )))| = | (ζ * (t h 0 (m 0 )))| this implies | (ζ * (t h (m 0 )))| > | (ζ * (t h 0 (m 0 )))
| Then, by continuity, we have also

| (ζ * (t h (m)))| > | (ζ * (t h 0 (m))|
for any m sufficiently near m 0 . It results that the index h cannot be carried, because it not corresponds with minimum of

| (ζ * (t h (m)))| when h varies. Whether ζ * (t h (m 0 )) = ζ * (t h 0 (m 0 )),but t h (m 0 ) = t h 0 (m 0 ),
and then, by continuity also, it's impossible that h being hold again by the last selections, if m is sufficiently approached by m 0 . It is clear, by making so for all the h, distinct from h 0 , that we must have h = h 0 for all the points of a neighborhood sufficiently small of m 0 . As t h (m) is holomorphic function in a neighborhood of m, and ζ * (t, m) is holomorphic in (t, m), the composited function ζ * (t h (m), m) is holomorphic in the disk minus the center. Finally, it is extended by continuity, as an holomorphic function in all the disk. It is clear that its trace on the real axis in a neighborhood of m 1 is the affix of the highest contact point of ζ * (∆ m ) with its most left tangent. In fact, for m real neighboring m 1 , the first selection made on the t j (m) amounts to carry only those which are real (for, in this case, the equation (3.1) allows real roots), since there is at least those which correspond to the point the most on the left on ζ * (∆ m ). Then, by the second selection, we carry only those which correspond effectively to the point the most on the left, and, finally, we keep only those which correspond to these point the most left, and the highest on ζ * (∆ m ). It results from which proceed, that the highest contact point of ζ * (∆ m ) with its most to the left vertical tangent, depends analytically m.

It results from which preced, that the highest contact point of ζ * (∆ m ) with its most to the left vertical tangent, depends analytically m. It is clear that we can show with the same manner, that the lowest contact point of ζ * (∆ m ) with its most to the left vertical tangent, depends analytically m. More precisely, we've proved that the two extreme contact points (the highest and the lowest of ζ * (∆ m ) with its most to the left tangent, are holomorphic functions of m in an open neighborhood in the complex plan of certain half real straight ]m ω , +∞[, whose origin depends on ω. Moreover, it emerges from the proof that the difference of these two functions is pure imaginary (its real part is zero), and therefore, it is constant, specially on ]m ω , +∞[. More generally, we can say that the most on the left vertical tangent , has a fixed number of contact points and their mutual distances are constant. We'll see further that this number reduce to 1. But, first, we've to prove the following proposition : Proposition 2. It is impossible that, from a point -λ, exterior to ζ * (∆ m ) situated on the most on the left vertical tangent, we could see ζ * (∆ m ) under an angle which exceeds a right angle by a small angle (by cutting on the curve a small arc) when the slope of ∆ m goes from m to m + dm.

Proof. We are leading to introduce the function

(3.2) Φ t,d (s) = [λ + ζ * (ω + s) + d exp(tse -iθ )] -1
where d and t are two positive parameters, s = re iθ represent a ∆ m parametrization of polar angle θ. The logarithmic derivative with respect to r = |s| is

(3.3) Φ t,d (s) Φ t,d (s) = - e iθ ζ * (ω + s) + dte rt λ + ζ * (ω + s) + de rt
Reasoning by absurd, if we suppose that, going from m to m + dm, the angle under which we see from -λ the image of ∆ m by ζ * been lightly more than a right angle, holds too for the image by z → ζ * (z) + d exp(tze -iθ ) if d is choosen sufficiently small. Moreover, it's a matter of the same right angle for ζ * (∆ m ) because they have the same horizontal side. Let us designate by s , s the curvilinear abscissa of points on ∆ m which have for images the extremities of the small arc cuted up on the curve image. We'll show that these gives a contradiction, if the parameters t and d are suitably choosen. For this, we are leading to consider the relative arrangement of Φ t,d (s) and Φ t,d (s) when s describes the small segment [s , s ], and thus when the representative point describes the small arc which is outside the right angle. For the choice of the parameters, it's a matter of satisfying all at once the two following conditions :

(1) de tr (where r = |s | is supposed > r = |s |) could be made as small as we want, in such a way the space difference (horizontally measured) between the image curves of ∆ m by ζ * (ω + s) + d exp(tse -iθ ) been infinitely small in a neighborhood of the contact point of ζ * (∆ m ) with the most of the left vertical tangent. (2) de tr could made as large as we want with respect to |ζ * (ω +s)| in such a way that the numerator argument of the second number of the above formula be infinitely small, always for r between r and r . Let us precise this: Given , M two positive numbers, arbitrary small, M arbitrary large, we've dte rt = de r t .te (r-r )t we suppose t sufficiently large for the second factor to be superior to M. -1 . A such t been fixed, we choose d in such a way that de tr = . In these conditions, the angle-space between Φ t,d (s) and Φ t,d (s), space which is given by the argument of Φ t,d (s) Φ t,d (s) could be made as near as we want the argument of -1 λ + ζ * (ω + s) + d exp(tse -iθ ) when ζ * (ω + s) + d exp tse -iθ describes the arc included between the two tangents taken from the point -λ. By taking as the origin the horizontal tangent, the argument of λ + ζ * (ω + s) + d exp(tse -iθ ) caries from 0 to π/2 + δ in the case where -λ is above the contact point with the vertical tangent, and varies from 0 to -π/2 -δ in the case where -λ is below. We can obviously confine ourself to the first case.

Figure 1

We've oversimplified on the Figure 1: The arc (1) described by λ+ζ * (ω +s)+d exp(tse -iθ ) . The arc (2) described by (λ + ζ * (ω + s) + d exp(tse -iθ )) -1 i.e, Φ t,d (s). Let us follow the advance of Φ t,d (s), from the contact point with the real axis, in the decreasing r way. We can distinguish two phases, corresponding respectively to these half quadrant : (Φ t,d (s)) and (Φ t,d (s)) increase for been in accordance with the fact Φ t,d (s) been then in the 3rd quadrant, the functions (Φ t,d (s)) and (Φ t,d (s)) are decreasing: since r decrease, (Φ t,d (s)) and (Φ t,d (s)) increase. On the arc situated in the 2nd half quadrant, (Φ t,d (s)) decrease, while (Φ t,d (s)) continuous to increase, which is still in accordance with the fact that, Φ t,d (s) been this time in the 4th quadrant, the function (Φ t,d (s)) is increasing while the function (Φ t,d (s)) is decreasing: since r decrease, (Φ t,d (s)) decrease and (Φ t,d (s)) increase. That contradiction comes about the arc which gets out the quadrant. That's for Φ t,d (s) is so in the 1st quadrant, and then (Φ t,d (s)) been positive, (Φ t,d (s)) had to don't step decreasing, while obviously, it has at least to come through an episode of increasing.

We've seen that it follow from the proposition 1, that the number of the contact points of ζ * (∆ m ) with its most on the left vertical tangent, is fixed when m changes, and that their mutual distances remained constant. Thanks to the proposition 2, this result can be precised by the following manner: Proposition 3. The contact point of ζ * (∆ m ) with its most the left tangent, is unique.

Proof. For that, we note firstly that we've analogous results about the highest horizontal tangent to ζ * (∆ m ). But furthermore, in this case, we easily se that theses results are good for any apex ω of the quadrant as far as it is on the half real positif axis. But, if we fix m, and we give analogous proof to these given higher, taking for this time as variables (t, ω), we can establish the analyticity with respect to ω, of the contact point (unique or not) of ζ * (∆ m ) with highest horizontal tangent, and, the unvarying of the number of the contact points and their mutual distances. As the image curve ζ * (∆ m ) tends to be confused with the point 1, when ω → ∞, we conclude that there is only one contact point. Then returning to the situation examined firstly, we observe that the contact point of ζ * (∆ m ) with its most to the left vertical tangent, be infinitely neighboring of the contact point with the highest horizontal tangent. That's because the proposition 2 excluded other contact points.

As we've seen above, the contact point of ζ * (∆ m ) with its most on the left vertical tangent is infinitely approaching the contact point of the curve with its highest horizontal tangent, so that we've nearly a retrogression point. It must be supposed moreover that it's under favor of a retrogression point that ζ * (∆ m ) could have for the first time, a most on the left vertical tangent. From this, m increasing, the curve couldn't progress to the left so as to avoid the application of the proposition 2, then, so as the most on the left point and the highest point remained infinitely neighboring.

Proposition 4. The image by ζ * of the quadrant of apex ω, ω ≥ 1/2 is entirely contained in the upper half plane, unless the quadrant contains a z 0 such as ζ * (z 0 ) is real and ζ * (z 0 ) = 0.

Proof. We suppose 1/2 ≤ ω < 1, the upper boundary 1 is taken for being in the conditions of the preceding propositions. But its obvious that all we have to do is to establish the result for all ω ≥ 1/2, for the image of the quadrant are included some into others when ω grows. Let us follow the advance towards the left of the contact point M of ζ * (∆ m ) with the most left vertical tangent. A such tangent surely exist as soon as m is large enough, since ζ * (∆ m ) coming through the neighboring of the origin: as we've pointed out highly, it cuts the vertical ζ * (ω) in at least two points and necessary has a vertical tangent in certain points of the arc bounded by these two points. We reason by absurd, by supposing that ζ * (∆ m ) comes, for the first time, to touch the real axis in a point, the real axis is then the lowest horizontal tangent to ζ * (∆ m ) : we'll note it T B 0 . Let -λ 0 be the point where T B 0 meet the most left vertical tangent T G 0 . Given to m a growth dm, let T B be the new position of the lowest horizontal tangent to ζ * (∆ m+dm ),-λ the point where it meets T G 0 , T G the most left tangent to ζ * (∆ m+dm ) taked from -λ. (T B , T G ) is then the angle under which we see the curve ζ * (∆ m+dm ) from the point -λ. It exceeds on right angle the angle (T G 0 , T G ) which cuts on ζ * (∆ m+dm ) a little arc, if dm is sufficiently small. There is however a situation where the preceding proof is not possible: It is when -λ 0 is precisely the point where ζ * (∆ m ) comes to touch the real axis. That occurs only if it is retrogression point (which corresponds then to a point z 0 of ∆ m 0 so that ζ * (z 0 ) is real and ζ * (z 0 ) = 0. But as soon as the real axis is crossed by ∆ m images for m > m 0 , in order to avoid that proposition 2 been applied again, that leads to a contradiction, we have to suppose that ζ * (∆ m ) has, in the inferior half-plan, only one small arc, getting round nearly the point ζ * (z 0 ), and which the lowest point is nearly confused with the most left point. The first one is then the contact point of ζ * (∆ m ) with its lowest horizontal tangent, a unique contact point as for the highest horizontal tangent. The second one must be supposed infinitely neighboring of the trace on the real axis of the most left vertical tangent.

Lemma 1. The first quadrant which has its image which crosses the real-axis has for apex ω 0 , and ω 0 is the orthogonal projection of a double zero of ζ * . It's the unique zero of ζ * in the upper half-plane.

Proof.

(1) By considering at the beginning -ω being fixed, the first value of m for which the point on the left of ζ * (∆ m ) has an x-coordinate = ζ * (ω), it cannot have a point of ζ * (∆ m ) below the real axis, nor on, (apart from ζ * (ω) and 1). In fact : let us call z 1 (m, ω) the antecedent of the point on the left, z 2 (m, ω) that of the point low, supposed to exist. It is known that z 1 , z 2 are analytical functions (and holomorphic in (m, ω) in the field of complexes. We saw for z 1 in the course of demonstration of proposition 1, and, for z 2 , it is c1ear that one has a similar demonstration. Under the particular conditions where one placed itself, we have

(ζ * (z 1 (m, ω))) = ζ * (ω) (1) ζ * (z 2 (m, ω)) = 0 (2)
The equation ( 2

m 0 for ω = ω 0 . Now ∂ ∂m (ζ * (z 1 (m, ω))) = ∂ ∂t (ζ * (z 1 (m, ω))). ∂t ∂m + ∂ ∂σ (ζ * (z 1 (m, ω))). ∂σ ∂m with ∂ ∂t (ζ * (z 1 (m, ω)
)) 0 = 0 (according to equation (3.1)). On the other hand ( ∂σ ∂m ) 0 = 0 in accordance with the unicity of the value of σ (or t) corresponding to the point of contact with ζ * (∆ ω 0 ,m 0 ) of the vertical tangent on the left (cf proof of equation

(3.1)) So one has ∂ ∂m (ζ * (z 1 (m, ω))) 0 = 0 iff ∂ ∂σ (ζ * (z 1 (m, ω))) 0 = 0 or ∂ ∂t (ζ * (z 1 (m, ω))) 0 = 0 (conditions of Cauchy-Rie- mann), which, taking into account ∂ ∂t (ζ * (z 1 (m, ω))) 0 = 0 is equivalent to ζ * (z 1 (ω, m
)) 0 = 0 ie, the point on the left of ζ * (∆ ω 0 ,m 0 ) is not a retrogression point. Thus, let us suppose at first, that one in this case.

Then, as one saw higher, (1) is solved in the neighborhood of ω 0 by m = m(ω), analytical and such as m(ω 0 ) = m 0 . Consequently, the point low of ζ * (∆ ω 0 ,m 0 ) describes a continuous arc, arc which cannot be reduced to a point since wary while remaining nearest to the vertical tangent on the left, which has abscissa = ζ * (ω). Because, by other hand, it is a retrogression point of ζ * (∆ ω,m(ω) ), thus a point where the derivate = 0 (condition 2 of the system), one has a contradiction with the fact that zeros of a holomorphic function = 0, of a single variable, are isolated. One concludes from it, that examinate situation doesn't take place.

Let us consider now the case ζ * (z 1 (ω 0 , m(ω 0 ))) is a retrogression point. It will be sufficient to show that one can always find a sequence of points ω n → ω 0 , such as the corresponding sequence m(ω n ) (it is clear that, in all the cases, it corresponds to any ω, starting from a certain value, a single value of m, such as the equation is verified) has a limit, and for which

ζ * (z 1 (ω n , m(ω n ))) is not a retrogression point of ζ * (∆ ωn,m(ωn) ).
In fact, under these conditions, these last curves, not having a point in the lower half-plane, by continuity, it will be similarly for ζ * (∆ ω 0 ,m(ω 0 ) ).

For that, one reasons by absurd. Thus, let us suppose that for any sequence One knows also it is necessarily a retrogression point. z 0 (m, ω) thus verified

ω n → ω 0 we have ζ * (z 1 (ω n , m(ω n ))))
ζ * (z 0 (m, ω)) = 0 (1) ζ * (z 0 (m, ω)) = 0 (2)
Given (ω, m(ω)) the general solution of (1) (m(ω) understood here as the single value of m corresponding to ω, so that ζ * (∆ ω,m ) intersect for the first time the real axis) Let us designate (ω 0 , m(ω 0 )) a particular solution. One will show that the function m(ω) is continuous.

• First, it is easy to see that it is an increasing function. Indeed, if ω < ω 0 , the sector limited by ∆ ω,m 0 because it contains the sector limited by ∆ ω 0 ,m 0 , has already points of its image in the lower half plane. It is thus for some m < m 0 that ζ * (∆ ω,m(ω) ) intersect for the first time the real axis. For that, one will reason by the absurd by supposing ζ * (z 0 ) < 0. One take ω < 1/2. If the image of ∆ m by ζ * did not surround the origin underneath, for m rather large, it would result from it as we will see it with proposition 5 -then ζ does not have zero in the quadrant of apex ω, which is in contradiction with existence of zeros on the axis z = 1/2 It is via its initial arc (which starts in ζ * (ω)) that ζ * (∆ m ) passes below the origin while cutting the real-axis obliquely, because, any other manner is not allowed for him under the proposition 2. But, with the limit (ω = -∞), also this manner of surrounding the origin becomes impossible, because the curve image of horizontal, admitting the origin as initial point, with a tangent of slop varying from 0 to +∞, could surround the origin underneath only while crossing the real-axis on the right of origin, which, once again, is not allowed by proposition 2.

We can add the following argument: Let show that the angle ϕ whose the extremity radius vector described the image of the sides of the horizontal band [0, h] is = multiple of 2π, as it would being if the band contained zero of -on the other hand, admits the origin as double-point, with tangent of polar angles respectively 0, θ 0 , 0 < θ 0 < π/2. One have: ϕ = π(corresponding to the arc between 1 and α in the upper half-plane) + θ 0 (corresponding to the arc between α and 0) +(-θ 0 ) (corresponding to the final loop of the curve) = π. One can conclude ζ * (z 0 ) = 0.

Before coming to, with proposals 5 and 6, the proof itself of the Riemann hypothesis one introduces below two questions witch do not have a direct connection with it, and where the interest will appear only later.

(1) Disposition of ζ * (∆ ω,m 0 ) in the neighborhood of the retrogression-point ζ * (z 0 ).

When ω varies from 0 to -∞, the half-straight z 0 ω turn -π/2. Then, because ζ * (z 0 ) = 0, and the first derived of ζ * (z) at z 0 , the non-null one, is even, more precisely of order 2 (otherwise, ζ * (∆ ω,m 0 ) would have points in the lower half-plane, in the neighborhood of the origin), the retrogression tangent turns 2 × (-π/2) = -π around ζ * (z 0 ). That does not leave him an other choice only: tend to being directed in the direction of the negative half real-axis when ω → ω 0 and in the opposed direction when ω → -∞.

A particular intermediate situation is that where z 0 ω has a polar angle = -3π/4 ; in which case, the retrogression tangent is directed as the positive imaginary axis. (2) Behavior of ζ * (∆ ω,m ) for m > m 0

It is clear that for m > m 0 , the image of the sector of apex ω, with sides having respective slopes 0 and m, contains the image of the sector of same origin-side, and with extremity-side of slope m 0 . One concludes from it that the application of proposition 2 has impossible being for the tirst values ofm > m 0 . ln order to clarify that, best is to place itself in the limit case ω = -∞. ∆ ω,m 0 is then the z 0 horizontal Hz 0 . ζ * (Hz 0 ) is the limit of ζ * (∆ ω,m 0 ) when ω → -∞.

Its retrogression-point is exactly at the foot of the vertical tangent on the left. In fact, let us consider the situation for ω = -∞, sufficiently far. Let us call θ 1 the polar angle of the first half-straight of origin ω, having its image crosses the ζ * (ω) vertical. Let us call θ 0 the polar angle of the first half-straight having its image which crosses the realaxis at 0, retrogression-point as one know. It arises from the lemma demonstration that one has θ 0 > θ 1 . Now, the retrogression tangent has a polar angle = 2θ 0 , and one saw that this angle → 0 when ω → -∞. It results from it that, so θ 1 → 0. Therefore, when ω → -∞, the images of the both half straight tend to have their tangents coincided at 0. What' s more, their images coincides. In fact :

The contrary implies that the point on the left of ζ * (Hz 0 ) is strictly on the left of imaginary axis. So as weIl, the analogous point of ζ * (∆ ω,m 0 ) for ω < 0 in the neighborhood of -∞. Then, in accordance of the proposition 2, the point low of the arc going to become the retrogression arc of ζ * (Hz 0 ), shouldn't to exist : in fact, the proposition 2 can be easily extended to the points with horizontal tangent and concavity turned upwards. It will be able to apply here, since the point concerned, tending towards 0, is clearly isolated of the left vertical tangent at ζ * (∆ ω,m 0 ). From there, ie. for horizontal with ordinates > z 0 the low point of their image by ζ * could by continuity, follow displacement towards the left, of the vertical tangent on the left. It is precisely that which makes impossible the application of proposition 2, the point -λ being with the right hand side of the point low (the point -λ must be necessarily placed on the negative imaginary demi-axis, and consequently on the right of the point low of the curve). Let us follow the progression of the ζ * image of the horizontal when its ordinate increase from that of Hz 0 . It can happen that for a certain position of the horizontal the point low of its image coincide again with the foot of the vertical tangent on the left, on the horizontal tangent lowest. It is then a retrogression-point. Let us call z 1 its antecedent (by analogy with z 0 , the antecedent of the retrogression point of ζ * (Hz 0 )). One will see that this analogy goes, in fact, further :

ζ * (z 1 ) is the point low on all the ζ * (∆ ω,z 1 ) curves, ω < ω 1 < ω 0 , ω 1 = z 1 .
It is a retrogression point for there curves, with a tangent whose polar angle varies from 0 (for ω → -∞) to π (for ω → ω 1 ) : One bases oneself on the fact that the point low of ζ * (∆ ω,z 1 ) (or ζ * (∆ ω,m 1 ) by putting m l =slope of ωz 1 ), depends analytically on ω, which is shown of same way as, for instance, the analytie of the high point of ζ * (∆ ω,m ), when, m remaining constant, ω varies. Because, on the other hand, the function holomorphic ζ * take necessarily the value 0 at this point (without what, proposition 2 applies again, and led to a contradiction), its results of what proceeds, that the point concerned can only be fixed. It's a matter of the same point ζ * (z 1 ), which is the point low of ζ * (Hz 1 ), as one see where ω → -∞. The both other assertions result from this, easily, and so the fact that z 1 verifies: z 1 < z 0 , z 1 < z 0 . Clearly, if the ordinate of the horizontal, continue to increase, the preceding processes can be repeated. Unless it would not be a position of the horizontal above Hz 0 , for which the point low of its ζ * -image coincides with the foot of the vertical tangent on the left, these is a sequence z 1 , z 2 , . . . , z k , . . ., placed more and more high, and more and more on the left, whose ζ * -images are the point of ζ * (∆ ω,m k ) (ω < ω k = z k , m k = slop of ωz k ) and are retrogression points for these last.

U (z) = log |(1 -z)ζ * (z)| -i arg((1 -z)ζ * (z)) (log |(1 -z)ζ * (z)|) 2 + (arg((1 -z)ζ * (z))) 2 (where arg((1 -z)ζ * (z)) indicates in reality arg(1 -z) + arg(ζ * (z)))
shows that there is a discontinuity of (U (z)) on the real axis, at the point 1. We'll avoid it by surrounding by a half-circle centered in 1, of arbitrarily small radius, in the upper half-plane. Thus, we are assured that arg(l -z) varies continuously between 0 and -π, in the closed quadrant except for a small half disc, and, in consequence, U (z) is continuous in this domain and holomorphic inside (except of course for points where log((1 -z)ζ * (z)) = 0, i.e at the eventual zeros of ζ * (z) in the quadrant). Let us verify that U (z) satisfied the required conditions: -We have

U (ω) = [log((1 -ω)ζ * (ω))] -2 . 1 1 -ω - ζ * (ω) ζ * (ω) By using the expansion ζ * (z) = r≥0 γ r (z -1) r
where the cosfficients (reals) verified r≥0 |γ r | < 1 (see [START_REF] Ellison | Les nombres premiers[END_REF] p.74, ex 2-4), we have, easily:

ζ * (ω) = r≥1 rγ r (ω -1) r-1
increased on [1/2, 1] by r≥1 r |γ r | (1/2) r-1 and then by

(1 -γ 0 ) sup r≥1 (1/2) r-1 = 1 -γ 0
where γ 0 is the Euler constant, and elsewhere ζ * (ω) is bounded by

γ 0 -(1 -γ 0 ) sup r≥1 (1/2) r = γ 0 -(1 -γ 0 )/2 It result that : 1 1 -ω - ζ * (ω) ζ * (ω) > (1 -1/2) -1 - 1 -γ 0 γ 0 -(1 -γ 0 )/2 is a positive number and consequently U (ω) > 0. -The expression log((1 -z)ζ * (z)) has its real part log |(1 -z)ζ * (z)| which tends to +∞ when z → ∞ on ∆ m , and in consequence U (z) → 0, in other word U (∞) = 0. Elsewhere U (ω) = log(1 -ω)ζ * (ω), with ζ * (ω) < ζ * (1) (= γ 0 ) for the function ζ * (z) is increasing on [ω,
+∞], its derivative being positive on this interval, as it results from the hypothesis, by virtue of the proposition 4: (if in a point of the interval, the derivative was negative, the image on the quadrant by ζ * , had some points below the real axis). Thus, we have U (ω) < 0, and then U (∞) > U (ω).

-Finally, as to the existence of the images-points at left of the vertical of U (ω), this contingency will be examined in the proof which follows.

We can then proceed to a study of U (∆ m ), analogous to that one about ζ * (∆ m ), at least in the sector of the quadrant included between the real axis and the first half-straight issued from ω that goes through a singularity of U (z), i.e a zero of ζ. More exactly, to avoid that there was more than one zero of ζ, we'll replace, in case of need, ω by a suitable point, arbitrarily neighboring at its left. Let us call ρ 0 , the unique zero of ζ, be on the halfstraight. There are three contingencies when z → ρ 0 , on the half straight: or U (z) → +∞, or U (z) → -∞, or U (z) → ±∞ that correspond respectively to the case where :

(log

((1 -z)ζ * (z))) → 0 + , or → 0 -, or → 0 ± That is to say |(1 -z)ζ * (z)| → 1 by
superior values, by inferior values or by the both. We still can, in case of need, replacing ω by an appropriate point arbitrarily neighboring its right, choicen in the manner to modify the inclination of the tangent to the half straight image by (1 -z)ζ * (z) at the point 1, image of ρ 0 , so that we had the third eventualities. Thus U (∆ m 0 ) has two branchs which went to infinity, one by the left, the other by the right. We will prove that it gives rise to a contradiction. In fact, it results in particular that U (∆ m ) has its most left point which tends to infinity at left when m → m 0 . Now, as well as we have seen precedently for ζ * (∆ m ), the most left point and the highest point of U (∆ m ), for m sufficiently neighboring of m 0 , are infinitely neighboring, they are nearly to form a retrogression point.

Its results that U (∆ m 0 ) admits in U (ρ 0 ) a retrogression point, i.e, it has two infinite branchs asymptot to an oblique directed to the direction of U (ρ 0 ). This is in contradiction with the initial hypothesis on the infinite branch of U (∆ m 0 ). This argument by absurd shows that ζ has no zero in the quadrant. We'll establish, by an analogous process to whese been used for proposition 5 the : Proposition 6. The complex zeros of ζ are all on the axis (z) = 1/2.

Proof. It is naturally to the function U (z) that we call again, as it was the case for the proof of proposition 5. This function already defined in the sector of apex ω, ω < ω 0 , inc1uded between the real axis and the half-straight ωz 0 , can be prolonged analytically in an additional sector of sides ∆ m 0 , ∆ m 1 , m 1 being the value of m for that ζ * (∆ m ) come to touch (ie with a contact point) for the first time ζ * (∆ m 0 ) on the arc inc1uded between ζ * (ω) and 0, in such a way that it form around the origin a loop composed of one arc of each of the two curves. The formula above imply that the laces which form around the origin by the image of sectors S ω,k , surround the origin once. For the previous formula, it has been supposed that ζ * (∂S ω,k ) forme a single cycle. But, it is not certainty. However, we note again log ω,k (z) the function log ζ * (z) -2ikπ, understood that it's not necessary in the k e sector that this function is defined, but, in a sector with an index, possibly = k. In fact, the concerned formula is not necessary for the demonstration of the Riemann hypothesis. The preceding construction holds for any ω, ω < ω 0 , ω = -∞ included. In this last case, the sectors are replaced by horizontal strips. This operation being made, we'll show analogously as for the quadrant of apex ω θ 0 , that none of the interior point to each of the preceding sectors can be a zero of ζ : let ρ be such a point, supposed interior to S ω,k . It is obvious that it is still interior to each of S ω ,k sectors, for ω sufficiently neighboring of ω. It is also obvious that, in the same conditions, the diverse prolongations U ω ,k defined a same function in ω S ω ,k . It's to this function, or rather to these deduces from by replacing the log ω ,k (. . .) by log ω ,k (. . .) + 2ikπ that we'll apply the indicated proceeding, as it's for this latest that the eventual zeros of ζ in the considerate domain, arc singularities, the function tending to infinite in their neighboring. The only points at left of the quadrant of apex ω 0 , which don't give way to the preceding proceeding, and which consequently, could be eventually the zeros of ζ, are the points of intersection of the extremities sides of the sectors S ω,k , for all ω, ω < ω 0 , for any fixed k (but to don't know if in all theses points the derivative of the functions U ω,k is positive, we'll take only points sufficiently neighboring on the left of 1/2 ). In fact, for we cannot apply to it the indicated proceeding, the point must be, for each fixed ω, on the extremity side of a certain S ω,k , with an index k, which, depending continuously on ω, is locally constant, and independent on ω by connexion, while the corresponding function, by continuity also, have a well defined index (not necessarily = k). Let us show that such points are necessarily on the axis (z) = 1/2 Given, in fact, a complex zero p of ζ, which is not on this axis, let us suppose it at its right. We know moreover that the complex zero of ω, ρ 1 , which has the smallest ordinate (positive), is unique and is placed on the axis. The straight which joins the two zeros ρ 1 and ρ cuts then the real axis in a point ω 1 , at left of 1/2. We can use ω 1 and the points ω of its neighborhood, for it is sufficiently near 1/2, ρ been, as we know, highly placed in the half-vertical strip. Making use of the sectors S ω,k whose extremities sides pass respectively through ρ 1 and ρ, we see that we must have: on the one hand k = k 1 , because ρ 1 and ρ are the extremity side of the sector of apex ω 1 , but on the other hand, k = k 1 because the sectors of apex ω, ω = ω 1 , are distinct.

Complements

One proposes in what follows, to prolong the method used for the demonstration of the Riemann-hypothesis (proposition 6), in order to specify the nature of the network formed by the intersection-points of the sectors S ω,k , for k fixed, ω varying. One saw that some of them are the zeros of the function ζ in the upper half-plane, and that they are the intersectionpoints of the sides extremity, for ω near to 1/2, on its left. One will see that it is so for other points. First, one frees oneself from the condition posed initially on ω. For that, it is sufficient to remark that, for instance, the proposition 5 -proof is based on the behavior of the function U (z) = (log(1 -z)ζ * (z)) -1 in the neighborhood of a possible zero of ζ in the sector S ω,0 . Now this behavior is not modified by a rotation of the figure, i.e. by the multiplication of U (z) by a constant. Thus, the property for a point s to be intersection of the extremity sides of the sectors S ω,k , is generalizable, first locally, i.e. in the neighborhood of any point ω < (s), then, by connexity, on the whole, i.e. for all ω < (s). Now, one seeks to establish the Proposition 7. In addition to the zeros of ζ, are also intersection-points of the sides of sectors S ω,k for all ω < 1/2 and k fixed:

(1) The zeros of ζ * previously shown, and which are of same nature that z 0 (cf sketch after the previous lemma). One will see that their number is finished, and they are placed on the right of the axis (z) = 1/2. (2) An infinite sequence of zeros of ζ * , and placed on the axis.

Proof.

(1) It is sufficient to show that any of these zeros cannot be inside a sector S ω,k . For that one reasons by the absurd supposing, for instance, that the zero z 1 is inside the sector S ω,k , where is defined the holomorphic function

U k (z) = (log k (1 -z)ζ * (z)) -1 (:= Z) z 1 is a regular point for U k (z) because U k (z 1 ) = [log(1 -z 1 )ζ * (z 1 )] -2 . 1 1 -z 1 - ζ * (z 1 ) ζ * (z 1 ) (4.1) = [log((1 -z 1 )ζ * (z 1 ))] -2 (1 -z 1 ) -1 = 0 One can thus reverse the relation U k (z) = Z, in the neighborhood of z 1 , by (4.2) z -z 1 = n≥1 a n (Z -Z 1 ) n or z = U -1 k (Z).
But, it is possible also to reverse it in two stages: -One writes it (1 -z)ζ * (z) = exp(Z -1 ), and substitutes

U -1 k (Z) to z in 1 -z ζ * (z) = (1 -U -1 k (Z)) -1 . exp(Z -1 ) := T Then on the one hand ζ * (z) = T is reversed by z -z 1 = n≥1 b n ((T -T 1 ) 1/2 ) n
with b 1 = 0 (because ζ * (z 1 ) = 0 and ζ * (z 1 ) = 0 (like for z 0 as one saws it in sketch (after the previous lemma))), in a small disk centered in T 1 , deprived of a ray, in the T -plane.

-On the other hand

(1 -U -1 k (Z)) -1 . exp(Z -1 ) = T is expanded as T -T 1 = n≥1 c n (Z -Z 1 ) n in a small disk centered in Z 1 .
If r is the first integer for which c r = 0, the disk is deprivated of the r arcs, of origin Z 1 , and extremity on the border. So, in crossing some or other of theses arcs, (T -T 1 ) 1/2 changes determination. i.e is exchanged for its opposite. Now, in accordance with (4.2)

n≥1 b n ((T -T 1 ) 1/2 ) n
has to remain unchanged. This implies that n odd b n ((T -T 1 ) 1/2 ) n ≡ 0 on these arcs. So, the holomorphic-function defined by n odd b n ((T -T 1 ) 1/2 ) n is ≡ 0 on a continuum, and, consequently ≡ 0, so, all its coefficients = 0, in particular b 1 = 0, hence, the contradiction. The two other properties are obtained by a same argument that used for the demonstration of Riemann-hypothesis (proposition 6):

-They are not on the left of the axis (z) = 1/2, because, if such were the case for z j , there will be a contradiction in connection with the indices of the sectors S ω,k , having z j on their extremity-side, containing respectively the zeros ρ i and ρ i of ζ.

-They are in a finished number, because if they were an infinity, they would necessarily tend towards ∞ in the vertical half-strip ]1/2, ω[, and then, by supposing z j placed enough high so that the straight joining it to ρ i , ρ i intersect the real axis (respectively in ω i , ω i on the left of 1/2) one has a contradiction analogous to that of the previous case. Thus, the z j are placed lower that the first zero ρ i of ζ. They verifies: (z 0 ) < (z 1 ) < . . . < (z n 0 ) < (ρ 1 ) with, as one knows ω 0 = (z) > (z 1 ) > . . . > (z n 0 ) > 1/2

(2) One needs the following lemma.

Lemma 2. Given a sector S ω,k ω < 1/2, of which the sides ∆, ∆ , intersect the axis (z) = 1/2 respectively in s, s , which are not retrogression-point for the ζ * -image of ∆, ∆ (i.e. ζ * (s) = 0, ζ * (s ) = 0). If s is on the extremity sides of all the sectors S ω,k , ω < 1/2, having the same index k that (∆, ∆ ), then, it is of same for s, towards the origin-side of theses.

Proof. Let us call p, p the points respectively on ∆, ∆ which the ζ * -image is the point of contact by which closes the cycle around the origin, formed by one arc of ζ * (∆) and one arc of ζ * (∆ ). Thus, on crossing p on the right on ∆ , log k ζ * (z) changes its determination. That implies that p coincides with s , because, if not, there is a contradiction with the fact that log k ζ * (z) does not change its determination in p , regarded as a point of the sector S ω,k , of apex ω 1 , near to ω, on its left. It results from it that ζ * (p)(= ζ * (p )) = ζ * (s ) and is thus a fixed point (i.e. independent of ω). Now, p depends continuously on ω. Because the property of the isolated zeros of a holomorphic function of one variable, = 0; one has necessarily the point p fixed, and consequently, it coincided with s, on the axis (z) = 1/2, (here, the argument is similar to that used for the z j in the 1 • ) to show that p is not at left of the axis (s) = 1/2, on the one hand, and, on the other hand, it is not, also, in the right. Reasoning by absurd, let us tend ω towards 1/2, it results from hypothesis that log ζ * (z) has the same determination on all the points of the vertical half axis (z) = 1/2, above of s (= ρ ). Now, clearly, this determination on these points varies : it is the log k ζ * (z) determination for k arbitrary large, for example, these corresponding to ζ-zeros with large ordinate.

Let us return to the demonstration of proposition 7, point 2. Given a ζ-zero ρ, one identifies ρ with the s of the lemma because ζ * (ρ) is = 0 (in fact, if there where ζ * (ρ) = 0, the rebroussement tangent to the curve image ζ * (ωρ) would make a turn when ω described ]-∞, 1/2[, and could take, thus a vertical position. There would be, then an image curve neighbouring with horizontal tangent and concavity turned to the bottom, near the contact point, which is impossible, according to the proprosition 2 extended (cf remark made in connection with a similar situation p 12) Then, one thing or the other:

• ζ * (s) is also = 0, and then, according to the lemma 2, s is the intersection point of the origin side of all the sectors S ω,k , k fixed, and it satisfies ζ * (s) = ζ * (ρ) (let us recall that ζ * (ρ) ∈ C(1, 1)). This process can continue with the secteur S ω,k-1 if it satisfies with the lemma conditions, and so on. . .. But it is clear that it must cease, before coming at the sector with extremity-side passing through the zero ρ witch precedes, otherwise, one would lead ζ * (ρ ) = ζ * (ρ) thus to 1 ρ -1 = 1 ρ-1 or ρ = ρ(!) • That leads us to consider the other alternative ; namely the sector S ω,k satisfies to ζ * (s ) = 0, but ζ * (s) = 0 There the demonstration of proposition 7 is finished. We give below some details about the new situation. In fact, the sector S ω,k does not have this only configuration. This is always accompanied by another (at least) whose the origin-side passes by some point p, with (p) ≥ 1/2 independent on ω. That which one distinguished higher is such as p coincided with s. To the various configurations of S ω,k , it correspond a partition of the interval ]-∞, 1/2[, ω describing such or such partial interval, according to the configuration considered. Thus, that which one distinguished has, among the partial intervals who are reserved for him, the interval with end at 1/2. In fact, it is this possibility for ω to tend towards 1/2, which makes that the point p coincided with s (situation similar to that treated in the lemma).

1 • 2 •

 12 Behaviour of ζ(s) in the half-plane s > Real zeros of ζ(s) -Existence of an infinity of zeros of ζ(s) on the axis s = 1/Development of ζ * (s) in series of (s -1) • Symmetry of the nonreal zeros of ζ(s) relatively to the axis s = 1/2 • Nonexistence (established by computer) of zeros of ζ(s) in the critical strip up to large ordinate, apart from the axis. Thus the role of these families (family of functions connected to ζ(s), family of properties of ζ(s)) is determinating in the proof of Riemann hypothesis. The same goes for its direct consequences :

  ) is expressing the fact that the point low of ζ * (∆ m ) is necessary a retrogression point, because the application of proposition 2, interdict that is a regular point. Let us designate by (m 0 , ω 0 ) a solution of (1). If ∂ ∂m (ζ * (z 1 (m, ω))) 0 = 0 then (1) is solved in the neighborhood of (m 0 , ω 0 ) by m = m(ω) where m(ω) is an analytical function, and takes the value

  is a retrogression point of ζ * (∆ ωn,m(ωn) ). It is obvious that there is at last one of there sequences for which m(ω n ) is bounded. One can extract a convergent sequence, ζ * (z 1 (ω n , m(ω n ))) which corresponds to him, is a convergent sequence where the derivate = 0. It is the sought contradiction. (2) ω being always fixed, now let us consider the first value of m for which ζ * (∆ m ) crosses for the first time, the real-axis. Let us call ζ * (z 0 (m, ω)) this point of contact with the real-axis. One will show that z 0 (m, ω) does not depend on ω, and that ζ * (z 0 (m, ω)) also. Since ζ * (z 0 (m, ω)) is the point low of ζ * (∆ m ) it depends analytically on (m, ω).

•

  The function m(ω) is surjective : In fact, let us designate Z 0 the point of ∆ ω 0 ,m 0 whose the ζ * image is the point of contact of ζ * (∆ ω 0 ,m 0 ) with real axis. Supposing m < m 0 . Let us sweep the higher half plane by half straight of slope fixed m, and origin varying in ]-∞, ω 0 [. It is clear that if ω is sufficiently on the left of ω 0 , the sector limited by ∆ ω,m will going to contain Z 0 , and consequently its will have already points in the lower half-plane. Necessarily, there was before a position of ω for which ζ * (∆ ω,m ) intersect for the first time, the real axis.• One can now conclude that, m(ω) being an increasing and surjective function of ]-∞, .[ → ]0, .[ is necessarily continuous. Its results from it that z 0 (m, ω(ω)) is continuous. On account of equation (2), one concludes that, if z 0 (m, ω(ω)) were not fixed, one would have a contradiction with that ζ * ≡ 0 in all the points of a continuous arc described by z 0 (m, ω(ω)).(3) Profiting of the fact that z 0 is independent of ω, let us tend ω towards -∞. That is possible in fact: at first, the equalities ζ * (-2n) = 1/2n + 1 ∀n ∈ N, imply that there are intervals as for as we want on the negative real axis in wich ζ * is increasing and ζ * positive. Then, this last property extends at all the points of R, because ∀ω∃ω < ω where ζ * (ω ) is positive, and which is thus the apex of a sector of the higher half-plane, of origin-side carried by the real axis and of which the image by ζ * is very whole in the upper half-plane (provided that the sector does not contain the point z 0 ) -What implies that ζ * (ω) is positive throughout the real axis. ω thus tending towards -∞, because ζ * (ω) tends toward 0 by decreasing positive values, and ζ * (∆ m,ω ) crosses for the first time the real axis only one point on the left of ζ * (ω) (the point ζ * (z 0 )), there is ζ * (z 0 ) ≤ 0. We show that one has in fact ζ * (z 0 ) = 0(and thus z 0 is double zero of ζ * ).

  ζ. ζ * -image of origin-side is the segment [0, 1]. The image of second side: -on the one hand, circumvents ζ * (z 0 ) by an arc contained in the lower half-plane. Let us indicate by α, α < ζ * (z 0 ), its left extremity.

Proposition 5 .

 5 Il ω, 1/2 < ω < 1, is the apex of a quadrant in which ζ * has no double zero, ζ has no zero in this quadrant. Proof. It is clear that the proceeding used for the study of the image of the quadrant by ζ * , can be applied to any function f , having analogous properties, especially the following ones: -Derivative strictly positive in the apex ω of the quadrant. -Existence of images-points on the left of the f (ω)-vertical. -Existence of a limit f (z) when z → ∞ on any half-straight issued from ω, inside the quadrant, and f (∞) > f (ω). Besides, on the real half straight [ω, +∞], f (z) > f (ω), f (z) ≥ 0. Let us consider in particular the function U (z) = (log(1 -z)ζ * (z)) -1 defined as follows: Taking the principal determination of log in the cutted plane in the negative half real axis, log ζ * (z) has a meaning for z in the quadrant, by virtue of the proposition 4. In other hand, log(1 -z) is obviously well defined. Setting log((1 -z)ζ * (z)) = log(1 -z) + log ζ * (z) we have an holomorphic function in the quadrant, and then, it's also the case for U (z), outside of the points where log((1 -z)ζ * (z)) = 0. The formula

Figure 2 ∆

 2 Figure 2

If φ 0 is its initial value (that which corresponds to ω = 1/2) and ∆θ, ∆θ , the respective variation of the polars angles of ωρ and ωρ , one have φ = φ 0 + (2∆θ -∆θ ). The first phase finishes when ω comes in ω 1 , such as φ = φ 1 = -π. It is then with the turn of p 2 to intervene, ω decreasing from ω 1 to ω 2 . ω 2 is = -∞.In fact : Let us call ψ, ∆τ , ∆τ the quantities analogous to φ, ∆θ, ∆θ whose intervened in this second phase. One have : ψ = ψ 1 + (2∆τ -∆τ ) with -π < ψ < 0. At the same time ∆φ have to remain < -π, φ = -π + β with β < 0. One have then 0 > ψ > -π > φ = -π + β, and therefore

is > 0 because s 1 nearest to s than the foot of the bissectrix of the angle 1 2 ω 1 s on the [1/2, s ]. In fact, when ω → -∞, the foot of the bissectrix tends about the middle of the side [1/2, s ] which is clearly below the zero ρ of ζ, taking into account the law of well known distribution of the zeros of on the axis (z) = 1/2. But, ψ -φ =angle of curves images of s ω and p 2 ω-angle of curves images of s ω and s 1 ω =angle of curves-image of s 1 ω and p 2 ω = 2 p 2 ωs 1 which → 0 when ω → -∞. Here the contradiction. One have similar arguments to show that this process continues indefinitely.


Nicolas Siauve

C Lormel

Romain Marion

Julien Dardenne

Fabien Sixdenier

Electromagnetic Characterization of Biological Tissues with Particle Swarm Optimization

The objective of this research is the electromagnetic characterization of biological tissues in radiofrequency domain. In this aim, a measurement bench including a vector network analyzer and an open-end coaxial probe was introduced. The complex permittivity identification of the middle under testing is performed by solving an inverse problem. A quadratic fitness function that represents the difference between reflection coefficient measurement and reflection coefficient acquired with the normalized aperture admittance is minimized. The optimization is based on a heuristic algorithm: particle swarm. The approach is validated on well known dielectric middle and then tested on biological tissues.

I. INTRODUCTION

Numerical simulations of electric field distribution in the human body exposed to radiofrequencies waves require the knowledge of dielectric tissues properties [START_REF] Siauve | Optimization of 3D SAR distribution in local hyperthermia[END_REF]. In the radiofrequency domain, several experimental techniques can be used for measuring permittivity (ε * =ε r -iε i =ε r +iσ/ε 0 ω): free space, resonant technique, transmission/reflection line and open-ended coaxial [START_REF] Venkatesh | An overview of dielectric properties measuring techniques[END_REF]. In our study, the open-ended coaxial probe has been selected for its suitability for liquid or soft materials such as biological materials without specific preparation. Temperature can be easily controlled and after the bench calibration several samples can be rapidly treated. This point is crucial since biological samples can be easily altered or desiccated. The bench measurement provides an electric Scattering (S) parameter or reflection. The permittivity is identified on the base of an inverse method that using Particle Swarm Optimization (PSO) procedure for minimizing the quadratic difference between the measured and the calculated reflection from a direct model. In the first part, the global procedure measurement is described and the method to identify the permittivity of the Middle Under Testing (MUT) is exposed. In the second section, the proposed approach is validated on MUT known and then tested on biological tissues.

II. PROCEDURE

The characterization contents two parts, the reflection coefficient measurement from probe aperture/MUT interface and the identification of permittivity from the reflection coefficient measured. A sensitivity analysis has been performing so as to optimize the experiments. Calibration, sensitivity and identification procedure are in C-language.

A. Measurement bench and calibration

The experimental radiofrequency bench is composed of a Vector Network Analyzer (VNA), an open-ended coaxial probe and an optical fiber thermometer. The probe is a semirigid coaxial line (UT85) mounted on a N connector. The probe is 200 mm long, inner and outer radius are respectively a=0.255mm and b=0.84mm. The reflection coefficient (S 11 parameter) of the MUT is obtained with VNA in the frequency range fixed (0.5 to 4 GHz) and for each frequency imposed by the linear step.

To determine the correct reflection coefficient of MUT, it is necessary to apply to the measured parameter * m , 11 S some correction coefficients [START_REF] Scheffler | Broadband microwave spectroscopy in Corbino geometry for temperatures down to 1.7K[END_REF]. These coefficients are obtained from three additional measurements, on well known middles, respectively j=short circuit (

1 S * m , 11
), air (ε r =1, ε i =0) and de-ionized water (ε s =25, ε ∞ =4.3, τ=170ps) (1). The corrected S 11,c,MUT is then calculated for each frequency [START_REF] Siauve | Optimization of 3D SAR distribution in local hyperthermia[END_REF].

* S * R D * j , c , 11 j , c , 11 j , m , 11 S E 1 S E E S (1)

B. Inverse problem

The identification of the MUT permittivity is based on the normalized aperture admittance [START_REF] Ritz | Analysis of broadband microwave conductivity and permittivity measurements of semiconducting materials[END_REF]: The Fitness Function (FF) to minimize is given by (4). [START_REF] Ritz | Analysis of broadband microwave conductivity and permittivity measurements of semiconducting materials[END_REF] This minimization problem is solving with PSO technique. PSO involves simulating social behavior among individuals (particles) "flying" through a multidimensional search space, each particle representing a single intersection of all search dimensions [START_REF] Eberhart | A new optimizer using particle swarm theory[END_REF]. Particles in a swarm move in discrete steps based on their current velocity, memory of where they found Electromagnetic Characterization of Biological Tissues with Particle Swarm Optimization N. Siauve (1) , C. Lormel (1) , R. Marion (1) , J. Dardenne (1,2) and F. Sixdenier (1) Université de Lyon, Lyon, F-69622, France (1,2) Université Lyon 1, Villeurbanne, F-69622, France (1) -INSA Lyon, Villeurbanne, F-6921, France (2) CNRS, UMR 5005, Laboratoire Ampère (1) -CNRS UMR 5220, INSERM U630, CREATIS-LRMN (2) nicolas.siauve@univ-lyon1.fr their best fitness value, and a desire to move toward where the best fitness value was found so far by all of particle during a previous iteration (Fig. 1). Each particle has a position and a velocity: their values are randomized initially. The position with highest fitness score in each iteration is set to be the entire swarm's global best (gbest) position, towards which other particles move. In addition, each particle keeps its best position that it has visited, known as the particle's personal best (pbest). Each particle n of the swarm is defined as a potential solution to the problem. The particle dynamics are governed by the following rules which update particle positions x n and velocities:

t x gbest 2 rand c t x pbest 1 rand c wv v n 2 n n 1 n n (5)
After a new velocity for each particle, the position is calculated according to

t v x x n n 1 n
. Where x n is the current position of particle n, t the time step equal to 1, x pbest is the best position attained by particle n, x gbest is the swarm's global best position, v n is the velocity of particle n, w is a random inertia weight, c 1 and c 2 are two positive constants, called the social and cognitive parameter, rand1 and rand2 are random numbers between 0 and 1.

III. VALIDATION AND RESULTS

All the measurements presented in the following are realized on MUTs at 23°C 1 000 frequencies linearly spaced. The swarm size is fixed to 10 particles. In a purpose of comparison an ethanol MUT has been tested and our proposed approach has been compared to Debye model (6) with these parameters ε ∞ =4.3, ε s =25 and τ=170ps (Fig. 2). Validation results are presented, a good agreement is found between the Debye model and our model approach results, the impact of calibration is also studied (Fig. 2). Then our approach is used on pork skin tissues (Fig. 3), since a sensibility analysis with reduced sensitivity functions has shown the ability to easily identify both imaginary and real of the permittivity in the range of frequencies by using the imaginary part or the phase of the reflection coefficient.

IV. CONCLUSIONS AND PERSPECTIVES

Several perspectives of this work are considering. Firstly, a comparison in term of efficiency between the Levenberg Marquardt and the genetic algorithm will be done with application of the sensitivity conclusions. Secondly, the monolayer approach presented in this paper will be enhanced for applications on bi-layered or multi-layered MUT and thirdly a micro coaxial probe will be developed.

V. ACKNOWLEDGMENT

,

  the distance between the source point and the field point, ρ' and ρ are respectively the radial coordinates of these points at the aperture of coaxial probe. The reflection coefficient is related to the normalized probe admittance (3) with Y 0 =1/50.

Fig. 1 .

 1 Fig.1. The Particle Swarm Optimization algorithm.

Fig. 2 .

 2 Fig.2. Validation on ethanol sample with and without calibration.

Fig. 3 .

 3 Fig.3. Complex permittivity of pork skin tissues.

This work is wholly supported by French National Agency (ANR) through Jeunes Chercheuses Jeunes Chercheurs Program (Project BioRFmod n°ANR-06-JCJC-0124).

VI. REFERENCES