Frederick R. Allen. 16th October 2017

Transcription

1 A Proof of the Riemann Hypothesis using a Subdivision of the Critical Strip defined by the family of curves Rl[ξ(s)] i =0 which intersect the Critical Line at the roots ρ i of ξ(s). Frederick R. Allen 16th October 2017 ABSTRACT. Rouché s Theorem is applied to the critical strip to show that the only zeros of the Riemann functions ξ(s) and ζ(s) lie on the critical line σ = 1 2. An infinite subdivision of the critical strip is made to prove that ξ(s) has no zeros within the interiors of the regions where 1 2 < σ < 1. The regions are separated by boundary curves defined as Rl[ξ(s)] i = 0 which intersect the critical line at the roots ρ i. There are no zeros on these curves for 1 2 < σ 1. The Riemann Hypothesis is proved to be true. 1 Introduction This paper is dependent on the work of H.M.Edwards[1] for the theory of the ξ(s) function of Riemann, although other papers are relevant; the more important are listed in the References. 1.1 The Riemann Zeta function. The Riemann Zeta function ζ(s) is a function of the complex variable s = σ + it, defined in the half plane σ > 1 by the absolutely convergent series: ζ(s) = 1/n s (1) n=1 Riemann[2] showed that ζ(s) extends by analytical continuation to the whole complex plane, with only a simple pole at s = 1, with residue 1, and that ζ(s) satisfies the functional equation: ζ(s) = 2 s π s 1 sin 1 2sπΓ(1 s)ζ(1 s) (2) 1

2 Riemann introduces the function ξ(s)) as: which satisfies the functional equation and has the same zeros as ζ(s). ξ(s) = 1 2 s(s s 1)π 2 Γ( s 2 )ζ(s) (3) ξ(s) = ξ(1 s) (4) 2 The function ξ(s) Riemann obtains the equation ξ(s) = 1 2 s(1 s) 2 1 ψ(x)(x s 2 + x (1 s) 2 ) dx x (5) where ψ(x) is the theta function ψ(x) = n=1 e n2 πx (6) Let lx = 1 2 log e x and s = σ + it = 1 + y + it (7) 2 Use of a local coordinate y = σ 1 2 in the critical strip, simplifies subsequent analysis. It follows that ξ(s) = 1 2 s(1 s) 2 1 2ψ(x)(cosh lx.y cos lx.t + sinh lx.y sin lx.t)x 3 4 dx = s(s 1)(I 1 + ii 2 ) where and I 1 = I 2 = 1 1 ψ(x)(cosh lx.y cos lx.t)x 3 4 dx ψ(x)(sinh lx.y sin lx.t)x 3 4 dx 2

3 With the definitions: U = y 2 t and V = 2yt, (8) s(s 1) = ( y + it)( y + it) = y2 t i.2yt = U + iv, 4 then with the change of variable u = 1 2 logx, and Since I 1 = 2 I 2 = ψ(e 2u )e u 2 cosh (yu) cos (tu)du (9) ψ(e 2u )e u 2 sinh (yu) sin (tu)du. (10) ξ(s) = (U + iv )(I 1 + ii 2 ) then ξ(s) = UI 1 V I 2 + i(v I 1 + UI 2 ) (11) 3 The zeros of ξ(s) and the Riemann Hypothesis By the definition of ξ(s), the zeros of ξ(s) coincide with those of ζ(s) in the critical strip defined as 0 σ 1 and all t. The Riemann Hypothesis is that all the roots of ξ(s) lie on the line σ = 1 2. The trivial zeros of ζ(s) occur when s is a negative even integer. Hardy first showed that ζ(s) has an infinity of zeros on the critical line Rls = 1 2. Hadamard and de la Valleé Poussin proved independently that ζ(s) has no zeros on the line Rls = 1, or y = 1 2, in the notation of this paper. Hence, ξ(s) also has no zeros on Rls = 1. Their work is described at length by Titchmarsh in Reference [4]. From the Equation(11) above for ξ(s), Rl[ξ(s)] = UI 1 V I 2 and Im[ξ(s)] = V I 1 + UI 2 (12) For a given y in the critical half-strip, 0 < y < 1 2, Rl[ξ(s)] and Im[ξ(s)] have individual zeros which are interleaved. No calculations have so far shown that Rl[ξ(s)] and Im[ξ(s)] 3

4 can be zero simultaneously in the critical strip. From their definitions, I 1 and U are even functions of y and t, whilst I 2 and V are odd functions of y and t. It follows that, if a zero of ξ(s) exists within the critical strip, there must also exist three more zeros which are reflections in the axes y = 0 and t = 0. In this paper, the criitical strip is defined as the quarter strip 0 < y < 1 2, t > 0. To prove the Riemann Hypothesis it is necessary and sufficient to show that there are no zeros of ξ(s) in this particular choice of reducd critical strip. 4 Rouché s Theorem and Application. 4.1 Rouché s Theorem This theorem[3] states that, if f(s) and g(s) are two functions, regular within and on a closed contour C on which f(s) does not vanish and also g(s) < f(s) on C, then f(s) and f(s) + g(s) have the same number of zeros within C. The theorem can be used to prove that two functions have the same number of zeros within a given contour C. This suggests a method of proving the Riemann hypothesis when f(s) + g(s) = ξ(s) and f(s) is chosen to have no zeros on or within C. Define g(s) = ξ(s) f(s), then f(s) and f(s) + g(s) = ξ(s) will have the same number of zeros within C if the inequality g(s) < f(s) is true on C. We refer to this inequality as the Rouché Inequality. 4.2 Application to the Critical Strip. If C is a contour enclosing a region of the critical strip 0 < y < 1 2, 0 < t and f(s) = Rlf + i.imf, then the Rouché Inequality is ξ(s) f(s) < f(s). This is equivalent to: or (Rlξ Rlf) 2 + (Imξ Imf) 2 < Rlf 2 + Imf 2 Rlξ 2 + Imξ 2 < 2(Rlξ.Rlf + Imξ.Imf) (13) It is necessary to find a function f(s), regular within and on C, which does not vanish on C and satisfies the Rouché Inequality. A range of choices for f(s) has been examined 4

5 but no suitable function has been found to treat the situation when the contour C encloses the complete critical strip. There are a number of ways in which the critical strip can be sub-divided so that Rouché s Theorem can be applied to each subdivision separately. In the next section functions f 1 (s) and f 2 (s) are introduced, which satisfy the Rouché Inequality in alternate subdivisions of the complete critical strip, with the boundary curves chosen to be Rl[ξ(s)] i = 0, which intersect the critical line at the roots ρ i. 5 Subdivision of the Critical Strip. 5.1 The Regions R i,i+1 The function F (s) = Rl[ξ(s)] is used to define a family of curves F i (s) = 0 which subdivide the critical strip. F i (s) = 0 is the member of the family chosen to pass through the i th root of ξ(s) at t = ρ i on the critical line y = 0 where F i (s) = 0. The chosen subdivision of the critical strip is illustrated in Figure1. The first region R 0,1 is defined by the y axis, the curve F 1 (s) = 0 and segments of the lines y=0 and y = 1/2, which are intersected by the first two boundaries. All other regions R i,i+1, i = 1,2,... are defined by the curves F i (s) = 0 and F i+1 (s) = 0 and the corresponding segments of the lines y = 0 and y = 1 2. F (s) alternates in sign as t increases from 0 to infinity. F (s) is > 0 in region R i,i+1 when i is even and is < 0 when i is odd. This behaviour of F (s) is clear for the initial regions of the critical strip, starting with F 0 (s) > 0 in region R 0,1. It is assumed in this paper that the behaviour of Rl[ξ(s)] is such that subdivisions can be defined using all the roots ρ i. If Rl[ξ(s)] has a turning point which is also a zero on y=0, then two adjacent regions would have the same sign for Rl[ξ(s)]. This would not prevent the application of Rouché s Theorem to those regions. 5.2 Analysis of a typical Region R 3,4. For purposes of illustration, the region R 3,4 is taken as shown in Figure 1, in which the function F (s) is < 0. The contour C 3 is defined by the segments of the lines y = 0 and y = 1 2, together with two 5

6 Figure 1: The subdivision of the critical strip by the boundary curves R[ξ(s)] i = 0, i=1,2,... The Contour C 3 in Region R 3,4 is shown thus: The curves R[ξ(s)] i = 0 have small negative gradient in 0 < y < 1 2. For regions beyond the critical strip, with y increasing, more curvature is evident, as shown in Figure 4 of Reference[9].. 6

7 curves taken infinitesimally close to F 3 (s) = 0 and F 4 (s) = 0, respectively. It follows that, since F (s) < 0 throughout the region, it will be satisfied on the chosen contour C 3. The definition of C 3 ensures that the zeros of ξ(s) at t = ρ 3 and t = ρ 4 are excluded from the contour C 3. 6 Choice of f(s) and the Rouché Inequality for all Regions of the Critical Strip. 6.1 Choice of the functions f 1 (s) and f 2 (s). If f(s) is chosen to equal f 1 (s) = c 1, where c 1 is a constant > 0, the inequality becomes ξ(s) c 1 < c 1 (14) Similarly, if f(s) is chosen equal to f 2 (s) = c 2 where c 2 is a constant < 0, the inequality is ξ(s) c 2 < c 2 (15) Although constants, f 1 (s) and f 2 (s) can be regarded as regular functions which do not vanish on the contour C 3 defined in Section 5.2 for the particular case of Region R 3,4. They also have no zeros within the contour C Proof that the Rouché Inequality is satisfied on the Contour C i in all Regions Inequalities. Let ξ(s) = α + iβ, where α = Rl[ξ(s)] and β = Im[ξ(s)], then the inequalities to be satisfied are, (α c 1 ) 2 + β 2 < c 2 1 for α > 0, (α c 2 ) 2 + β 2 < c 2 2 for α < 0, or α 2 + β 2 < 2αc 1 for α > 0 and c 1 > 0 (16) α 2 + β 2 < 2αc 2 for α < 0 and c 2 < 0. (17) If ξ(s) is bounded, values of c 1 and c 2 can be found so that the inequalities are satisfied at all points on each contour C i. 7

8 6.2.2 Choice of c 1 and c 2 The boundary curves Rl[ξ(s)] i = 0 intersect the critical line y=0 at the roots t = ρ i. The intersections of the same curves with the axis y= 1 2, are defined here as t = τ i. It is found that τ i < ρ i due to the small negative gradients of the boundary curves(section 8.2). For example, ρ 1 = and τ 1 = ; ρ 2 = and τ 2 = For the region R i,i+1, it is sufficient to find a bound for ξ(s) in the disk s 1 2 R i, where R 2 i = ρ2 i+1 (18) Edwards[1] has shown that, for R i sufficiently large, the estimate ξ(s) R R i i holds throughout the disk s 1 2 R i. If α mi is the minimum value of α on the contour C i, it is necessary that c 1 > ξ(s) 2 /2α mi or c 2 > ξ(s) 2 /2α mi (19) for i even or odd, respectively. α mi > 0. If the constants c 1 and c 2 are chosen to satisfy the inequality The choice of contour C i in each region ensures that c 1 R 2R i i /2α mi or c 2 R 2R i i /2α mi, (20) respectively, the Rouché Inequalities (16) and (17) are then satisfied in all regions of the subdivided critical strip. 7 Conclusion from the analysis of the Regions R i,i+1 Rouché s theorem has been applied to each region in turn where the contour C i excludes points on the boundary curves Rl[ξ(s)] i = 0. It has been shown that no zeros of ξ(s) can exist in the Regions R i,i+1 i = 0, 1, The boundary curves F i (s) = 0. The analysis so far, using a subdivison of the critical strip, proves the Riemann Hypothesis for the individual regions, but excluding the boundary curves on which F i (s) = 0. It remains to show that no other zeros of ξ(s) can exist on these curves which pass through the known zeros t = ρ i on the critical line. 8

9 8.1 The functions Rl[ξ(s)] andim[ξ(s)] The function ξ(s) is analytic, hence Rl[ξ(s)] and Im[ξ(s)] are harmonic functions, both satisfying Laplace s equation i.e. Rl[ξ(s)] yy +Rl[ξ(s)] tt = 0 and Im[ξ(s)] yy +Im[ξ(s)] tt = 0. It follows that Rl[ξ(s)] and Im[ξ(s)] cannot have local maxima or minima, but can have critical points which are also saddle points. It is convenient to use the notation introduced in Section 6.2.1, which is: Let α = Rl[ξ(s)] and β = Im[ξ(s)]. (21) α y = α y and α t = α t for derivatives of α with respect to y and t. Similarly for derivatives of other functions. On the critical line y=0 bounding the critical strip, α passes through an infinity of zeros of ξ(s) whilst β = 0 for all t. There are critical points of α between the roots t = ρ i where the derivatives α t = 0 and β t = 0 are both zero, which are also saddle points with: (22) α 2 yy + α 2 yt > Gradients of the boundary curves. The boundary curves Rl[ξ(s)] i =0 pass through the roots ρ i and the derivatives w.r.t. y and t are, from Equation (12) of Section 3, α y = 2yI 1 + UI 1y 2tI 2 V I 2y (23) and α t = 2tI 1 + UI 1t 2yI 2 V I 2t (24) These equations show that, since α y and α t are non-zero and have the same sign, the gradients of the boundary curves are finite and negative. There are also no critical points where both α y and α t are zero. It is known that there are no zeros of ξ(s) on the line y = 1 2, segments of which form part of the contour C i in each region R i,i+1. However, α and β can each have a zero on y = 1 2, but not simultaneously. Where the boundary curve F i(s) = 0 intersects y = 1 2, α = 0 and β > 0. On a boundary curve Rl[ξ(s)] i = 0, β is either positive or negative, according as i is odd or even. The modulus of β increases monotonically as y increases from y = 0 to y =

10 8.3 The region R 0,1 The region R 0,1 is special with one boundary curve as the y axis, where t=0. It can be shown that there are no zeros of ξ(s) on the y axis. 8.4 The regions R i,i+1 for i > 0 If a zero of ξ(s) exists at y = y 0 in 0 < y < 1/2 on F i (s), then β, which is non-zero at y = 1/2 on F i (s), would become zero at y = y 0, before passing through a maximum or minimum at y = y c, then zero again on the critical line. Since β = Im[ξ(s)] is harmonic, y = y c, which is a critical point with β y = 0 and β t = 0, must also be a saddle point where: The Cauchy-Riemann equations for ξ(s) are: β 2 yy + β 2 yt > 0 (25) α y = β t and α t = β y (26) at any point on a boundary curve in 0 < y < 1/2. It follows that, at the critical point y = y c, α y = 0 and α t = 0. Zero derivatives of α are inconsistent with the result that each boundary curve has a finite and negative gradient. The assumption that a zero can exist on a boundary curve is therefore false. 9 Conclusions. The known zeros of ξ(s) on the critical line σ = 1 2 or y = 0, have been used to define a family of curves Rl[ξ(s)] i = 0 which subdivide the critical strip into an infinity of regions R i,i+1, i=0,1,2.... Rouché s Theorem is applied to each region in turn to prove that no zeros of ξ(s) lie within these regions, which exclude the boundary curves. R 0,1 is a special region, with the σ axis as one boundary on which there are no zeros of ξ(s). It is also shown that no zeros of ξ(s) can lie on the boundary curves Rl[ξ(s)] i = 0 for σ > 1 2. Therefore, all the zeros of ξ(s) must lie on the critical line σ = 1 2, and the Riemann Hypothesis is true. 10

11 References [1] H.M.Edwards, Riemann s Zeta Function, Dover edition [2] B.Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Gesammelte Werke, Teubner, Leipzig, [3] E.T.Copson, An Introduction to the Theory of Functions of a Complex Variable. OUP London [4] E.C.Titchmarsh, The Theory of the Riemann Zeta Function, Second Edition 1986, Clarendon Press - Oxford. [5] T.J.I A. Bromwich, An Introduction to the Theory of Infinite Series. Third Edition 1991 AMS Chelsea Publishing. [6] E.T.Whittaker and G.N.Watson, A Course of Modern Analysis, CUP [7] E.C.Titchmarsh, The Theory of Functions, OUP London [8] J.Hadamard, Sur la distribution des zeros de la fonction ζ(s) et ses consequences arithmétiques. Bull.Soc.Math.France 24, , [9] J.M.Hill and R.K. Wilson, X-rays of the Riemann Zeta and Xi functions. School of Mathematics and Applied Statistics, University of Wollongong, NSW 2522, Austrailia. [10] The CMI Millenium Meeting Collection, Collége de France, Paris May 24-25, Springer, 11