A Proof of the Riemann Hypothesis using Rouché s Theorem an Infinite Subdivision of the Critical Strip. Frederick R. Allen ABSTRACT. Rouché s Theorem is applied to the critical strip to show that the only zeros of the Riemann functions ξ(s) ζ(s) lie on the critical line σ = 2. A function η(s) is introduced, regular with the same zeros as ξ(s) in the critical strip. An infinite subdivision of the critical strip is made to prove that ξ(s) has no zeros within the interiors of the regions where 2 < σ <. The regions are separated by boundary curves, which are also zero-free, due to the harmonic properties of Rl[ξ(s)] Im[ξ(s)]. The Riemann Hypothesis is proved to be true. Introduction The work of this paper is based mainly on the work of H.M.Edwards[], although other books papers have been studied; the more important are listed in the References. The Riemann Zeta function ζ(s) is a function of the complex variable s = σ+it,defined in the half plane σ > by the absolutely convergent series: ζ(s) = /n s n= Riemann[2] showed that ζ(s) extends by analytical continuation to the whole complex plane, with only a simple pole at s =, with residue. He showed that ζ(s) satisfies the functional equation: ζ(s) = 2 s π s sin 2sπΓ( s)ζ( s) Riemann introduces the function ξ(s)) as: ξ(s) = 2 s(s )π s 2 Γ( s 2 )ζ(s)
which satisfies the functional equation has the same zeros as ζ(s). 2 The function ξ(s) Riemann obtains the equation ξ(s) = 2 where ψ(x) is the theta function s( s) 2 ξ(s) = ξ( s) ψ(x) = n= ψ(x)(x s 2 + x ( s) 2 ) dx x e n2 πx Let lx = 2 log e x s = σ + it = 2 + y + it. Use of a local coordinate y = σ 2 critical strip, simplifies subsequent analysis. in the It follows that ξ(s) = 2 s( s) 2 2ψ(x)(cosh lx.y cos lx.t + sinh lx.y sin lx.t)x 3 4 dx = 2 + s(s )(I + ii 2 ) where I = I 2 = ψ(x)(cosh lx.y cos lx.t)x 3 4 dx ψ(x)(sinh lx.y sin lx.t)x 3 4 dx With the definitions: U = y 2 t 2 4 V = 2yt, s(s ) = ( 2 + y + it)( 2 + y + it) = y2 t 2 4 + i.2yt = U + iv 2
With the change of variable u = 2 logx Since then I = 2 I 2 = 2 ψ(e 2u )e u 2 cosh (yu) cos (tu)du ψ(e 2u )e u 2 sinh (yu) sin (tu)du ξ(s) = 2 + (U + iv )(I + ii 2 ) ξ(s) = 2 + UI V I 2 + i(v I + UI 2 ) These equations are fundamental in the following analysis. 3 The zeros of ξ(s) the Riemann Hypothesis By definition, the zeros of ξ(s) coincide with those of ζ(s) in the critical strip defined as σ all t. The Riemann Hypothesis is that all the roots of ξ(s) lie on the line σ = 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 = 2. Hadamard de la Valleé Poussin proved independently that ζ(s) has no zeros on the line Rls = or y = 2 in the notation of this paper. Hence, ξ(s) also has no zeros on Rls =. Their work is described at length by Titchmarsh in Reference [4]. From the above equation for ξ(s) Rl[ξ(s)] = 2 + UI V I 2 Im[ξ(s)] = V I + UI 2 It follows that necessary sufficient conditions for ξ(s) to be zero are: where I = I = I I 2 = I 2 U 2(U 2 + V 2 ) I 2 = V 2(U 2 + V 2 ) 3
As t increases, I 2t 2 I 2 y t 3 For a given y in the critical half-strip, < y < 2, Rl[ξ(s)], Im[ξ(s)] have individual zeros which are interleaved. No calculations have so far shown that Rl[ξ(s)] Im[ξ(s)] can be zero simultaneously in the critical strip. From their definitions, I I are even functions of y t, whilst I 2 I 2 are odd functions of y 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 = t =. In this paper, the critical strip is defined as the quarter strip < y < 2, t >. To prove the Riemann Hypothesis it is necessary sufficient to show that there are no zeros of ξ(s) in this particular choice of reducd critical strip. 4 Rouché s Theorem Application. 4. Rouché s Theorem This theorem[3] states that, if f(s) g(s) are two functions, regular within on a closed contour C on which f(s) does not vanish also g(s) < f(s) on C, then f(s) 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) f(s) is chosen to have no zeros on or within C. Define g(s) = ξ(s) f(s), then f(s) 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 < y < 2, t f(s) = Rlf + i.imf, then the Rouché Inequality is ξ(s) f(s) < f(s). This is equivalent to (Rlξ Rlf) 2 + (Imξ Imf) 2 < Rlf 2 + Imf 2 4
which is Rlξ 2 + Imξ 2 < 2(Rlξ.Rlf + Imξ.Imf) It is necessary to find a function f(s), regular within on C, which does not vanish on C satisfies the Rouché Inequality. A range of choices for f(s) has been examined 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 subdivided so that Rouché s Theorem can be applied to each subdivision separately. In the next section functions f (s) f 2 (s) are introduced, which satisfy the Rouché Inequality in alternate subdivisions of the complete critical strip. 5 Functions η(s), f (s), f 2 (s) F (s) 5. A new function with the same zeros as ξ(s) In Section 3 the necessary sufficient conditions for a zero of ξ(s) are shown to be: I = I I 2 = I 2 where I I 2 are simple functions of y t. A new regular function η(s) is defined as: η(s) = I I + i.(i 2 I 2 ) which is now used in place of ξ(s) for the application of Rouchés Theorem. It has the same zeros as ξ(s) in the critical strip. 5.2 The Rouché inequality with η(s) in place of ξ(s). Define g(s) = η(s) f(s), where f(s) is a regular function with no zeros on or within a contour C, chosen to satisfy the condition g(s) < f(s) on C. This definition of g(s) implies that f(s) + g(s) = f(s) + η(s) f(s) = η(s) It follows that η(s) f(s) have the same number of zeros within C, provided that the inequality g(s) < f(s) is satisfied, which is η(s) f(s) < f(s) on contour C. 5
5.3 Functions f (s), f 2 (s) F (s). If f(s) is chosen to equal f (s) = I +i.i 2, the inequality becomes I i.i 2 < I +i.i 2 or I 2 + I 2 2 < I 2 + I 2 2 Then, if a function F (s) is defined as F (s) = I 2 + I 2 2 (I 2 + I 2 2) where s = 2 + y + it, the inequality can be written as F (s) >. In a similar way, if f(s) = f 2 (s) = I i.i 2, the inequality becomes F (s) < f (s) f 2 (s) are regular functions which do not vanish on the contour C to be defined in Section 6.2. They also have no zeros within the contour C. The reason for these definitions of f (s), f 2 (s) F (s) will be evident in the discussion of the sub-division of the critical strip of the choice of contour C which follows. 6 Subdivision of the critical strip. 6. The Regions R i,i+ The function F (s) = I 2 + I2 2 (I2 + I2 2 ) defines a family of curves F i(s) = which subdivide the critical strip. F i (s) = 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 =. F i (s) = at the root ρ i because I = I I 2 = I 2 at any zero of ξ(s), as derived in Section 3. The chosen subdivision of the critical strip is illustrated in Figure. When t=, I > I I 2 > I 2. Both I I are monotonic decreasing functions of t. The sign of I I alternates as t passes through the roots ρ i on the line. Since the values of I I are dominant, F (s) oscillates in a similar way in step with I I. The first region R, is defined by the y axis, the curve F (s) = segments of the lines y= y = /2, which are intersected by the first two boundaries. All other regions R i,i+, i =,2,... are defined by the curves F i (s) = F i+ (s) = 6
the corresponding segments of the lines y = y = 2. F (s) alternates in sign as t increases from to infinity. F i (s) is > in region R i,i+ when i is even is < when i is odd. 6.2 Analysis of a typical region R i,i+. For purposes of illustration, the region R 3,4 is taken as shown in Figure, in which the function F (s) is <. A contour C is defined by the segments of the lines y = y = 2, together with two curves taken infinitesimally close to F 3 (s) = F 4 (s) =, respectively. It follows that, since F (s) < throughout the region, it wil be satisfied on the chosen contour C. The definition of C implies that the zeros of ξ(s) at t = ρ 3 t = ρ 4 are excluded from C. As described in Sections 5.2-5.3, with the choice f(s) = f 2 (s), the Rouché Inequality is satisfied at all points on within C. Hence, the functions η(s) ξ(s) have no zeros within C. In a similar way, for regions in which F (s) >, with f(s) = f (s), the Rouché Inequality is again satisfied. 6.3 The boundary curves F i (s) = F i,i+ (s) =. 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) =. 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. The function ξ(s) is analytic, hence Rl[ξ(s)] Im[ξ(s)] are harmonic functions, both satisfying Laplace s equation i.e. Rl[ξ(s)] yy +Rl[ξ(s)] tt = Im[ξ(s)] yy +Im[ξ(s)] tt =. It follows that Rl[ξ(s)] Im[ξ(s)] cannot have local maxima or minima. It is known that there are no zeros of ξ(s) on the line y = 2, segments of which form part of the contour C in each region R i,i+. However, Rl[ξ(s)] Im[ξ(s)] can each have a zero on y = 2, but not simultaneously. 6.3. The region R, The region R, is special with one boundary curve as the y axis, where t=. It can be shown that there are no zeros of ξ(s) on the y axis. 7
Figure : Subdivision of the critical strip choice of Contour C in Region R 3,4 shown thus: - - - -. The boundary curves are formed by the family F i (s) =, where i =, 2,.... 8
6.3.2 The regions R i,i+ for i > There are three cases to consider for points on the line y = /2 (i) Rl[ξ(s)] = Im[ξ(s)] non-zero (ii) Rl[ξ(s)] non-zero Imξ[(s)] = (iii) Rl[ξ(s)] Im[ξ(s)] are both non-zero. In Case(i), if a zero of ξ(s) were to exist at y = y in < y < /2 on F i (s), then Im[ξ(s)], which is non-zero at y = /2 on F i (s), would pass through a local maximum or minimum as y decreases to, before becoming zero on the line y=. Case(ii) is analagous to Case(i) except that Rl[ξ(s)] replaces Im[ξ(s)]. The same argument applies because Rl[ξ(s)] = at the point where F i (s) intersects the critical line y = at the zero t = ρ i of ξ(s). In Case(iii), the same argument can be used for either Rl[ξ(s)] or Im[ξ(s]) to show that no zero of ξ(s) can be present on any boundary curve F i (s) = for i=,2,... < y < /2. 6.4 Conclusions from the analysis of Section 6. It has been shown that there are no zeros of ξ(s) in all the subdiivisions of the critical strip, interior to the boundary curves F i (s) =. Furthermore, it has been shown that there can be no zeros of ξ(s) on the boundary curves other than on the critical line y =, from which it follows that the Riemann Hypothesis is true. 7 The limiting forms of I I 2 as t. 7. Notation The integrals defined in Section 2 are: I = 2 ψ(e 2u )e u 2 cosh(yu) cos(tu)du I2 = 2 for < y < /2 < t <, where ψ(x) = n= e n2 πx ψ(e 2u )e u 2 sinh(yu) sin(tu)du 9
Define then f () (u) = 2ψ(e2u )e u 2 cosh(yu) f () 2 (u) = 2ψ(e2u )e u 2 sinh(yu) I = f () (u) cos(tu)du I 2 = The differential coefficients with respect to u are: where r =, 2,... f (r) It is convenient to define I,N = 2 f () 2 (u) sin(tu)du (u) = dr f () (u) du r f (r) 2 (u) = drf () 2 (u) du r f (N) (u) cos sin.tu.du I 2,N = 2 f (N) 2 (u) sin cos.tu.du for N=,,2,... The upper or lower function applies, according as N is even or odd, respectively. This notation implies that 7.2 Series expansions of I I2. I = I I 2 = I 2 Now Integration by parts for t > gives: = I = [ f () (u)sin(tu) t ] f () (u) cos(tu)du t [ ] lim f () u (u)sin(tu) = t [ ] lim f () u (u)sin(tu) = t f () (u) sin(tu)du
Hence, Similarly, In this case I 2 = I = t [ ] f () 2 (u)cos(tu) + t t f () (u) sin(tu)du [ ] lim f () u 2 (u)cos(tu) = t t f () 2 () I 2 = t f () 2 () + t f () 2 (u) cos(tu)du f () 2 (u) cos(tu)du After repeated integration by parts to give three terms plus a remainder, I = t 2 f () () + t 4 f (3) () t 6 f (5) () I 6 t 6 I 2 = t 3 f (2) 2 () + t 5 f (4) 2 () t 7 f (6) 2 () + I 28 t 8 In general I = N ( ) n f (2n ) () t 2n + ( ) N I,2N t 2N I 2 = n= N ( ) n f (2n) 2 () n= t 2n+ + ( ) N+ I 2,2N+2 t 2N+2 7.3 The functional dependence of f (r) () f (r) 2 () on the coordinate y. By repeated application of Leibnitz s theorem for the successive differential coefficients of a product, in terms of the differential coefficients of its factors, with r odd k even, r () = f (r) k= λ r k yk
with r even k odd. r 2 () = f (r) k= λ r k yk The coeficients λ r k are given by ( ) r λ r k = k 2 r k. The final expressions for I I 2 are: with k even with k odd. I 2 = I = N n= N n= ( ) n 2n 2 ( ) n 2n k= k= λ 2n k y k t 2n + ( )N I,2N t 2N λ 2n k yk t 2n+ + ( )N+ I 2,2N+2 t 2N+2 When N=4 the expansions are: I = 2 t 2 ( 8 + 3 2 y2 ) t 4 + ( 32 + 5 4 y2 + 5 2 y4 ) t 6 ( 28 + 2 32 y2 + 35 8 y4 + 7 2 y6 ) t 8 + I,8 I 2 = y t 3 (y 2 + 2y3 ) t 5 + ( 3 6 y + 5 2 y3 + 3y 5 ) t 7 ( 6 y + 7 4 y3 + 7y 5 + 4y 7 ) t 9 I 2, 7.4 Calculations of I I 2 t 8 t Calculations show that I I 2 converge rapidly towards I I 2 as t increases for y 2.This indicates that the necessary sufficient conditions for a zero of ξ(s) are approached as t. The relationships between I, I 2 I,I 2 are described in the next section. 2
8 Series expansions of I I 2. 8. Power series in Limits as t t The functions I (s) I 2 (s), defined in Section 3, can be expressed as power series in /t with coefficients which are polynomials in the variable y. After some algebraic manipulation, it is found that, with k even with k odd. I (y, t) = I 2 (y, t) = n= n= ( ) n 2n 2 k= ( ) n 2n k= λ 2n k y k t 2n λ 2n k yk t 2n+ These expansions are identical to those obtained for I I 2, except that after N terms the remainders are different. This significant result implies that, if I I 2 are written in the form then I = (Nterms) + R (N) I 2 = (Nterms) 2 + R 2 (N) I = (Nterms) + R (N) I 2 = (Nterms) 2 + R 2 (N) (Nterms) = (Nterms) (Nterms) 2 = (Nterms) 2 It follows that I I = R (N) R (N) I 2 I 2 = R 2 (N) R 2 (N) The remainders after N terms, are, R (N) = ( ) N I,2N t 2N R 2 (N) = ( ) N+ I 2,2N+2 t 2N+2 3
R (N) = n=n+ 2n 2 ( ) n k= λ 2n k y k t 2n R 2 (N) = where k is even for R (N) is odd for R 2 (N). As N R (N) R 2 (N). n=n+ 2n ( ) n k= λ 2n k yk t 2n+ From Section 7. the definitions of I,N I 2,N are given in terms of infinite integrals. Hence, the remainders R (N) R 2 (N) can be written as: R (N) = ( ) N.2 f (2N) (u) cos sin.tu.du t 2N R 2 (N) = ( ) N.2 f (2N+2) 2 (u) sin cos.tu.du t 2N+2 for N=,2,3... The upper or lower function applies, according as N is even or odd, respectively. The Riemann-Lebesque theorem[7] implies that the two infinite integrals both tend to zero as t tends to infinity. It follows that (I I ) (I 2 I 2 ) both tend to zero as t tends to infinity. 8.2 Limiting forms of (I I ) (I 2 I 2 ). For given values of y t, I (y, t) = I (y, t) R (N) + R (N) I 2 (y, t) = I 2 (y, t) R 2 (N) + R 2 (N) In the limit, as N, I = I + lim N ( )N I,2N t 2N I 2 = I 2 + lim N ( )N I 2,2N+2 t 2N+2 This can be written as I = I + R ( ) I 2 = I 2 + R 2 ( ). For s = 2 + y + it to be a zero of ξ(s) in the critical strip, the remainders R ( ) R 2 ( ) must be zero simultaneously. 4
9 The functions η(s) ξ(s). The function η(s) introduced in Section 5., can be expressed as: η(s) = I I + i(i 2 I 2 ) = lim N ( )N+ (I,2N t 2N + i.i 2,2N+2 t 2N+2 ) Numerical integration of the infinite integrals shows rapid convergence as N is increased. Whereas the definition of η(s) is η(s) = I I + i.(i 2 I 2 ), the Riemann function ξ(s) can be expressed in terms of I, I ; I 2, I 2 as follows: 2.Rl[ξ(s)] = I I + I 2 I 2 (I ) 2 + (I 2 ) 2 2.Im[ξ(s)] = I I 2 I I 2 (I ) 2 + (I 2 ) 2 The necessary sufficient conditions for both η(s) ξ(s) to be zero are, as shown earlier in Section 3 for ξ(s). I = I I 2 = I 2, It is possible to express either ξ(s) or η(s) as linear functions of the components of the other function, with coefficients U,V or I, I 2, which are simple functions of y t, thus: Rl[ξ(s)] = U.Rl[η(s)] + V.Im[η(s)] Im[ξ(s)] = V.Rl[η(s)] U.Im[η(s)], 2 Rl[η(s)] = I.Rl[ξ(s)] I 2.Im[ξ(s)] 2 Im[η(s)] = I 2.Rl[ξ(s)] + I.Im[ξ(s)]. 5
Conclusion A regular function η(s) has been introduced with the same zeros as ξ(s) in the critical strip. The known zeros of ξ(s) on the critical line σ = 2 have been used to define a family of curves F i (s) = which subdivide the critical strip into an infinity of regions R i,i+, i=,,2....rouché s Theorem is applied to each region in turn to prove that no zeros of ξ(s) lie in these regions, excluding the boundary curves F i (s) =. R, is a special region, with the σ axis as one boundary on which there are no zeros of ξ(s). Since ξ(s) is a regular function of s, Rl[ξ(s)] Im[ξ(s)] are harmonic functions which both satisfy Laplace s equation. It follows that ξ(s) cannot be zero on a boundary curve except at t=ρ i on the critical line. The analysis shows that the zeros of ξ(s) must all lie on the critical line σ = 2, that the Riemann Hypothesis is true. 6
References [] H.M.Edwards, Riemann s Zeta Function, Dover edition 2. [2] B.Riemann, Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse, Gesammelte Werke, Teubner, Leipzig, 892 [3] E.T.Copson, An Introduction to the Theory of Functions of a Complex Variable. OUP London 95. [4] E.C.Titchmarsh, The Theory of the Riemann Zeta Function, Second Edition 986, Clarendon Press - Oxford. [5] T.J.I A. Bromwich, An Introduction to the Theory of Infinite Series. Third Edition 99 AMS Chelsea Publishing. [6] E.T.Whittaker G.N.Watson, A Course of Modern Analysis, CUP 99 [7] E.C.Titchmarsh, The Theory of Functions, OUP London 95. [8] J.Hadamard, Sur la distribution des zeros de la fonction ζ(s) et ses consequences arithmétiques. Bull.Soc.Math.France 24, 99-22, 896 [9] K.Knopp, Theory Application of Infinite Series. Dover Edition 99. [] E.T. Copson, Asymptotic Expansions. Cambridge Tracts in Mathematics No.55 CUP 24. [] J.M.Hill R.K. Wilson, X- rays of the Riemann Zeta Xi functions. School of Mathematics Applied Statistics, University of Wollongong, NSW 2522, Austrailia. [2] The CMI Millenium Meeting Collection, Collége de France, Paris May 24-25, 2. Springer, http://www.claymath.org 7