THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY
BENOIT CLOITRE
Abstract. In this article I present my last updates on a topological approach to the Riemann hypothesis based on the theory of functions of good variation. This is my first attempt to demonstrate the truth of RH. The Riemann hypoth- esis appears to be the consequence of arithmetical, combinatorial, analytical, algebraical and topological phenomena.
Foreword Now I (almost) fully agree with the opinion of Ricardo Pérez Marco [Mar]: “Reviewing the analytic progress towards the Riemann Hypothesis is quite frus- trating. The pathetic attempts to enlarge the ridiculous zero free region in the critical strip is a perfect example of what brute force can do without fully exploit- ing fundamental arithmetic aspects of the problem. And we can go on and on, with such examples, where the goal is becoming more and more on improving the epsilons, which evidences the fact that the gap towards the conjectured results will never be filled by such methods without some very original input. Reviewing the algebraic progress towards the Riemann Hypothesis is equally frustrating. Only the toy mod- els for non-transcendental functions have been dealt with. Pushing such methods to transcendental zeta functions has been tried and seems well out of scope. The Grothendieckian approach to divinize and solve with a trivial corollary, does not seem well adapted to the analytic nature of the problem (indeed to any hard con- crete analytic problem). The rich structure of the universe of zeta-functions, allows to combinatorially construct L-functions with a complete lack of analytic informa- tion. This explains the proliferation of interrelated conjectures that after all is just another evidence that some analytic results are missed. One may bet that the real hard problem is the Riemann Hypothesis for Riemann zeta-function, and all the vast other generalizations would fall from the techniques, in a pretty similar way, or with some extra not so hard technique” I say "almost" because the philosophy of Alexander Grothendieck, explained in texts that I have read here and there devoted to his topoï, used as bridges between mathematical theories [Car], have influenced the evolution of my recent thoughts. This led me to a quasi affirmation: not only RH is a topological property but it is a “trivial” corollary of some relation between homeomorphic topological spaces of functions (corollary 68). So I would like to say that the sea has risen around the problem and softened it to the point where it can now be picked like a ripe fruit. That said I won’t use category theory and topoï since my construction in- volves classical general topology (topological spaces, homotopy, homeomorphisms and invariants) which seems sufficient to express my thoughts.
Date: Sunday, January 14th 2018. 1 THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 2
1. Notations and definitions In this section I give notations and definitions which will be commonly used in the paper. Definition 1. Sums
If (an)n∈ is a sequence and g :]0, 1] → R is a function then I define N P • A(x) := 1≤k≤x ak P • A1(x) := 1≤k≤x akk P k • Ag(x) := 1≤k≤x akg x Definition 2. Function of good variation (FGV)
I say that a bounded function g :]0, 1] → R satisfying g(1) 6= 0 is a FGV of index α (g) if α (g) is the greatest real value such that for any β < α (g) there exists a non zero constant c(β) such that: −β −β • Ag(n) = n ⇒ A(n) ∼ c (β) n (n → ∞) Definition 3. Dirichlet series and L functions
If (an) is a sequence then L(s, a) denotes the Dirichlet series n∈N P −s • L(s, a) := n≥1 ann Definition 4. Mellin transform
If g :]0, 1] → R is Riemann integrable I consider the following Mellin transform M[g] which is the meromorphic continuation of the integral defined for If g? is a meromorphic function such that there exists g :]0, 1] → R continuous on ? 1 −z−1 the left such that g (z) = 0 g(t)t dt then g is unique and called the inverse Mellin transform of g? and´ I use the notation • g = M−1[g?] and if G? is a space of functions having an inverse Mellin transform then I define −1 ? •M (G ) = {g}g?∈G? Definition 7. Broken harmonic functions (BHF) A bounded function g is a BHF if it exists a real positive sequence (rn)n≥1 satis- fying 1 = r1 > r2 > r3 > ... > r∞ = 0 and such that for any n ≥ 1 we have • rn+1 < x ≤ rn ⇒ g(x) = snx THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 3 where sn > 0 is an increasing sequence of reals such that ∀i ≥ 1 we have risi ≤ M for a constant M > 0. Definition 8. Ingham function The Ingham function is defined by 1 • Φ(x) = x x 1 It is a BHF for which ri = i , si = i and M = 1. Here a plot of the Ingham function. Figure 1 Plot of Φ Definition 9. Generalised Ingham functions Let u = (un)n≥1 then the generalised Ingham function associated to u is defined by P 1 • Φu(x) = x 1≤k≤x−1 uk kx It is easy to see that it is a BHF. The Ingham function is obtained with u1 = 1 and un = 0 for n ≥ 2. Here a plot of the generalised function Φχ4 where χ4 = 1, 0, −1, 0, 1, 0, .... Figure 2 Plot of Φχ4 Definition 10. The Hardy-Littlewood-Ramanujan (HLR) criterion A function g satisfies the HLR criterion (and I say that g is HLR) if for any β ≥ 0 we have the property −β 1−ε • Ag(n) = n ⇒ ∀ε > 0 limn→∞ ann = 0 Definition 11. The Riemann functional equation. A function g which is Riemann integrable on ]0, 1] satisfies a Riemann functional equation (and I say that g is RFE) if its Mellin transform satisfies the property THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 4 g?(z) • • Z≥m denotes the set of integers greater or equal to m. •F denotes the space of FGV (according to definition 2). •FHLR denotes the subspace of F of functions which are HLR (according to definition 10). •FRFE denotes the subspace of F of functions which are RFE (according to definition 11). •FAff denotes the space of functions defined on ]0,1] which are bounded, affine by parts and continuous on the left. •H denotes the set of arithmetical functions h satisfying the following prop- erties: – h is multiplicative – h(n) nε (Ramanujan condition) P −s – L(s, h) := n≥1 h(n)n converges absolutely for 2. Lemmas In this section I state several lemmas. Some proofs are given in the appendices and some of them are classical and are omitted. Lemma 13. The Ingham summation formula n n X jnk X X u = u k k d k=1 k=1 d|k Proof. Classical. Lemma 14. Mellin transform of the Ingham function Φ. We have: 1 ζ(1 − z) Φ?(z) := Φ(t)t−z−1dt = ˆ0 1 − z Proof. See APPENDIX 1 Lemma 15. Mellin transform of generalised Ingham functions Φu. We have: 1 ? −z−1 ζ(1 − z)L(1 − z, u) Φu(z) := Φu(t)t dt = ˆ0 1 − z Proof. See APPENDIX 1 Lemma 16. Asymptotic property of a Dirichlet convolution THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 5 If f is multiplicative and 0 < f(n) ≤ 1 for n ≥ 1 then we have X n µ ? f(n) := µ f(d) = O(1) d d|n Proof. See APPENDIX 2. Lemma 17. Two sums involving the divisors If f(n) is defined for n ≥ 1 by n X X j n k f(n) = u(k) v(i) ik k=1 1≤i≤n/k then we have X n X n u (d) v = µ (f(d) − f(d − 1)) d d d|n d|n Proof. See APPENDIX 2. Lemma 18. Dirichlet inverse of a completely multiplicative function If w is completely multiplicative then w has a Dirichlet inverse given by w−1(n) = µ(n)w(n) . Proof. Classical. Lemma 19. Dirichlet inverse of a multiplicative function If u is multiplicative then the Dirichlet inverse u−1 is also multiplicative. Proof. Classical. Lemma 20. Inverse of multiplicative functions preserve the Ramanujan condition If u is multiplicative and satisfies u(n) = O(nε) then its Dirichlet inverse satisfies also u−1(n) = O (nε). Proof. see APPENDIX 2. Lemma 21. The Ingham function is HLR −β −1 More precisely for any β ≥ 0 we have AΦ(n) = n ⇒ an = O(n ). Proof. See APPENDIX 3 Lemma 22. Some generalised Ingham functions are HLR These Ingham functions are HLR X 1 Φ (x) = x u(k) u kx 1≤k≤1/x where u = χ is a Dirichlet character or u is multiplicative and satisfies the Ra- ε manujan condition un = O (n ). Proof. See APPENDIX 3. Lemma 23. Dirichlet convolution of multiplicative functions THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 6 If u and v are a multiplicative functions then the Dirichlet convolution w = u ? v is also a multiplicative function. In particular if u is a multiplicative function, so m does u for any m ∈ Z≥0. Proof. Classical. Lemma 24. Inverse Mellin transform Using the definition 5 we get (when the integral converges) c+i∞ −1 ? 1 ? ds M [g ](x) := g(x) = g (s) s 2πi ˆc−i∞ x where g is chosen so that it is continuous on the left. Thus it is unique. Proof. It follows from the usual inverse Mellin transform. Lemma 25. Inverse Mellin transform of a product ? ? If g1 and g2 have inverse Mellin transform g1 and g2 according to definition 1.5 ? ? then the product g1 g2 has an inverse Mellin transform defined for x ∈]0, 1] and given by the convolution integral 1 −1 ? ? x dt M [g1 g2 ](x) = g1(t)g2 ˆx t t Proof. It follows from the convolution theorem ([Deb] 8.3.18). Lemma 26. Perron formula Whenever the integral converges we have 1 c+∞ ds A(x) = L(s, a) s 2iπ ˆc−∞ x Proof. Classical. Lemma 27. Weighted Perron formula Whenever the integral converges we have for y ≥ 1 c+∞ 1 ? ds Ag(y) = L(s, a)g (s) s 2iπ ˆc−∞ y Proof. Classical. Lemma 28. The index lemma for generalised Ingham functions P 1 Let Φu(x) = x k≥1 uk kx and suppose α (Φu) ≤ 1 then we have the inequality α (Φu) ≤ η (Φu) Proof. See APPENDIX 4. Lemma 29. The index lemma for generalised Ingham functions which are RFE THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 7 P 1 Let Φh(x) = x k≥1 h(k) kx where h ∈ H then we have 1 1 α (Φ ) = ⇒ η (Φ ) = h 2 h 2 1 Proof. It follows from the lemma 28 yielding η (Φh) ≥ 2 and from the fact that we 1 have trivially η (Φh) ≤ 2 from lemma 15. Lemma 30. The stability of LH under multiplication The space of L functions LH is stable under multiplication. Namely we have 2 (h1, h2) ∈ H ⇒ L(s, h1)L(s, h2) ∈ LH Proof. Straightforward using lemma 23. Lemma 31. The stability of LH under positive exponentiation If h ∈ H then there exists a unique multiplicative function hz ∈ H such that z L(s, h) = L(s, hz) for any complex number z. It is given by the formula k k X z i1 + i2 + ... + ik Y jij hz p = h p i1 + i2 + ... + ik i1, i2, ..., ik i1+2i2+...+kik=k j=1 where Y is the generalised multinomial coefficient of order k. In particular X1,..,Xk if x > 0 is a real value we have x h ∈ H ⇒ L(s, h) ∈ LH Proof. Straightforward using the generalised multinomial theorem applied to the equation z X k −sk X k −sk hz p p = 1 + h p p k≥0 k≥1 3. Introduction Good variation theory (GVT) is in essence discrete and from the definition of FGV (definition 2) we see that the index of good variation is the key feature of the space of FGV. This remark will prove decisive in this article. The reader may consult various working papers on the subject that I have written since 2010 ([Cl1, Cl2, Cl3, Cl4, Cl5, Cl6]) where it is difficult to separate the good from the bad since I didn’t remove my traces from the Internet. In particular I attempted in [Cl5] to developp a topological approach to RH. Unfortunately I was unable to find the suitable continuous morphisms I was looking for. It is done here. In section 4 I start with two canonical example of FGV (theorems 32 and 33) and I recall the existence theorem (theorem 35) underlying somewhat the vastness of the space F. In section 5 I discuss the duality between GVT and complex analysis. Then using suitable dual topological spaces of functions I prove that RH is the central THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 8 problem i.e. RH generalisations are consequences of RH (corollary 44). However it is not enough to prove that RH is true for the zeta function nor for Dirichlet L- functions because an information is missing due to restrictions on the spaces that I consider. Hence building upon this approach I introduce in section 6 two new dual spaces and this time the method works for proving that the functions L(s, h) satisfy RH when h ∈ H (corollary 53). However I still can’t prove it is true for the zeta function. That’s why I extend the definition of FGV in section 7 so that in section 8 I can prove that RH is true for the zeta function (corollary 68) modulo a Tauberian theorem of Delange. 4. The space F and some canonical examples Before going further it is worth to exhibit concrete and important examples of FGV since it is not obvious to see that non trivial FGV exist. The two follow- ing theorems (theorem 32 and theorem 33) provide constructive proofs that some interesting functions are FGV whereas the third theorem (theorem 35) is an exis- tence theorem showing that F is somewhat big and could contain many interesting subspaces. 4.1. Concrete examples. The following theorem shows that continuous affine functions are FGV. These functions are the simplest functions in Faff . Theorem 32. Affine functions are FGV Let g(x) = c1x + c0 where c0, c1 > 0 are reals. Then g is a FGV of index c α(g) = 0 c1 + c0 ? c1 c0 and letting g (z) = 1−z − z we have −1 β < α (g) ∧ A (n) = n−β ⇒ A(n) ∼ n−β (n → ∞) g βg?(β) −α(g) −α(g) −α(g) Ag(n) = n ⇒ A(n) = (1 − α(g)) n log n + O n Proof. See APPENDIX 5. The theorem below shows that functions having infinitely many discontinuites can be FGV. Theorem 33. Some BHF are FGV of index 1 and they are HLR. More precisely let λ ∈ Z≥2 and let b− log x c gλ(x) = xλ log λ then gλ is a BHF with Mellin transform 1 λz−1 − 1 g?(z) = λ 1 − z λz − 1 −β and supposing Agλ (n) = n we get THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 9 1 −β β < 1 ⇒ A(n) ∼ − ? n (n → ∞) βgλ(β) ( lim sup nA(n) = λ β = 1 ⇒ n→∞ lim infn→∞ nA(n) = 1 hence gλ is a FGV of index α (gλ) = 1 according to the one-sided definition of FGV. Moreover gλ satisfies the HLR criterion. Proof. See APPENDIX 6. b− log x c Remark 34. When λ ≥ 2 is an integer value the functions gλ(x) = xλ log λ are hidden generalised Ingham functions. Indeed it is not hard to see that X 1 g (x) = x b(k) λ kx 1≤k≤x−1 P n where b(n) = d|n c(d)µ d and c is the integer sequence given by • c(1) = 1 • c(k) = (1 − λ−1)k ⇔ k is a power of λ • c(k) = 0 otherwise 4.2. The existence theorem for FGV. The space F contains many functions as shown by the theorem 35 below. Theorem 35. Riemann integrable functions are FGV More precisely let g :]0, 1] → R be Riemann integrable, continuous at 1 and satisfying g(1) 6= 0. Then we have g ∈ F. Proof. See APPENDIX 7 for a sketch of proof. 5. The Riemann Hypothesis is the central problem In this section I prove that the good variation index of the generalised Ingham function X 1 Φ (x) = x h(k) h kx 1≤k≤x−1 where h ∈ H (see definition 12) is invariant. A we shall see this means that the Riemann hypothesis for the zeta function can be seen as the central problem which substantiates somewhat the insight of Brian Conrey who writes in [Con] Over the years striking analogies have been observed between the Riemann zeta function and other L-functions. While these functions are seemingly independant of each other, there is a growing evidence that they are all somehow connected in a way that we do not fully understand...There is a growing body of evidence that there is a conspiracy between L-functions- a conspiracy which is preventing us from solving the Riemann hypothesis. THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 10 5.1. Duality. I recall that in [Cl5] I have begun to developp a topological approach to RH but I didn’t succeed. The main reason was that I considered the space FAff only where it is very hard (if not impossible) to exhibit directly continuous morphisms having the suitable properties I was looking for. This was not the right space to work with. Then I felt that a dual space where morphisms are easier to build should exist and should be the key. Thus thanks to the Mellin transform I succeeded to construct two dual spaces of functions which are homeomorphic and such that one of them possesses trivial topological properties so that we could infer non trivial topological properties in the other space. This allows me to say that the truth of the Riemann hypothesis implies the truth of the generalised RH and many other generalisations. In some way this approach underlines that there is indeed a conspiracy among L-functions and the method developped below prevents us to prove that RH is true! An additional ingredient will be required as shown later in section 6. First of all I must emphasize that there is a duality between GVT and complex analysis. It is clear that GVT has something to do with complex analysis because the lemmas 26 and 27 translate in analytic terms the definition of FGV and show the proeminent role of Dirichlet series and of the Mellin transform and its inverse. It is the Perron’s duality: c+∞ 1 ? ds Ag(x) = L(1 − s, a)g (s) s 2iπ ˆc−∞ x 1 c+∞ ds A(x) = L(1 − s, a) s 2iπ ˆc−∞ x Secondly the HLR criterion acts as a bridge between GVT and complex analysis as shown by the following important lemma. Lemma 36. The index lemma in FAff Suppose g ∈ FAff satisfies g(0)g(1) 6= 0 then we have g ∈ FHLR ∩ FRFE ⇒ α (g) = η (g) but the converse doesn’t (always) hold as seen in [Cl5]. Proof. The proof will appear in a forcoming paper [Cl7] and is based on Perron −β duality and the abscissa of convergence of L(s, a) when Ag(x) = x . Let me present a concrete example illustrating and sheding light on this lemma. b− log x c Example 37. The functions gλ(x) = xλ log λ introduced before are HLR if and only if λ ≥√2 is an integer value (see APPENDIX 5 for a proof). If for instance we √ 1 √ take λ = 2 I recently proved that α g 2 = 2 and g 2 ∈/ FHLR. Remark 38. In [Cl4] (section 3, p.139) I state the anti-HLR conjecture underlying the crucial role of the HLR criterion regarding RH and provide a more sophisticated example using the well know Heilbronn-Davenport zeta function which satisfies a Riemman functional equation but violates the RH. THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 11 5.2. Unifying the discrete and the continuous. In passing it is also worth to mention that GVT unifies somewhat the discrete and the continuous1. This is the equivalence principle that I stated few years ago. Suppose that g is a Riemann integrable FGV of good variation index α (g) an consider the discrete-continuous equation n X k n t a g = f(t)g dt = n−β k n ˆ n k=1 0 then when β < α (g) we have the same asymptotic formula for the discrete and the continuous solutions n X n 1 a ∼ f(t)dt ∼ − n−β (n → ∞) k ˆ βg? (β) k=1 0 5.3. The index of good variation is a topological invariant. I introduce now our two dual spaces of functions: a space of complex functions and a space or real functions in that order. ? Definition 39. The space FΦ and FΦ ? FΦ denotes the space of meromorphic functions defined by ? ? ? ? FΦ = {g | g (s) = Φ (s)L(1 − s, h)}h∈H ? ζ(1−s) where Φ (s) = 1−s is the Mellin transform of the Ingham function (see definition 8). FΦ denotes the space of real functions defined on ]0, 1] by −1 ? FΦ= M (FΦ) where M−1 denotes the inverse Mellin tranform (see definition 6) so that from the lemma 15 we get FΦ = {g | g = Φh}h∈H P 1 where Φh(x) = x 1≤k≤x−1 h(k) kx is a generalised Ingham function. We are ? now ready to state a theorem saying that two elements of FΦ are homotopic. ? ? ? ? ? Theorem 40. Homotopy in FΦ. For any pair of functions (g1 , g2 ) ∈ FΦ × FΦ ? ? ? ? there is a continuous morphism in FΦ between g1 and g2 . In other words g1 and ? ? g2 are homotopic in FΦ. ? ? ? ? Proof. If (g1 , g2 ) ∈ FΦ×FΦ then by definition there exist (h1, h2) ∈ H×H such that ? ? ? ? we have g1 (s) = Φ (s)L(1 − s, h1) and g2 (s) = Φ (s)L(1 − s, h2). Next let us define ? ? 1−x ? x ? ? the map T : [0, 1] → FΦ by T (x) = (g1 ) (g2 ) then we have T (0) = g1 ,T (1) = g2 ? and 0 < x < 1 ⇒ T (x) ∈ FΦ (for this last point we have used the lemmas 30 and ? 31). Thus since T is continuous we have found a continuous morphism between g1 ? and g2 . 1“It must be fifteen or twenty years ago when, flicking through the modest volume that holds Riemann’s complete works, I was struck by a passing comment that he makes. He remarks that it could well be that the ultimate structure of space is “discrete”, and that the “continuous” representations that we observe are perhaps simplified (excessively so, perhaps, in the long run. . . ) versions of a more complex reality; that for the human mind, “continuity” is easier to grasp than “discontinuity” [. . . ]. In a strictly logical sense, it has traditionally been the discontinuous that has served as a technical method for approaching the continuous”. Alexandre Grothendieck [Gro]. THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 12 We can now state the theorem I was looking for in [Cl5]. Theorem 41. Homotopy in FΦ. For any pair of functions (g1, g2) ∈ FΦ × FΦ there is a continuous morphism in FΦ between g1 and g2. ? ? Proof. Let gi = M(gi). Let T : [0, 1] → FΦ be defined as above in theorem 40. −1 Then M (T ) is a continuous morphism between g1 and g2. Although the following lemma is a trivial consequence it is of considerable im- portance. ? Lemma 42. The spaces FΦ and FΦ are homeomorphic topological spaces. ? Proof. It is easy to see that FΦ and FΦ are topological spaces and that the ap- −1 ? ? plication M : FΦ → FΦ and the inverse M : FΦ → F Φ are continuous and bijective. 5.3.1. The index of good variation is a topological invariant in FΦ. By construction ? we have built trivial topological properties in FΦ . Namely the properties: g?(z) (1) lim g(x) 6= 0 ⇔ lim zg?(z) 6= 0 x→0 z→0 it is less obvious to describe the dual property of the topological property (1) in ? the space FΦ. Nevertheless the main characteristic of any function in FΦ is the Riemann functional equation and the main characteristic of any function in FΦ is its good variation index. It means that this last one must be a topological invariant as well and must take the form of a constant value. So we have the following table of correspondence of invariants ? Invariant in FΦ Invariant in FΦ Riemann functional equation Good variation index is constant Simple pole at zero Nonzero limit at zero So we can state the following theorem. Theorem 43. (g1, g2) ∈ FΦ × FΦ ⇒ α (g1) = α (g2). 1 Remark. So far we can’t say that the constant value is 2 . It could be zero since one can construct dense subspaces of FGV such that zero is the good variation index of any function in this subspace (an invariant) and such that the Mellin transform has a trivial symmetry (no zero at all in the complex plane). 2The proof of this fact is easy. It is also a simple case of the “fundamental correspondence” of Flajolet stating that there is a precise correspondence between the asymptotic expansion of a function at zero (or infinity) and poles of its usual Mellin transform in a left (resp. right) half-plane (see [Fla]). THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 13 5.3.2. RH is the central problem. Let us see now why I say that RH is the central problem when we stay in the realm of FΦ. Corollary 44. If RH is true for ζ it is true for L(s, h) whenever h ∈ H. Proof. Since (Φ, Φh) ∈ FΦ × FΦ from theorem 42 we have α (Φ) = α (Φh) 1 Next if RH is true for ζ it means that η (Φ) = 2 and Φ is HLR from lemma 21 so 1 that lemma 36 gives α (Φ) = 2 = α (Φh). Since Φh ∈ FHLR as well from lemma 22 we can apply lemma 29 yielding 1 η (Φ ) = α (Φ ) = h h 2 1 and the function ζ(1 − s)L(1 − s, h) has no zero in the half plane 6. The Riemann hypothesis for L(s, h) In this section I introduce another space of meromorphic functions and I derive that RH is true for L(s, h) when h ∈ H in the same spirit than in section 5. Alas it is not sufficient to prove that RH is true for the Riemann zeta function for the moment. 6.1. A step beyond. Let us consider the meromorphic function 1 1 1 g?(z) = − 0 2 1 − z z which is the simplest meromorphic function sharing the main topological properties of Φ? (i.e. a Riemann functional equation and a simple pole at zero and one) and such that the function x + 1 g (x) := M−1 [g?](x) = 0 0 2 is in FAff ∩FHLR and satisfies g0(0)g0(1) 6= 0. Then similarly as before I define two dual spaces of functions F ? and F as follows. g0 g0 Definition 45. F ? = {g? | g?(s) = g?(s)L(1 − s, h)} and F = M−1 F ? . g0 0 h∈H g0 g0 It is easy to describe the space Fg0 as shown by the following lemma. n P h(n) o Lemma 46. We have Fg0 = g | g(x) = n≤x−1 n g0 (nx) . h∈H P a(n) 1 −c+i∞ ? s Proof. From lemma 27 we have n≤x−1 n g (nx) = 2πi −c−i∞ g (s)L(1−s, a)x ds −1 ? P a(n) ´ hence M (g (s)L(1 − s, a)) = n≤x−1 n g (nx). THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 14 P χ4(n) Example. Here the plot of g(x) = n≤x−1 n g0 (nx) P χ4(n) Figure 4 g(x) = n≤x−1 n g0 (nx) Lemma 47. Properties of Fg0 . We have Fg0 ⊂ FAff ∩ FHLR ∩ FRFE . Proof. Straightforward for Fg0 ⊂ FAff ∩ FRFE . For Fg0 ⊂ FHLR it is left to the reader. The space Fg0 has still a deep arithmetical meaning which should be explored to uncover new landscapes in prime number theory as shown by the following remark where it is question of a mock Moebius function having relations with primes. Remark 48. Consider the Dirichlet character χ4 = 1, 0, −1, 0, 1, 0, −1, 0, ... and de- fine the function gχ4 wich is in Fg0 by 1 X χ4(n) g (x) = (nx + 1) χ4 2 n n≤x−1 µ(n) Recalling that the Moebius function is given by the sequence an = n defined by A (n) = n−1 let a be another sequence defined by the same recursion A (n) = Φ gχ4 −1 n−1 n and let bn = 2 n!an. This produces an odd integer sequence: (bn)n≥1 = {1, −1, 5, −15, −489, −2865, 35685, −135135, −5897745, ...} yielding a characterisation of odd composite numbers: • n | bn ⇔ n is an odd composite number More interestingly there are these two properties allowing us to characterise primes modulo 4: • (2n + 1) | (b2n+1 − 1) if and only if 2n + 1 is a prime of form 4k + 1 • (2n + 1) | (b2n+1 − 2n) if and only if 2n + 1 is a prime of form 4k + 3 THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 15 In the APPENDIX 9 I provide a graphic supporting the fact that A(n) n−1/2+ε 1 meaning that α (gχ4 ) = η (gχ4 ) = 2 so that RH would be true for L(s, χ4). In the APPENDIX 10 I provide an alternative experimental support. But let’s go back to our subject. Theorem 49. Homotopy in F ? . For any pair of functions (g?, g?) ∈ F ? × F ? g0 1 2 g0 g0 there is a continuous morphism in F ? between g? and g?. g0 1 2 Proof. Similar to theorem 39. Theorem 50. For any pair of functions (g1, g2) ∈ Fg0 × Fg0 there is a continuous morphism in Fg0 between g1 and g2. Proof. Similar to theorem 40. Lemma 51. The dual spaces F ? and F are homeomorphic topological spaces. g0 g0 Proof. Similar to lemma 41. 6.2. The index of good variation is a topological invariant in Fg0 . Again we have built trivial topological properties in F ? . Namely for g?∈ F ? we have g0 g0 the properties: ? g (z) ? (1) Theorem 52. (g1, g2) ∈ Fg0 × Fg0 ⇒ α (g1) = α (g2). Corollary 53. The Riemann hypothesis is true for L(s, h) when h ∈ H. Proof. We have g?(s) and g?(s) = g?(s)L(1 − s, h) which are in F ? for h ∈ H. 0 h 0 g0 1 Next from theorem 52 we get α (g0) = α (gh). Hence since α (g0) = 2 from theorem 1 1 32 (taking c0 = c1 = 1) we get α (gh) = 2 . Then the lemma 36 yields η (gh) = 2 meaning that RH is true for L(s, h) when h ∈ H. Note that the conspiracy of L functions is still working since we are unable to derive that RH is true for the zeta function with the pair of spaces F ? , F and g0 g0 the theorem 52. However going further and generalising the definition of FGV will allow us to derive that RH is true for zeta. THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 16 7. Towards the original problem 7.1. FGV of slow variation at zero. The boundedness condition at zero in definition 2 of FGV is superfluous and we can consider FGV behaving like log(x)m as x → 0 in order to take into account Mellin transforms of FGV having a multiple pole at 0 (and so L functions having a multiple pole at 1) . Here my goal is just to prove that RH is true for the Riemann zeta function but this generalisation of the definition of FGV is required to prove that many more functions than L(s, h) with h ∈ H satisfy the RH (in fact the whole Selberg class as discussed briefly in the concluding remarks). First of all I recall the definition of a function of slow variation. Definition 54. (After Karamata, [Kor], p.177). L : R+? → R is a function of slow variation if we have: L(tx) ∀x > 0 lim = 1 t→∞ L(t) Then I extend the definition 2 of a function of good variation. Definition 55. Extended definition of functions of good variation. I say that g :]0, 1] → R satisfying g(1) 6= 0 is a FGV of index α (g) if we have: • g(x) ∼ L(x)(x → 0) where L is slowly varying. • g is bounded on any intervall ]ε, 1] when 0 < ε < 1. • α (g) is the greatest real value such that for β < α (g) ∧ βg?(β) 6= 0 we have 1 A (n) = n−β ⇒ A(n) ∼ − n−β (n → ∞) g βg? (β) I extend also the definition of spaces of functions given in definition 12 to func- tions of slow variation at zero. Let us start with one of the simplest example of FGV of logarithmic growth at zero. Example 56. The function g(x) = 1 − log(x) has Mellin transform given by 1 − z g?(z) = z2 which has a double pole at zero and is a FGV according to the definition 55 which satisfies α (g) = η(g) = 1. See APPENDIX 8 for a constructive proof of this fact. 7.2. Some lemmas and definitions. Lemma 57. The lemma 36 is still valid. Proof. The proof will appear in [Cl7]. ? ? ? ? ? λ −1 ? Definition 58. Fζ = g | g (s) = g0 (s)ζ (1 − s) 0≤λ<1 and Fζ = M Fζ where (ζ?) λ is the meromorphic continuation of the function defined by ζ?(z)λ = ζ(z)λ in the half-plane λ + v − 1 τ (pv) = λ v Proof. Similar to lemma 46 and using lemma 31. Remark 60. For the real power of the zeta function the general lemma 31 is not v required. Indeed the formula for τλ(p ) is well known when one uses the Selberg- Delange method (see for instance[Ten], chapitre “La méthode de Selberg-Delange”) which involves complex powers of zeta. P τ1/2(n) Example. Here the plot of g(x) = n≤x−1 n g0 (nx) in figure 5. P τ1/2(n) Figure 5 g(x) = n≤x−1 n g0 (nx) Lemma 61. We have Fζ ⊂ FAff ∩ FHLR ∩ FRFE . Proof. Straightforward for Fζ ⊂ FAff ∩FRFE . For Fζ ⊂ FHLR it follows from lemma 22. Before stating the lemma 63 which shows that functions in Fζ are of slow variation at zero I need to recall the Delange Tauberian theorem (see for instance[Ten], chapitre “Théorèmes taubériens”) which is a generalisation of the Wiener-Ikehara theorem ([Kor]p. 124). P −s Theorem 62. Delange Tauberian theorem. Let F (s) := n≥1 fnn denotes a Dirichlet series with nonnegative coefficients and convergent for Rs > 1. Assume (i) F(s) is analytic for all points on X y λ−1 f ∼ (log y) (y → ∞) n Γ(λ) n≤y Lemma 63. Functions in Fζ are of slow variation at zero. P τλ(n) More precisley let gλ(x) = n≤x−1 n g0 (nx) and 0 < λ ≤ 1. Then we have the two following properties: THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 18 1 P τ(n) 1 2 (1) λ = 1 ⇒ g1(x) ∼ 2 n≤x−1 n ∼ 4 log(x) + γ log(x) + C (x → 0) 1 P τλ(n) −1 λ (2) 0 < λ < 1 ⇒ gλ(x) ∼ 2 n≤x−1 n ∼ 2λΓ(λ) log(1/x) (x → 0) P √ Proof. For property (1) it follows from n≤y τ(n) = y (log y + 2γ − 1) + O ( x) (cf. [Ten], chapitre “Ordres moyens”) and Abel summation. For property (2) it follows from Delange Tauberian theorem 62 and Abel summation. The reader may notice that there is a discontinuity between the properties (1) and ? (2) when λ → 1. This is the reason why I consider λ < 1 not λ ≤ 1 and Fζ is a semi open topological space. We are now ready to tackle RH for the Riemann zeta function. 8. The Riemann hypothesis for the zeta function ? ? ? ? ? Theorem 64. Homotopy in Fζ . For any pair of functions (g1 , g2 ) ∈ Fζ ×Fζ there ? ? ? is a continuous morphism in Fζ between g1 and g2 . ? ? ? ? 2 Proof. If (g1 , g2 ) ∈ Fζ ×Fζ then by definition there exist (λ1, λ2) ∈]0, 1[ such that ? ? ? λ1 ? ? ? λ2 we have g1 (s) = g0 (s)ζ (s) and g2 (s) = g0 (s)ζ (s) . Next let us define the map ? ? 1−x ? x ? ? T : [0, 1] → Fζ by T (x) = (g1 ) (g2 ) then we have T (0) = g1 ,T (1) = g2 and ? 0 < x < 1 ⇒ 0 < (1 − x)λ1 + xλ2 < 1 ⇒ T (x) ∈ Fζ Theorem 65. Homotopy in Fζ . For any pair of functions (g1, g2) ∈ Fζ × Fζ there is a continuous morphism in Fζ between g1 and g2. Proof. Similar to theorem 41. ? Lemma 66. The spaces Fζ and Fζ are homeomorphic topological spaces. Proof. Similar to lemma 42. 8.1. The index of good variation is a topological invariant in Fζ . Again ? ? ? we have built trivial topological properties in Fζ . Namely for g ∈ F ζ we have the properties: g?(z) (1) ? Invariant in Fζ Invariant in Fζ Riemann functional equation Good variation index is constant 1+λ ? −λ 1 ∃λ ∈ [0, 1[, limz→0 z g (z) 6= 0 ∃λ ∈ [0, 1[, limx→0 log(1/x) g(x) 6= 0(= − 2λΓ(λ) ) So we can state the following theorem. Theorem 67. (g1, g2) ∈ Fζ × Fζ ⇒ α (g1) = α (g2). Corollary 68. The Riemann hypothesis is true for ζ. THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 19 Proof. Taking λ1 = 0 and for instance λ2 = 1/3 we get from theorem 67 α(g0) = ? 1/3 1/2 = α g1/3 . Next from lemma 57 we have η g1/3 = 1/2 meaning that (ζ ) has no zero in the half-plane 1 A(n) = A(bntc)dg(t) + n−β ˆ0 which looks more like a variational problem. Moreover there is in one hand an ex- plicit formula for A(n) using the spectrum of zeros of g? and in the other hand there is a variational equivalence to RH where the usual Mellin transform and the classical explicit formula play a role ([Bro], p.166, 9.6). Then it seems realistic to compare good variation theory to Noether theory since we are dealing with RH which is known to be deeply connected to physics [Gran]. One can add that representations of Lie groups are connected to random matrices [Col] which are experimentaly connected to RH [Bou] and that Noether theorem relies on Lie groups. I guess also that the theorems 43, 55 and 67 are just the tip of an iceberg. Indeed let us consider the space of meromorphic functions ? ? ? ? F = {g | g (s) = g0 (s)L(1 − s)}L∈S where S denotes the Selberg class. Then using the same method and additional technical results (such as an extension of the theorem 62) we can prove that RH is true for all functions in S. Surely there must be some Grothendieckian statement encompassing these par- ticular cases. In an other hand good variation theory allows us to go further in number theory since we could tackle other big conjectures related to nontrivial ze- ros such as the simplicity or the linear independance over the rationals. This will be discussed in a forcoming paper [Cl8]. 3 Pn It is an equation of the type x(n) = f(n) + j=0 y(n, j)x(j)(n ≥ 0) where y(., .) and f(.) are known functions and x(.) is unknown. Cf. for instance [Dib] for results and other references related to these equations. THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 20 References [Bou] Paul Bourgade and Jonathan P. Keating, Quantum chaos, random matrix theory and the Riemann zeta function, Séminaire Poincaré XIV (2010) [Bro] Kevin Broughan, Equivalents of the Riemann Hypothesis, Volume Two: Analytic Equiva- lents, Encyclopedia of Mathematics and its Application 165, Cambridge University Press (2017) [Car] Olivia Caramello, Unifying theories: Toposes as bridges [Ras] Olexandr Rasin and Peter Hydon, Conservation laws for NQC-type difference equations, J. Phys. A: Math. Gen. 39 (2006) [Cl1] Benoit Cloitre, A tauberian approach to RH, Arxiv (2010) [Cl2] Benoit Cloitre, Broken Harmonic Functions (2014) [Cl3] Benoit Cloitre, Good Variation Theory, long version (2015) [Cl4] Benoit Cloitre, Good variation theory: a Tauberian approach to RH, International Journal of Mathematics and Computer Science, Vol.2, (2016) [Cl5] Benoit Cloitre, The space of functions of good variation (2017) [Cl6] Benoit Cloitre, Quelques conjectures nouvelles en théorie des fonctions à bonne variation, On researchgate (2017) [Cl7] Benoit Cloitre, On two fundamental lemmas in good variation threory, in preparation (2018) [Cl8] Benoit Cloitre, Good variation theory and qualitative properties of nontrivial zeros, in preparation (2018) [Col] Benoit Collins and Piotr Sniady, Representations of Lie groups and random matrices, Trans- actions of the American Mathematical Society, Volume 361 (2009) [Con] Brian Conrey, The Riemann Hypothesis, 2003 [Deb] Lokenath Debnath and Dambaru Bhatta, Integral Transforms and Their Applications, Chapman and Hall/CRC (2014) [Dib] J. Diblik et al, On the existence of solutions of linear Volterra difference equations asymp- totically equivalent to a given sequence, Applied Mathematics and Computation 218 (2012) [Fla] Philippe Flajolet et al, Mellin transforms and asymptotics: Harmonic sums, Theoretical Computer Science, Volume 144, Issues 1–2, Pages 3-58 (1995) [Gran] Andrew Granville, Nombres premiers et chao quantique, Gazette 97, Société Mathématique de France, (2003) [Gro] Alexandre Grothendieck, Récoltes et semailles. Chapitre 2. Promenade à travers une oeuvre ou l’Enfant et la Mère. § 2.20. Coup d’oeil chez les voisins d’en face, p. 80 (1983-1985) [Kor] Jacob Koorevar, Tauberian theory: a century of developpements, Springer, Volume 329 (2004) [Kos] Yvette Kosmann-Schwarzbach, Les théorèmes de Noether: Invariance et lois de conservation au vingtième siècle, deuxième édition, Les éditions de l’école polytechnique, 2006. [Mar] Ricardo Pérez Marco, Notes on the Riemann Hypothesis, Jordanas sobre los problemas del Milenio, Barcelona, (2011) [Noe] Emmy Noether, Invariante Variationsprobleme, Nachr. d. König. Gesellsch. d. Wiss. zu Göttingen, Math-phys. Klasse, 235–257 (1918) [Ten] Gerald Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Belin, 4ème édition (2014) THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 21 APPENDIX 1 Proof of lemmas 14 and 15 By definition of the Mellin transform for X jnk w(n) = u k k 1≤k≤n we get 1 X w(n) − w(n − 1) Φ? (z) = u 1 − z n1−z n≥1 Pn P and since we have w(n) = k=1 d|k ud from lemma 13 the above equality becomes P 1 X d|n ud 1 X (u ? 1)(n) ζ(1 − z)L(1 − z, u) Φ? (z) = = = u 1 − z n1−z 1 − z n1−z 1 − z n≥1 n≥1 P −s where L(s, u) = n≥1 unn . The Mellin transform of the Ingham function is obtained with L(s, u) = 1 and we get ζ(1 − z) Φ?(z) = 1 − z THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 22 APPENDIX 2 Proof of lemmas 16, 17 and 20 Lemma 16. It is easy to see that X n b(n) := µ f(d) d d|n is the multiplicative function given by v v−1 bpv = f (p ) − f p Q αi Next letting n = pi where pi are the distinct primes in the factorisation of n we have using the fact 0 ≤ f(n) ≤ 1 αi αi−1 −1 ≤ f (pi ) − f pi ≤ 1 hence we get Y αi αi−1 bn = f (pi ) − f pi = O(1) Lemma 17. If f(n) is defined for n ≥ 1 by n X X j n k f(n) = u(k) v(i) ik k=1 1≤i≤n/k then we have by lemma 13 n n b k c X X X f(n) = u(k) v(d) k=1 j=1 d|j thus we get n n−1 n−1 b k c b k c X X X X X f(n) − f(n − 1) = u(n)v(1) + u(k) v(d) − v(d) k=1 j=1 d|j j=1 d|j next we have n n−1 b k c b k c X X X X jnk n − 1 v(d) − v(d) 6= 0 ⇔ − > 0 ⇔ k | n k k j=1 d|j j=1 d|j so that we get with the convention f(0) = 0 X X 0 X f(n) − f(n − 1) = u(d) v(d ) = u(d)(v ? 1)(n/d) = (u ? v ? 1)(n) d|n d0|n/d d|n Finally by Möbius inversion it follows that X n X n u (d) v = (f(d) − f(d − 1)) µ d d d|n d|n THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 23 Lemma 20. We have u−1(1) = 1 and the recursive formula for n ≥ 2 X n u−1(n) = − u−1(d)u d d|n,d n−1 X u−1 (pn) = − u−1 pk u pn−k k=0 We assume that ∀ε > 0, u pn−k = O pε(n−k). Suppose also that we make the reccurence hypothesis ∀ε > 0, u−1 pk = O pεk for any p ≥ 2 prime and any k ≤ n − 1 then we get n−1 n−1 −1 n X −1 k n−k X ε k ε (n−k) ε n ∀ε > 0, u (p ) ≤ u p u p p 2 p 2 = np 2 k=0 k=0 ε n next n p 2 for any ε > 0 and any p ≥ 2 hence we get −1 n εn u (p ) p Q αi and the reccurence hypothesis is true for all n. Thus letting n = pi we have finally −1 Y εαi ε u (n) pi = n THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 24 APPENDIX 3 Proof of LEMMA 21 and 22 The Ingham function is HLR. This is true for the case β = 0 since we have trivially in this case a1 = 1 and an = 0 for n ≥ 2. So I consider β > 0. Then from lemma 13 we have n n X jnk X X X ka = da ⇒ da = n1−β − (n − 1)1−β k k d d k=1 k=1 d|k d|n Therefore by Möbius inversion we get X n na = µ d1−β − (d − 1)1−β n d d|n next we have d1−β − (d − 1)1−β = (1 − β)d−β + O d−1−β thus we get X n X n na = (1 − β) µ d−β + µ O(d−1−β) n d d d|n d|n P n −β Now since β > 0 we have, on one hand, from the lemma 16 d|n µ d d = O(1) P −1−β and, on the other hand, n≥1 n converges toward the finite value ζ(1 + β). P n −1−β Consequently we get d|n µ d O(d ) = O(1). As a result, nan = O(1) and Φ is HLR. The generalised Ingham functions Φχ are HLR. From lemma 13 we have X n X n A (n) = n−β ⇒ da (d) χ = µ d1−β − (d − 1)1−β Φχ d d d|n d|n and we know from above that for β ≥ 0 we have X n µ d1−β − (d − 1)1−β = O(1) d d|n 0 P n 1−β 1−β Hence letting a (n) = na(n) and b(n) = d|n µ d d − (d − 1) we get from above and lemma 18 (Dirichlet characters are completely multiplicative) X n n a0 ? χ(n) = b(n) ⇒ a0(n) = b ? χ−1(n) = b (d) χ µ d d d|n whence since |b(n)| ≤ C (for some C > 0) we get X n n |a0(n)| ≤ |b (d)| χ µ ≤ Cτ(n) nε d d d|n ε so that na(n) = O(n ) and Φχ is HLR. THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 25 Some other generalised Ingham function are HLR. The generalised Ingham P 1 function Φu(x) = x 1≤k≤1/x u(k) kx is HLR where u is any multiplicative func- tion satisfying the Ramanujan condition u(n) = O (nε) . Indeed from lemma 13 we have X n X n A (n) = n−β ⇒ da (d) u = µ d1−β − (d − 1)1−β Φu d d d|n d|n and from lemma 16 for β ≥ 0 we have X n µ d1−β − (d − 1)1−β = O(1) d d|n 0 P n 1−β 1−β Hence letting a (n) = na(n) and b(n) = d|n µ d d − (d − 1) we get X a0 ? u(n) = b(n) ⇒ a0(n) = b ? u−1(n) = b (n/d) u−1(d) d|n whence since b(n) = O(1) and u−1(n) = O (nε) from lemma 20 we get for any ε > 0 X −1 X ε/2 ε/2 ε |na(n)| ≤ |b (n/d)| u (d) d n τ(n) n d|n d|n ε/2 ε since τ(n) n , consequently na(n) = O(n ) and Φu is HLR. THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 26 APPENDIX 4 Proof of LEMMA 28 −1 Let AΦu (n) = n then by the definition 2 of FGV en supposing α (Φu) ≤ 1 we have −1 −α(Φu)+ε AΦu (n) = n ⇒ A(n) n so that by Abel sommation we get n X 1−α(Φu)+ε akk n k=1 in an other hand it is easy to see using lemma 13 that we have X akk 1 A (n) = n−1 ⇒ = Φu ks ζ(s)U(s) k≥1 1 hence the function ζ(s)U(s) is analytic in the half-plane 1 − η (Φu) ≤ 1 − α (Φu) ⇒ α (Φu) ≤ η (Φu) THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 27 APPENDIX 5 Proof of theorem 32 −β More precisely if Ag(n) = n we have 7 cases to consider summarized in the ? c1 c0 following table where g (z) = 1−z − z is the little Mellin transform of g. Condition on β A(n) (as n → ∞) n−β −1−β β < α(g) − 1 − βg?(β) + O n n−β −α(g) β = α(g) − 1 − βg?(β) + O n log n n−β −α(g) α(g) − 1 < β < 0 − βg?(β) + O n β = 0 1 + O n−α(g) c0 n−β −α(g) 0 < β < α(g) − βg?(β) + O n β = α(g) (1 − α(g)) n−α(g) log n + O n−α(g) β > α(g) O n−α(g) 1 Proof. Without loss of generality, we take c1 = c0 = 1/2 so that α (g) = 2 and prove the formula of theorem 32 for the case β = α(g) − 1 = −1/2. The other formulas are proved similarly for any c1, c0 > 0 and any β. So let x 1 • g(x) = 2 + 2 1/2 • Ag(n) = n • h(n) = n−1 n3/2 − (n − 1)3/2 First we get the exact formula for any n ≥ 2 (details are omitted) n−1 ! (1/2) X k! A (n) = n1/2 ⇒ A(n) = h(n) + n 2 + h(k) g n! (1/2) k=2 k where (x)n = x(x + 1)...(x + n − 1) and it is easy to see that we have the 3 asymptotic formulas as n → ∞ or k → ∞ 3 3 h(n) = n−1/2 − n−3/2 + O n−5/2 2 8 (1/2) 1 Γ (1/2) n = n−1/2 − n−3/2 + O n−5/2 n! 8 1 k! 3 3 h(k) = − k−1 + O k−2 Γ (1/2) (1/2)k 2 16 hence plugging these 3 asymptotic formulas we get 3 3 A (n) = n1/2 ⇒ A(n) = n1/2 − n−1/2 log n + O n−1/2 g 2 16 THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 28 APPENDIX 6 Proof of theorem 33 Without loss of generality I take λ = 2 and I write g = g2. We get X X k X X A (n) = h(n) ⇒ a g = h(n) ⇒ 2j ka = nh(n) g k n k j≥0 2−j−1 log n b log 2 c X n A (n) = w + O(1) 1 2i j=0 n n n −β n −β Since w 2i = 2i 2i − 2i+1 + O(1) we obtain n n 1−β w = 1 − 2β + O(1) 2i 2i yielding log n b log 2 c β 1−β X i(β−1) A1(n) = 1 − 2 n 2 + O(log n) j=0 Pm i xm+1−1 1−2β 1−β Next since j=0 x = x−1 we get A1(n) = 1−2β−1 n + O(log n). Hence if β < 1 we have 1 − 2β A (n) ∼ n1−β (n → ∞) 1 1 − 2β−1 Now by Abel summation we get n X A1(n) A1(t) a = + dt k n ˆ t2 k≤n 1 whence THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 29 1 − 2β 1 A(n) ∼ 1 − n−β (n → ∞) 1 − 2β−1 β ? 1 1−2z−1 Since g (z) = 1−z 1−2z finally we get 1 β < 1 ⇒ A(n) ∼ − n−β (n → ∞) βg? (β) Case β = 1. Then A1(1) = 1 and for n ≥ 2 we have jnk A (n) = A − 1 1 1 2 hence we get the simple closed form formula log n A (n) = 1 − 1 log 2 yielding 1 log(n − 1) log n a(n) = − n log 2 log 2 1 Therefore a1 = 1, an = − n if n ≥ 2 is a power of 2 and an = 0 otherwise and we get in this case X 1 1 A(n) = 1 − j = log n 2 b log 2 c 2j ≤n 2 thus we have ( lim sup nA(n) = 2 β = 1 ⇒ n→∞ lim infn→∞ nA(n) = 1 Hence g is a FGV of index 1. It is also easy to see that nan = O(1) for any β ≥ 0 so that g satisfies the HLR criterion. THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 30 APPENDIX 7 Proof of theorem 35 Let g :]0, 1] → R be Riemann integrable and satisfying g(1) 6= 0. Then we have g ∈ F. Indeed for β < 0 with |β| large enough we have 1 A (n) = n−β ⇒ A(n) ∼ − n−β (n → ∞) g βg? (β) So that from the least upper bound property there is a maximum value α (g) such that for β < α (g) the previous asymptotic formula holds. THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 31 APPENDIX 8 I prove that the function g(x) = 1 − log x is a FGV of index α (g) = 1 according to definition 55. Indeed letting Ag(n) = h(n) we have A(1) = 1 and for n ≥ 2 it is easy to see that we have the first order recursion 1 A(n) = 1 + log 1 − A(n − 1) + h(n) − h(n − 1) n and that (xn)n∈Z≥0 satisfying xn+1 = unxn + vn can be written as follows n−1 ! n−1 n−1 ! Y X Y xn = uk x0 + uk vk k=0 k=0 i=k+1 Qn 1 Thus letting w(n) = k=2 1 + log 1 − k we get for n ≥ 2 n ! X h(k) − h(k − 1) A(n) = w(n) 1 + w(k) k=2 Now we have c w(n) ∼ (n → ∞) n with n Y 1 c = 0.45342... = lim n 1 + log 1 − n→∞ k k=2 Pn h(k)−h(k−1) −β so that k=2 w(k) converges when k (h(k) − h(k − 1)) k and β > 1. Hence if h(n) = n−β we have 3 cases to consider: −β −β −1 P∞ k −(k−1) (1) β > 1 ⇒ A(n) ∼ c (1 + C(β)) n (n → ∞) where C (β) = k=2 w(k) −1 −1 −1 −1 log n Pn k −(k−1) (2) β = 1 ⇒ A(n) ∼ − log(n)n +cn (n → ∞) since − c ∼ k=2 w(k) 1 −β −1 β −β (3) β < 1 ∧ β 6= 0 ⇒ A(n) ∼ − βg?(β) n + cn (n → ∞) since c(1−β) n ∼ Pn k−β −(k−1)−β k=2 w(k) ? 1−z where g (z) = z2 is the Mellin transform of g. Therefore g is a FGV of index 1. THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 32 APPENDIX 9 About the remark 48 and the sequence defined by A (n) = n−1. gχ4 Figure 6. n1/2A(n) THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 33 APPENDIX 10 The Dirichlet beta function is defined in n X (−1) X χ4(n) β(s) := = (2n + 1)s ns n≥0 n≥1 where χ4 is again the Dirichlet caracter modulo 4 {χ4(n)} = {1, 0, −1, 0, 1, 0, −1, 0, ...} n∈Z≥1 Then I define the function Θ on ]0, 1] by X χ4(k) Θ(x) = k 1≤k≤x−1 Here the plot of Θ on ]0, 1] showing that it is a BHF. Figure 7. Plot of Θ 1 1 As we can see Θ is constant on semi open intervall ] 2n+1 , 2n−1 ] and more pre- cisely we have 2n−1 n k−1 1 1 X χ4(k) X (−1) < x ≤ ⇔ Θ(x) = = 2n + 1 2n − 1 k 2k − 1 k=1 k=1 The index of good variation of Θ. We have β(1 − z) Θ?(z) = − z THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 34 next since RH is true for the Dirichlet beta function and that we have Θ ∈ FHLR (easy check) the lemma 36 yields 1 α (Θ) = 2 1 Experimental support. To support the fact that we have α (Θ) = 2 let us in- troduce a mock Moebius function (µΘ(n)) with the following recursion n∈Z≥1 n X µΘ(k) k 1 Θ = k n n k=1 and the mock Liouville lambda function (λΘ(n)) n∈Z≥1 n √ X λΘ(k) k b nc Θ = k n n k=1 As we shall see these two functions are connected to primes and so the Dirichlet beta function encodes informations about primes, therefore on zeros of the zeta function so that it is not surprising to read that L functions know about each other. Properties of µΘ. The first few values of µΘ are 1 1 1 1 1 5 1 5 4 1, − , , − , − , − , , − , − , , ... 2 6 12 4 5 42 56 72 45 and the sequence b given by b(n) = n!µΘ(n) is an integer sequence starting with 1, −1, 1, −2, −30, −144, 600, −720, −25200, 322560, ... The main properties of b allow us to characterise primes and primes of form 4k + 1 or 4k + 3. Namely we have: • b(n) > 0 modulo n if and only if n is 2, 4 or an odd prime of the form 4k+3. • b(n) > 0 modulo n2 if and only if n is 2, 4, 8, an odd prime or 2 times an odd prime of form 4k+1. Properties of λΘ. The first few values of λΘ are 1 1 1 3 7 2 1 1 1 1, − , , , − , − , , − , , , ... 2 6 6 10 30 21 28 36 15 and the sequence c given by c(n) = n!λΘ(n) is an integer sequence starting with 1, −1, 1, 4, −36, −168, 480, −1440, 10080, 241920, 2540160, 15724800, ... The main properties of c allow us to characterise primes only. Namely we have: • c(n) > 0 modulo n if and only if n is 2, 4 or an odd prime. • c(n) > 0 modulo n2 if and only if n is 2, 4, 8, 9, an odd prime or 2 times an odd prime. THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 35 Experimental support that RH is true for the Dirichlet β function. It is not hard to see that RH is true for the Dirichlet beta function if and only if we have MΘ(x) := P 1/2+ε P 1/2+ε 1≤k≤x µΘ(k) x and LΘ(x) := 1≤k≤x λΘ(k) x . Here a plot of the mock Mertens function MΘ and the mock Liouville summatory function LΘ (green) compared with the Mertens and the Liouville summatory function (red) 1/2+ε (figures 8 and 9). The figure 10 and the figure 11 support MΘ(x) x and 1/2+ε LΘ(x) x respectively. Figure 8. MΘ (green) vs M (red) Figure 9. LΘ (green) vs L (red) THE RIEMANN HYPOTHESIS IS A TOPOLOGICAL PROPERTY 36 M (x) √Θ Figure 10. Plot of x L (x) √Θ Figure 11. Plot of x 1 – L(1, h) 6= 0 L(1−z,h) – 1 we have F (s) = (1−s)λ + H(s)(λ > 0) where H is analytic at s = 1. Then 0 by the Dirichlet series