The Riemann Hypothesis is not true Jonathan Sondow, Cristian Dumitrescu, Marek Wolf To cite this version: Jonathan Sondow, Cristian Dumitrescu, Marek Wolf. The Riemann Hypothesis is not true. 2021. hal-03135456v2 HAL Id: hal-03135456 https://hal.archives-ouvertes.fr/hal-03135456v2 Preprint submitted on 3 May 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. The Riemann Hypothesis is most likely not true J:Sondow1 , C. Dumitrescu2, M. Wolf 3 1https://jonathansondow.github.io 2e-mail: [email protected] 3e-mail: [email protected] Abstract We show that there is a possible contradiction between the Riemann's Hypothesis and the theorem on the strong universality of the zeta func- tion. 1 Introduction In his only paper devoted to the number theory published in 1859 [32] (it was also included as an appendix in [12]) Bernhard Riemann continued analytically the series 1 X 1 ; s = σ + it; σ > 1 (1) ns n=1 to the complex plane with exception of s = 1, where the above series is a harmonic divergent series. He has done it using the integral Γ(1 − s) Z (−z)s dz ζ(s) = z ; (2) 2πi C e − 1 z where the contour C is 6 C ' - - & The definition of (−z)s is (−z)s = es log(−z), where the definition of log(−z) conforms to the usual definition of log(z) for z not on the negative real axis as the branch which is real for positive real z, thus (−z)s is not defined on the positive real axis, 1 see [12, p.10]. Appearing in (2) the gamma function Γ(z) has many representations, we present the Weierstrass product: 1 e−Cz Y ez=k Γ(z) = : (3) z 1 + z k=1 k Here C is the Euler{Mascheroni constant n ! X 1 C = lim − log(n) = 0:577216 :::: (4) n!1 k k=1 From (3) it is seen that Γ(z) is defined for all complex numbers z, except z = −n for integer n > 0, where are the simple poles of Γ(z). The most popular definition R 1 −t z−1 of the gamma function given by the integral Γ(z) = 0 e t dt is valid only for Re[z] > 0. Recently perhaps over 100 representations of ζ(s) are known, for review of the integral and series representations see [27]. The function ζ(s) has trivial zeros at s = −2n; n = 1; 2; 3;::: and nontrivial zeros in the critical strip 0 < Re(s) < 1. In [32] Riemann made the assumption, now called the Riemann Hypothesis (RH 1 for short in following), that all nontrivial zeros ρ lie on the critical line Re[s] = 2 : 1 ρ = 2 + iγ. Contemporary the above requirement is augmented by the demand that all nontrivial zeros are simple. The classical (from XX century) references on the RH are [39], [12], [17], [20]. In the XXI century there appeared two monographs about the zeta function: [6] and [7]. There was a lot of attempts to prove RH and the common opinion was that it is true. However let us notice that there were famous mathematicians: J. E. Littlewood [1], P. Turan and A.M. Turing [5, p.1209] believing that the RH is not true, see the paper \On some reasons for doubting the Riemann hypothesis" [18] (reprinted in [6, p.137]) written by A. Ivi´c.New arguments against RH can be found in [4]. When J. Derbyshire asked A. Odlyzko about his opinion on the validity of RH he replied \Either it's true, or else it isn't" [11, p. 357{358]. There were some attempts to prove RH using the physical methods, see [33] or [42]. We recall some conjectures linked to RH that were disproved in the past. Hasel- grove [15] disproved the P´olya's Conjecture stating that the \most" (i.e., 50% or more) of the natural numbers less than any given number have an odd number of prime factors, i.e. the function X L(x) := λ(n) (5) n≤x satisfies L(x) ≤ 0 for x ≥ 2, where λ(n) is Liouville's function defined by λ(n) = (−1)Ω(n): Here Ω(n) is the number of, not necessarily distinct, prime factors in the decompo- r1 rn sition n = p1 ··· pα(n), with multiple factors counted multiply: r = r1 + ::: + rn, i.e. 2 Ω(n) counts the total number of prime factors of an integer. From the truth of the P´olya Conjecture the RH follows, but not the other way around. The Haselgrove proof was indirect, and in 1960 Lehman [24] has found using the computer explicit counter{example: L(906180359) = 1. Many hopes to prove RH were linked to the Mertens conjecture. Let M(x) denote the Mertens function defined by X M(x) = µ(n); (6) n≤x where µ(n) is the M¨obiusfunction 8 < 1 for n = 1 µ(n) = 0 when p2jn r : (−1) when n = p1p2 : : : pr: From Mertens conjecture p jM(x)j < x (7) again the RH would follow. However in 1985 A. Odlyzko and H. te Riele [29] disproved the Mertens conjecture, again not directly, but later it was shown by J. Pintz [30] that the first counterexample appears below exp(3:21 × 1064). The upper bound has since been lowered to exp(1:59 × 1040) [22]. Some analogs of the RH were proved and some other were disproved. Andr´e Weil proved the Riemann hypothesis to be true for field functions [41], while the Davenport-Heilbronn zeta-function [10], which shares many properties with usual ζ(s), has zeros outside critical line and even to the right of Re(s) = 1, see [3]. In this paper we are going to present incompatibility of RH with the strong (also termed modern or enhanced) theorems on the universality of the zeta function. First such theorem was proved in 1975 by S.M. Voronin [40]. He wanted to prove the RH but instead he proved remarkable theorem on the universality of the Riemann ζ(s) function: Voronin's theorem: Let K be a compact subset of the strip D(1=2; 1) = 1 fs 2 C : 2 < Re(s) < 1g ⊂ C such that the complement of K is connected. Let f : K ! C be a continuous function on K which is holomorphic on the interior of K and is not zero in K. Then for any > 0 there exists a T (, f) such that 3 max f(s) − ζ s + + i T (, f) < . (8) js|≤r 4 Put simply in words it means that the zeta function approximates locally any smooth function in a uniform way. We will use the strong (modern or enhanced) version of (8) (see [23], [26]): 3 Strong version of the universality theorem: Let K be a compact subset of 1 the strip D(1=2; 1) = fs 2 C : 2 < Re(s) < 1g ⊂ C with connected complement, and let f(s) be holomorphic function in the interior of K and non-vanishing on K. Then for any > 0, 1 lim inf µ τ 2 [0;T ] : sup jζ(s + iτ) − f(s)j < > 0 (9) T !1 T s2K holds. Here µ(A) is the Lebesgue measure of the set A. We stress that in these theorems the RH is not assumed. Our reasoning will be based on the following Strong universality in modulus theorem: Consider the function F (s) = eAsζ(s); (10) where A > 0 is a positive constant. Let K be a compact (non{empty) subset of the 1 strip D(1=2; 1) = fs 2 C : 2 < Re(s) < 1g ⊂ C with connected complement and let f(s) be a holomorphic function in the interior of K and non { vanishing on K. Then for any > 0 1 lim inf µ τ 2 [0;T ] : sup jF (s + iτ)j − jf(s)j < > 0 (11) T !1 T s2K Proof: We have F 0(s) ζ0(s) = A + (12) F (s) ζ(s) Next consider the function f(s) g(s) = : (13) eAs The function ζ(s) has the strong universality property so we have: 1 lim inf µ τ 2 [0;T ] : sup ζ(s + iτ) − g(s) < > 0 (14) T !1 T s2K We have the equivalence As Ks ζ(s + iτ) − g(s) < , e ζ(s + iτ) − f(s) < e < M (15) where As σA A e = e < e =: M (16) 4 1 for 2 < σ < 1. So we have the implication: As As e ζ(s + iτ) − g(s)j < ) je ζ(s + iτ) − f(s)j < M (17) and finally: 0 jF (s + iτ)j − jf(s)j < M = : (18) and it is what we wanted to prove. We give some heuristics behind our reasoning. The horizontal (i.e. at fixed t) 1 plots of the jζ(s)j in the critical strip show a local minimum at σ = 2 (at nontrivial zeros) or close to it and to right of the critical line the modulus of the zeta increases 1 up to a local maxima. Sometimes this maximum is for σ 2 ( 2 ; 1) and sometimes it moves to the right of σ = 1, see Fig.1.