THE RIEMANN HYPOTHESIS: IS TRUE!!

Transcription

1 THE RIEMANN HYPOTHESIS: IS TRUE!! Ayman MACHHIDAN Morocco Abstract : The Riemann s zeta function is defined by = for complex value s, this formula have sense only for >. This function is analytically extended to hole complex plan (with a simple singularity on s=). The Zeta was first introdud by Euler, but Riemann proved it utility by establishing the relationship between the distribution of prime number and the localization of zeta s zeros. Riemann conjectured that all non-trivial zeros of the zeta functions are on the critical line = In this paper we will try to give a positive answer. INTRODUCTION In 859 Bernhard Riemann wrote a short paper titled Uber die Anzahl der Primzahlen unter einer gegebenen Grosse, which can be translated as On the Number of Prime Numbers less than a Given Quantity. As the title suggests, it deals with prime numbers and, in particular, with the prime counting function = = # / It was the only paper written by Riemann on number theory but it is considered, together with the Dirichlet s theorem on the primes in arithmetic progression, the starting point of modern analytic number theory. Riemann s aim was to provide an explicit formula for ; before him, Gauss already tried to find such a formula but he was only able to prove that the function π(x) is well approximated by the logarithmic integral =! "# log #

2 Gauss s estimate was motivated by the observation made by Euler about the divergen of the series ' = = In Euler s terminology, ' = log log, which was a consequen of the Euler s product formula for the harmonic series, So that log 2 = =. 0 And the right hand side is the sum of S plus convergent series. = Riemann was intentioned to find an explicit formula for the prime counting function, not only an estimate like Gauss did. In order to do that, he rtainly based his work on the exllent approximation (3) but, at the same time, he made use of a generalization of the Euler s product formula, introducing the most important function in analytic number theory, the Riemann zetafunction, defined as For = 9+# C. = The zeta-function converges absolutely for >, in which case we can generalize the Euler s product formula to =. 0, σ> The function plays a fundamental role inside Riemann s paper: some of its zeros appear in the explicit formula that connects to, they are the basic ingredient of the error term in Gauss s approximation (3). The zeros we are talking about are the so-called non-trivial zeros of, in contrast with the trivial zeros of ζ(s) which happens for = 2, 4, 6, : the non-trivial zeros, usually indicated with B = C+D, are infinite in number and they have B = C 0,, the region 0 < < is called critical strip and the most important open problem in number theory, the Riemann Hypothesis (RH), states that each non-trivial zero has β = 2, that is they are 2

3 all located along the critical line, =. RH was first conjectured by Riemann in his paper, where he wrote that it is probable that all non-trivial zeros have real part equal to. This paper presents an affirmation of Riemann Hypothesis. PRELIMINARY : In this section we will present definitions and nessary theorem (lemma) for establishing the Riemann Hypothesis.. THE NOTION OF GH-INJECTIVE COMPLEX FUNCTION The notion of injective complex function is well known; in this section we introdu a new vision for function of complex variable. Definition: a function I:K C C is called -injective if and only if verify one of the equivalent assertion: For all N, K, I = IN = N For all N, K, N I IN Remark: from the definition every injective function is -injective. The inverse is false, in fact: the exponential function is non-injective (indeed if = Q then = N+2R ), but it is well injective. 2. SEQUENCES OF ST-INJECTIVE FUNCTIONS. The Huwirtz s (Titchmarsh, 985) theorem proves that the limit of injective sequen of complex function is also injective. In this section we will prove that the limit of -injective sequen of complex function is also -injective. LEMME : Let Ω an open connected subset of C. Suppose that I V I in WΩ. And suppose that I V is not identically zero on Ω for big. Then I is identically zero on Ω. Proof. see (Titchmarsh, 985) 3

4 The main Theorem: Let Ω = I R [ an open interval of R open connected subsets of C. Suppose that I V I in WΩ. And suppose that I V are -injective on Ω for big enough. Then I is -injective on Ω, or constant. Proof. Suppose that there is \ > 0, such that for \ that I V is -injective sur Ω. Suppose that I is not constant on Ω. Let N,N^ Ω such that N^ N and let : By construction _ V _ in WΩ\N^. _ V N = I V N I V N^ et _N=IN IN^ Sin I V are -injective on Ω, then _ V is not zero on Ω\N^ and Ω\N^ is an open connected subset, by the predent lemma _ is not zero on Ω\N^, hen I is -injective on Ω. L HYPOTHESE DE RIEMANN Riemann Theorem: The Riemann zetas Function verify the following property: Proof: = 0, = 9+#,0 < 9 < 9 = 2 Denote by Ω =]0,[ R C open connected subset of C. The proof is divided in three parts: i. The sequen of function I = Converge to the Riemann zeta function. cd 20 R ii. We prove that the following sequen is -injective : I = cd 20 R For that let,n Ω, such that > N, on obtient 4

5 0ghQi cd I < i 2 R and 0ghQi cd I N > +2 R Q i It is clear that i cd i < i cd R R Q i then, I < 0ghQi cd i < 2 R < I N 0ghQi cd 2 R Q i < Thus the sequen of functions I N is well -injective on Ω 0ghQi cd +2 R Q i iii. The sequen of function I N is -injective on Ω. Thus by the fundamental theorem bellow the Riemann Zeta function is -injective on Ω. Sin the exist zeros (Gourdon, octobre 2004) of the Riemann zeta function in the critical line 9 = then from the result (i),(ii) and (iii) bellow and by definition of -injective we find that all non-trivial zero are in the critical line. BIBLIOGRAPHIE Gourdon, X. ( octobre 2004). The 03 first zeros of the Riemann zeta function, and zeros computation at very large height. Titchmarsh, E. C. (985). The Theory of Functions. Oxford University Press. 5