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The real primes and the Riemann hypothesis Jamel Ghanouchi Ecole Supérieure de Sciences et Techniques de Tunis [email protected] Abstract In this study, the Riemann problem is presented with highlights on history of the zeta function. Thereafter, the real primes, which constitute a novelty, are defined. It allows to generalize the Riemann hypothesis to the reals. A calculus of integral solves the problem. The proof is generalized to the integers by an elementary development. Keywords Primes ; Reals ; Riemann hypothesis Introduction The Riemann conjecture is a conjecture which has been formulated in 1859 by Bernard Riemann in the subject of the Riemann function zeta or . It is called the zeta Riemann function. This conjecture is one the seven problems of the millennium. The first result is the divergence of the harmonic serie. The second consist of the results of Leonard Euler. Finally, Bernard Riemann has presented the conjecture. The problem is reconsidered here as the real primes are defined for the first time. This definition will allow to generalize the hypothesis to the reals for which the solution is easier. Thereafter, the conjecture will be solved after an elementary development. Materials or methods 1 The main novelty of the study consists on the definition of the real primes. In fact, it is a generalization of the concept of prime to the reals. The discovery of the real primes will make easier the resolution of the hypothesis since a calculus of integral will be sufficient. The Riemann hypothesis The zeta function of Riemann is defined as follows n ( s) ( n 1 1 1 1 ) 1 s s ... s n 2 3 The first result is the divergence of the harmonic serie n 1 n 1 2 1 3 (1) ( ) 1 ... n 1 It has been proved in the middle age by Nicole Oresme. In the XVIII century, Leonard Euler has discovered the main proprieties of the function. In the 1730’s he conjectured after numerical calculus the following equality, which is often called the Basel problem. 1 1 1 2 ) 1 ... n2 22 32 6 n (2) ( n 1 Euler proved it in 1748 and introduced the function. He calculated its value for the positive even numbers. n (2k ) ( n 1 | B2k | (2 )2 k 1 1 1 ) 1 ... n2 k 22 k 32 k 2(2k )! Where B2k are the Bernoulli numbers. Thereafter, he proved in 1744 the Euler idendity where prime numbers are related to the function. n ( s) ( n 1 1 1 1 ) 1 s s ... s n 2 3 1 1 p s primes Consequently he deduced the divergence of the serie of the inverse of primes. 2 With Bernard Riemann, s can be complex number. Riemann proved the following formula s (1 s ) 2 s 2 2 ( ) ( s) 1 s ( ) (1 s) 2 Where (s) t s 1et dt 0 This formula demonstrates that this equation does not change if we replace s by 1-s. Thus it is symmetric | s 1 2 Riemann demonstrates that the only zeros in the R( s) 0 are the trivial zeros negative even numbers and that there is no zero in the R ( s ) 1 . The other zeros are the non trivial zeros. They are in the critical zone 0 R( s) 1 1 2 . Riemann conjectured they are all in the critical line R( s) . This conjecture is called the Riemann hypothesis. They calculated numerically one billion zeros of the Riemann they are all located in the critical line. Resolution of the Riemann hypothesis for the reals Definition A real number is compound if it can be written as p nj j where p j are primes j and n j are rationals. This decomposition in prime factors is unique. A prime real number or R-prime can be written only as p=p.1 . Thereafter, there are other real prime numbers like , e, ln(2) . Of course, it is a convention, because, if 2 is prime will be no more prime. It is equivalent in what will follow. Thus 2i q 1 q p p is compound. Also q 1 q p 1 p 1 is prime when p is prime and p 1 ( p 1)(2 p 1)1 (2 p 1)1...( p 1)1 i i 1 example. 3 is compound for p prime, for The approach of the Riemann hypothesis The Rieman hypothesis states that the non trivial zeros of the Riemann zeta function ( z) t 1 1 1 lie on the critical line iy . z 2 t 1 z t 1 t For t integer, Euler has proved that ( z ) the Euler identity. ( R primes Q For t real, it is 1 1 p z primes still 1 ( p primes N z ) , it is true and it becomes 1 dt 1 t1 z But But let 1 ( z ) the Riemann function for the ) z z 1 p t 1 z 1 reals and ( z ) the Riemann function for the integers. We will see below that ( z) 0 1 ( z) 0 Thus (2k ) 0 1 (2k ) lim( t t t eu ; u k N : t 2 ki lim(t 2 ki t t1 2 k 1 ) 0 lim(t1 2 k ) 1 t 1 2k e 2 kiu (e 2 ki ) u 1u ) 1 lim(t 3 6 ki t ) 1 lim(t 1 ki 2 t )e i 2 m 6 e i 2 m ' 6 lim(t ki ) t If m m ' lim(t ki ) e i 2( m m ') 6 t ; k N lim(t i 2 k ) 1 e i 2( m m ') 3 t e i 2 e i 2 m ' m m ' 3 4m ' m ' 1; m 4 lim(t ik ) 1; k N t Let k 2k ' 1 1 it is impossible m m ' . Another proof : k 2k ' lim(t ik ) 1 e i( m m ') 3 t e i 2( m m ') 3 e i 2 e i 6 m ' 3 m m' 1 2 But lim(t ) 1 lim(t ) lim(t ) 1 lim(t ) 1 and we know there exists 6 t 5 t t t 1 1 iyI 1 1 yI ; ( iyI ) 0 1 ( iyI ) 0 lim(t 2 ) 1 lim(t 2 ) it means t t 2 2 4 1 iy lim(t 2 t ) 1; y N Let us suppose there exists m different of 2 verifying 1 1 lim(t m ) lim(t 2 ) 1 lim(t t t lim(t m 2 2m t t ) 1 lim(t t 4 1m 2 4 m2 2m 1 1 1m 2 4 1 2m 1 1 ) 1 lim(t 2 m ) 1m 2 t 1 m2 ) 1 1 1 1 14 lim(t 4 m ) lim(t 8 ) t t Thus lim(t 1 2 m ( m 2) t 1 ) 1 m2 4 lim(t 1 2 m ( m 2) t ) 2 lim(t m ( m2 4) t ) 1 And 1 m lim(t ) 1 lim(t t 1 1 m ( m2 4) 2 t ) 1 lim(t 1 M1 t ) Hence 2M 1 m(m 2 4) 10 22 M 2 2M 1 ( M 12 4) 210 ... 2n M n 2M 1 ( M 12 4)...( M n 12 4) m(m 2 4)( M 12 4)...( M n 12 4) m.Pn .2n lim( Pn ) n 1 m And lim(t ) 1 ei 2u lim(t t t 1 Mn i 2n1 u ) 1 ei 2v e ( m 2 4)...( M n12 4) , n N with 2n u m(m2 4)...( M n 12 4)v 2n Pn v, n N It is impossible ! The initial hypothesis is false : m does not exist ! Hence m=2, is the only value for which lim(t t 1 iy 2 ) 1; y N . Hypothesis H : 5 n 1 H : n, m | lim(t m ) 1 lim(t 2 ) t t 2n m, m 2,1 n m lim(t 2nm 2m t ) 1 lim(t 2nm 2m t ) lim(t 1 2 m (2 n m ) t t m lim(t n (4 n 2 m 2 ) t ) lim(t 1 ) 1 lim(t ) lim(t t 1 2 m (2 n m ) 4 n2 m2 t 1 ) 14 n 2 m2 n m ) lim(t ) 1 t n m 4n 2 m 2 2 lim(t ) 1 t 2n m, m 2,1 n m Let the hypothesis H n H : n, m | lim(t m ) a lim(t iy ) t lim(t n iy m t lim(t ) 1 lim(t t 2nm 2m t lim(t t 1 ik 2 ) lim(t i ( k y ) t 2nm 2n t ) lim(t t ) n m ) a lim(t ) t m 1 2n ) 1 lim(t t m n m 2n ) n m lim(t ) 1 lim(t ) lim(t ) 1 a a 1 t 1 t2 iy t t 2n 1 m 1; y R 1 1 1 iy iy 1 1 1 it means that ( iy ) t 2 lim( (t 2 1)) 0 t 1 1 2 1 iy 1 iy 2 1 2 Let 1 x 1 1 iy x 1 1 t 2 1 1 ( x iy) 0 t1 x iy lim( (t 2 2 1)) lim( )0 x t 1 x iy t 1 x iy 2 1 x iy 1 Thus the non trivial zeros of the Riemann function for the reals lie in the critical line ! So the hypothesis is proved for the real numbers. The Riemann hypothesis is important because it gives information about the zeros of the 6 Riemann function and the distribution of those zeros are related to real primes ! The generalization to the integers As dt 1 1 1 1 1 z t 1 t z 1 z 1 t 1 t z Q t ' z Q \ N t ' z 1 z .B t 1 t ( 1 )0 primes N 1 1 ). B A z z p t 1 t Let now ( p primes N ( x iy ) If B is finite, the zeta function for the reals will be equal to zero. Else Let 1 ( z ) the Riemann function for the reals and ( z ) the Riemann function for the integers, if B and if A is finite and A 1 ( A ( z )) 2 A2 A2 ( z ) 2 2 A ( z ) A2 2 A ( z ) A ( z ) 0 ( An ( z ) A) 2 A2 A2 ( An ( z )) 2 2 An 1 ( z ) An 1 ( z ) 0, n | B A An 0 1 ( z ) B ( z ) An ( z ) 0 1 ( z ) 0 1 A And if 1 A 0 ( A1 ( z )) 2 A2 A2 ( z ) 2 2 A1 ( z ) A2 2 A1 ( z ) A1 ( z ) 0 ( A n ( z ) A1 ) 2 A2 A2 ( A n ( z )) 2 2 A n 1 ( z ) A n 1 ( z ) 0, n | B A A n 0 1 ( z ) B ( z ) A n ( z ) 0 1 ( z ) 0 Now if 7 0 A 1 ( A1 ( z )) 2 A2 A2 ( z ) 2 2 A1 ( z ) A2 2 A1 ( z ) A1 ( z ) 0 ( A2 n 1 ( z ) A2 ) 2 A4 A4 ( A2 n1 ( z )) 2 2 A2 n3 ( z ) A2 n3 ( z ) 0, n | B A A2 n 1 0 1 ( z ) B ( z ) A2 n 1 ( z ) 0 1 ( z ) 0 And if A 1 ( A ( z )) 2 A2 A2 ( z ) 2 2 A ( z ) A2 2 A ( z ) A ( z ) 0 ( A2 n 1 ( z ) A2 ) 2 A4 A4 ( A2 n 1 ( z )) 2 2 A2 n 3 ( z ) A2 n 3 ( z ) 0, n | B A A2 n 1 0 1 ( z ) B ( z ) A2 n 1 ( z ) 0 1 ( z ) 0 A 1 If now B and if A is finite and A 1 ( A ( z )) 2 A2 A2 ( z ) 2 2 A ( z ) A2 2 A ( z ) A ( z ) 0 ( An ( z ) A) 2 A2 A2 ( An ( z )) 2 2 An 1 ( z ) An 1 ( z ) 0, n | B A An 0 1 ( z ) B ( z ) An ( z ) 0 1 ( z ) 0 1 A And if 1 A 0 ( A1 ( z )) 2 A2 A2 ( z ) 2 2 A1 ( z ) A2 2 A1 ( z ) A1 ( z ) 0 ( A n ( z ) A1 ) 2 A2 A2 ( A n ( z )) 2 2 A n 1 ( z ) A n 1 ( z ) 0, n | B A A n 0 1 ( z ) B ( z ) A n ( z ) 0 1 ( z ) 0 And if 0 A 1 ( A1 ( z )) 2 A2 A2 ( z ) 2 2 A1 ( z ) A2 2 A1 ( z ) A1 ( z ) 0 ( A2 n 1 ( z ) A2 ) 2 A4 A4 ( A2 n1 ( z )) 2 2 A2 n3 ( z ) A2 n3 ( z ) 0, n | B A A2 n 1 0 1 ( z ) B ( z ) A2 n 1 ( z ) 0 1 ( z ) 0 And if A 1 ( A ( z )) 2 A2 A2 ( z ) 2 2 A ( z ) A2 2 A ( z ) A ( z ) 0 ( A2 n 1 ( z ) A2 ) 2 A4 A4 ( A2 n 1 ( z )) 2 2 A2 n 3 ( z ) A2 n 3 ( z ) 0, n | B A A2 n 1 0 1 ( z ) B ( z ) A2 n 1 ( z ) 0 1 ( z ) 0 A 1 8 And if A : B A 1 hence A m ( z ) m 1 An ( z ) n 1 0 Am n 1 B A 1 A( m n ) A( m n ) ( z ) m n 1 B ( z ) m n 1 A ( z ) m n ( 1)( m n ) A (m n) 1 mn2 1 A1( m n ) A B 1 B 2 ( z ) m n 1 A1 1 1 1 1 B 2 Am n BA 2 m n 1 m n A B A B A B mn B 1 ( A1 B) 1 A1 A2 B A 1 1 It is impossible ! A , A Thus A=0 And t 1 dt x iy 1 t1 x iy 1 1 x iy ( p primes N 1 ( x iy ) ).B ( p primes N 1 ( x iy ) ) A0 x 1 2 Thus the non trivial zeros of the Riemann funtion zeta lie in the critical line like for the reals ! It is the proof of the Riemann hypothesis ! Results The concept of prime has been generalized to the reals. It allowed to reconsider the Riemann hypothesis under a new point of view. Then, it has been proved for the reals after a calculus of integral and it has been generalized to the integers after an elementary calculs. Finally, the conjecture has been proved. Discussion As it has been said, the Riemann hypothesis is one of the most important problems of number theory. It is one of the seven problems of the millennium. Until now, the approaches never reconsidered the concept of prime as it has been done in this study. 9 In fact, this new approach will generalize the concept of prime to the reals. It constitutes a novelty. Therefore, the resolution will depend on a simple integral and an elementary calculus. Conclusion The generalization of the concept of prime is the main novelty of this study. It allows to generalize the hypothesis to the reals and to solve the conjecture. The Bibliography [1]R. J. Backlund, « Sur les zéros de la fonction ζ(s) de Riemann », CRAS, vol. 158, 1914, p. 1979–1981. [2]X. Gourdon, « The 1013 first zeros of the Riemann zeta function, and zeros computation at very large height » [3]J.P.Gram, « Note sur les zéros de la fonction ζ(s) de Riemann », Acta Mathematica, vol. 27, 1903, p. 289–304. [4]J.I.hutchinson « On the Roots of the Riemann Zeta-Function », Trans. AMS, vol. 27, no 1, 1925, p. 49–60. [5]A. M. Odlyzko, The 1020-th zero of the Riemann zeta function and 175 million of its neighbors, 1992. [6]J.Barkley Rosser, J. M. Yohe et Lowell Schoenfeld, « Rigorous computation and the zeros of the Riemann zeta-function.», Information Processing 68 (Proc. IFIP Congress, Edinburgh, 1968), Vol. 1: Mathematics, Software, Amsterdam, NorthHolland, 1969, p. 70–76. [7]http://fr.wikipedia.org/wiki/Edward_Charles_TitchmarshE.C.Titchmarsh, « The Zeros of the Riemann Zeta-Function », Proceedings of the Royal Society, Series A, Mathematical and Physical Sciences, vol. 151, no 873, 1935, p. 234–255. [8]E. C. Titchmarsh, « The Zeros of the Riemann Zeta-Function », Proceedings of the Royal Society, Series A, Mathematical and Physical Sciences, The Royal Society, vol. 157, no 891, 1936, p. 261–263. [9]A.M.Turing, « Some calculations of the Riemann zeta-function », Proceedings of the LMS, Third Series, vol. 3, 1953, p. 99–117. 10 [10]J. van de Lune, H.te Riele et D. T. Winter, « On the zeros of the Riemann zeta function in the critical strip. IV », Mathematics of Computation, vol. 46, no 174, 1986, p. 667–681. 11