The real primes and the Riemann hypothesis. Jamel Ghanouchi. Ecole Supérieure des Sciences et Techniques de Tunis.

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The real primes and the Riemann hypothesis Jamel Ghanouchi Ecole Supérieure des Sciences et Techniques de Tunis Jamel.ghanouchi@live.

1 The real primes and the Riemann hypothesis Jamel Ghanouchi Ecole Supérieure des Sciences et Techniques de Tunis bstract In this paper, we present the Riemann problem and define the real primes. It allows to generalie the Riemann hypothesis to the reals. calculus of integral solves the problem. We generalie the proof to the integers. The Riemann hypothesis hal-8586, version 9 - The Riemann conjecture is a conjecture which has been formulated in 859 by Bernard Riemann in the subjet of the Riemann funtion eta or. It is called the eta Riemann function. This function is defined as follows n ( s) ( )... s s s n 3 n The first result is the divergence of the harmonic serie n () ( )... n 3 n 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.

2 In the 73 s he conjectured after numerical calculus the following equality, which is often called the Basel problem. n () ( )... n 3 6 n Euler proved it in 748 and introduced the function. He calculated its value for the positive even numbers. n B k ( ) ( k) ( )... k k k n 3 ( k)! n k Where Bk are the Bernoulli numbers. hal-8586, version 9 - Thereafter, he proved in 744 the Euler idendity where prime numbers are related to the function. n ( s) ( )... n 3 p n s s s s primes Consequently he deduced the divergence of the serie of the inverse of primes. With Bernard Riemann, s can be complex number. Riemann proved the following formula s ( s) s s ( ) ( s) ( ) ( s) Where s t s t e dt () This formula demonstrates that this equation does not change if we replace s by -s. Thus it is symmetric s

3 Riemann demonstrates that the only eros in the Rs ( ) are the trivial eros negative even numbers and that there is no ero in the Rs ( ). The other eros are the non trivial eros. They are in the critical one Rs ( ). Riemann conjectured theu are all in the critical line Rs (). This conjecture is called the Riemann hypothesis. They calculated numerically one billion eros of the Riemann they are all located in the critical line. Resolution of the Riemann hypothesis for the reals hal-8586, version 9 - Definition real number is compound if it can be written as n j p j where j j p are primes and n j are rationals. This decomposition in prime factors is unique. prime real number can be written only as p=p.. Thus we define other real prime numbers like, e, ln(). Thus when p is prime and we have q p p q is compound. lso q p p q is prime p ( p )( p ) ( p )...( p ) compound for p prime, for example. i i i The approach of the Riemann hypothesis The Rieman hypothesis states that the non trivial eros of the Riemann eta function 3

4 ( ) t lie on the critical line iy t. For t integer, Euler has proved that ( ) ( ) t p p t, it is primes primes N the Euler identity. For t real, it is still true and it becomes dt t ( ) p t But there are the tivial eros : we have primes Q k ( k), k N and t t but if is the limit in the infinity, k lim( ), t t k N and hal-8586, version 9 - iy iy iy iy iy lim( t ) lim( t ) lim( t ) (lim( t ) lim( t )) a i i i i i ( iy) iy ( iy) iy iy lim( t at t) lim( t at ) lim( t ) a a a i i i it means that Let now iy ( iy) t iy iy x x x iy t t t t ( x iy) x x iy x iy x iy We have proved that the non trivial eros 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 eros of the Riemann function and the distribution of those eros are related to real primes! The generaliation to the integers 4

5 We have dt t t t t Q t ' Q \ N t '. B ( ). B t p t t primes N t Let now ( ) ( x iy) primes N p We have : if -<B< then the Riemann function for the reals is equal to ero, then x=/. Then, let for B<- and for B>. If we suppose that exists hal-8586, version 9 - alpha greater than and beta finite ( ). B ( ) p p primes N ( xiy) ( xiy) primes N ( ) ( xiy) p B ( )( B ) ( B ) Thus primes N B dt B B xiy t x iy B ( B ) B xiy t dt B B xiy t x iy B ( B ) xiy t For B in the infinity B dt B xiy t x iy B xiy t 5

6 B dt B xiy t x iy B xiy t It means that the Riemann function for the reals would never be equal to ero, it is impossible. Hence, dt t ( ). B ( ) x t x iy p p xiy xiy ( xiy) ( xiy) primes N primes N Thus the non trivial eros of the Riemann funtion eta lie in the critical line like for the reals! It is the proof of the Riemann hypothesis! hal-8586, version 9 - nother proof that = : Let ( ) the Riemann function for the reals and () the Riemann function fot the integers, if B ( ( )) ( ) ( ) ( ) ( ) n n n n ( ( ) ) ( ( )) ( ) ( ), n n ( ( )) ( ) ( ) ( n B ( ) B ( ) ( ) ( ) ) ( ) n n n n ( ( ) ) ( ( )) ( ) ( ), n B ( ) B ( ) ( ) ( ) n n ( ( )) ( ) ( ) ( ) ( ) ( ( ) ) n 4 4 n n3 n3 ( ( )) ( ) ( ), n B ( ) B ( ) ( ) ( ) n n ( ( )) ( ) ( ) ( ) ( ) n 4 4 n n3 n3 ( ( ) ) ( ( )) ( ) ( n n ), n B ( ) B ( ) ( ) ( ) 6

7 If hal-8586, version 9 - B ( ( )) ( ) ( ) ( ) ( ) n n n n ( ( ) ) ( ( )) ( ) ( ), n n B ( ) B ( ) ( ) ( ) n ( ( )) ( ) ( ) ( ) ( ) n n n n ( ( ) ) ( ( )) ( ) ( ), n B ( ) B ( ) ( ) ( ) n n ( ( )) ( ) ( ) ( ) ( ) ( ( ) ) n 4 4 n n3 n3 ( ( )) ( ) ( ), n B ( ) B ( ) ( ) ( ) n n ( ( )) ( ) ( ) ( ) ( ) ( ( ) ) ( ( )) ( ) n 4 4 n n3 n3 n n ( ), n B ( ) B ( ) ( ) ( ) If B ( ) ( ) 3 3 ( ( ) ( ) ) ( ) ( ) ( ) ( ) n n ( ) ( ) n n ( n) n ( n) n ( ( ) ( )) ( ) ( ) ( ) ( ), n ( ), Then = nd dt t ( ). B ( ) x t x iy p p xiy xiy ( xiy) ( xiy) primes N primes N 7

8 Thus the non trivial eros of the Riemann funtion eta lie in the critical line like for the reals! It is the proof of the Riemann hypothesis! Conclusion We have generalied the concept of prime to the reals. It allowed to prove the conjecture to the reals. Then, we have proved the Riemann hypothesis. The Bibliography []R. J. Backlund, «Sur les éros de la fonction ζ(s) de Riemann», CRS, vol. 58, 94, p []X. Gourdon, «The 3 first eros of the Riemann eta function, and eros computation at very large height» hal-8586, version 9 - [3]J.P.Gram, «Note sur les éros de la fonction ζ(s) de Riemann», cta Mathematica, vol. 7, 93, p [4]J.I.hutchinson «On the Roots of the Riemann Zeta-Function», Trans. MS, vol. 7, n o, 95, p [5]. M. Odlyko, The -th ero of the Riemann eta function and 75 million of its neighbors, 99. [6]J.Barkley Rosser, J. M. Yohe et Lowell Schoenfeld, «Rigorous computation and the eros of the Riemann eta-function.», Information Processing 68 (Proc. IFIP Congress, Edinburgh, 968), Vol. : Mathematics, Software, msterdam, North- Holland, 969, p [7]E.C.Titchmarsh, «The Zeros of the Riemann Zeta-Function», Proceedings of the Royal Society, Series, Mathematical and Physical Sciences, vol. 5, n o 873, 935, p [8]E. C. Titchmarsh, «The Zeros of the Riemann Zeta-Function», Proceedings of the Royal Society, Series, Mathematical and Physical Sciences, The Royal Society, vol. 57, n o 89, 936, p [9].M.Turing, «Some calculations of the Riemann eta-function», Proceedings of the LMS, Third Series, vol. 3, 953, p

9 []J. van de Lune, H.te Riele et D. T. Winter, «On the eros of the Riemann eta function in the critical strip. IV», Mathematics of Computation, vol. 46, n o 74, 986, p hal-8586, version 9-9

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