Proof of Bernhard Riemann s Functional Equation using Gamma Function

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1 Journal of Mathematic and Statitic 4 (3): 8-85, 8 ISS Science Publication Proof of Bernhard Riemann Functional Equation uing Gamma Function Mbaïtiga Zacharie Department of Media Information Engineering, Okinawa ational College of Technology, 95 Henoko, ago, 95-9, Okinawa Prefecture, Japan Abtract: Thi tudy how the ue of gamma function to prove the Riemann functional equation. Two approache had been ued to olve thi problem: firt the value of t in the definition of the gamma function had been changed to pi nu if only if igma i greater than zero in the comple plane. Secondly, the Poion ummation formula i ued to how that zeta ha a imple pole at = with reidue, we had found that Riemann zeta function depended intimately on propertie of gamma function, which wa a new gate for olving comple problem related to zeta function. Key word: Gamma function, pecific vertical line ITRODUCTIO Bernhard Riemann paper, Ueber die Anzahl der primzahlen unter einer gegebenen Gröe (On the number of prime le than a given quantity) wa firt publihed in Monatberichte de Berliner AKademie, in ovember 859. Thi tudy, jut i manucript page in length, introduced radically new idea to the tudy of prime number, idea which led, in 896, to independent proof by Hadamard and de la Valleé Pouin of the prime number theorem. Thi theorem, firt conjectured by Gau when he wa a young man, tate that the number of prime le than i aymptotic to /log(). Very roughly peaking, thi mean that the probability that a randomly choen number of magnitude i a prime i /log(). Riemann gave a formula for the number of prime le than in term the integral of /log and the root (zero) of the zeta function, defined by: ζ () = = = n 3 n n= n= He alo formulated a conjecture about the location of thee zero, which fall into two clae: the obviou zero -, -4, -6, -8 and thoe real part lie between and. He checked the firt few zero zeta function by hand and they atify hi hypothei. By now over.5 billion zero have been checked by computer. Very trong eperimental evidence, but in mathematical we require a proof. A proof give certainty, but, jut a important, it help u to undertand why a reult i true. It i in thi direction that we ued the gamma function () to proof Riemann functional equation, where the gamma can be thought of a the natural way to generalize the concept of factorial to non-integer argument uch a: For non-negative integer n, denote n! can be defined by: Correponding Author: Mbaitiga Zacharie, Okinawa ational College of Technology, Department of Media Information Engineering, 95 Henoko, ago, 95-9, Okinawa Prefecture, Japan Tel: Fa: n n! = r n = where, for n = the empty product i taken to be For every non-integer n Γ (n + ) = n! where, Γ i Euler gamma function who come up with a formula for uch a generalization in 79. At around the ame time, Jame Stirling independently arrived at a different formal but wa unable to how that it alway converged. The proce of our proof i compared to other tudie on the open literature on the functional equation. MATERIALS AD METHODS Riemann functional equation: The function ζ () can be continued analytically over the whole comple plane C and atifie the functional equation: ( ) Γ( ) ζ () = Γ( ) ζ( ) ()

2 where, Γ denote the gamma function. In particular, the function ζ () i analytic everywhere, for a ingle pole at = with reidue. Proof: Firt note that the functional Eq. enable propertie of ζ () for σ < to be inferred from propertie of ζ () for σ > a can be oberved from () the tudy of Riemann zeta function depend on propertie of the gamma function: t () e t dt Γ = (3) J. Math. & Stat., 4 (3): 8-85, 8 In Riemann publihed paper, he formulated a conjecture about the location of zero of the zeta function, which fall into two clae: the obviou zero -,-4,-6,-8 and thoe whoe real part lie between and. He added alo in a conjecture that the real part of the non obviou zero i eactly /. That i, they all lie on a pecific vertical line in the comple plane Fig.. If the Riemann conjecture i true, the value of alo lie on that pecific vertical line in the comple plane, and we need to find the pecific vertical line that Riemann referred to in hi paper a our tudy i going to baed on it. From Fig. we found that: Fig. : The geometric repreentation of z and it conjugate z in the comple plane. The ditance along the black line from origin to the point z i the modulu or abolute value of z. The angle θ i the argument of z In the ditribution of prime number, Riemann etended Euler zeta function to the entire comple plane that: ζ () = = = n 3 4 n (5) n= n = vertical ai pecific vertical line = That i, from the origin to poitive ai, thu the value of can be replaced by S/. ow the pecific vertical line in the comple plane i known, we can uppoe that σ >. Writing t = n with dt = n d then ubtituting t and dt in the gamma function Γ Eq. 3, n Γ ( ) = e ( n ) n d Γ( ) n = n ( n ) e d ( ) ( ) n = ( n n )e d n n e d n = e d n = Γ = n ( )n e d definition of the (4) 8 ow by multiplying both the left hand ide and right hand ide of Eq. 4 by: n= Γ = n ( ) n e d n= n= Γ = n ( ) n ( e )d n= n = Γ ζ = n ( ) () ( e )d n = (6) where, the change of order of ummation and integration i jutified by the convergence of: n= n e d Writing n () e d n = ω = (7)

3 J. Math. & Stat., 4 (3): 8-85, 8 it follow for σ > Eq. 6 become: ( ) () Γ ζ = ω()d + y ω(y)dy (8) Due to the location of the non-obviou zero value that lie on the pecific vertical line in the comple plane according to Riemann Fig, we can write that: d = = = y and dy Subtituting the value of y and dy in the econd term of the integral Eq. 8 from to ( ) ω( ) d Uing the integration propertie by invering the integration border from to intead from to. We have: ( ) ω( ) d ( ) = ω( )d = ω ( )d (9) Subtituting Eq.9 into the econd term of the right hand ide of Eq.8 and letting = and =, we have: ( ) () Γ ζ = ω()d + ω ( )d () ow we have to how that for every >, the function: n () e () n = θ = = + ω () n n e = e () n = Hence: ω = θ ( ) ( ) ( ) = θ = + + ω() ω ( ) = + + ω() ubtituting ω ( ) d We obtain: ω ω ( ) into Eq.9 where ( ) d = ( + + ω()) d = + + ω() d (3) = and = (4) It follow on combining Eq. 9 and Eq. 4 that for σ > Eq. 8 become: Γ( ) ζ ( ) = + + ( + ) ω()d (5) The integral on the right hand ide of Eq. 5 converge abolutely for any, and uniformly in any bounded part of the plane, ince ω () = O(e ) when +. Hence the integral repreent an entire function of, and the formula give the analytic continuation of ζ() over the whole plane. ote that the right hand ide of Eq. 4 remain unchanged when, i replaced by -, o that the functional Eq. follow immediately. Finally note that the function: Satifie the functional equation which can be written: θ ( ) = θ () ξ () = ( ) Γ( ) ζ () (6) 83

4 J. Math. & Stat., 4 (3): 8-85, 8 i analytic everywhere, ince Γ () ha non zero, the only poible pole of ζ () i at = with reidue a we have hown in the previou equation. It remain to etablih the functional equation Eq. for every >. Which the tarting point i the Poion ummation formula, that under certain condition on a function f (t) B iν t 'f (n) = f (t)e dt (7) A A n B ν= Where, ' denote that the term in the um A n B correponding to n = A and n = B are f (A) f (B) and repectively. Uing (7) with = -A and : t = B and f (t) e = The function z e i an entire function of the comple variable z and from Cauchy theorem, we have: (ui ν) du u = e du = A e Where: A = e dy = y () The functional Eq. now follow on combining Eq. -3 and the proof of Eq. i completed. RESULTS AD DISCUSSIO (3) We have: n t i t e e ν = e dt (8) n = ν= Letting, we obtain: n t i t e e ν = e dt (9) n = ν= Thi i jutified by the fact that: ( + ) t iν t e e dt and that: ν t = e co(νt)dt, t e co( ν) dt a. now writing t = u and uing (9), n u i u e e ν = e du n ~ ν= (u i ν ) ν = e du ν= ν (u i ν ) = e e du. ν= () () 84 For more than two thouand year, mathematic ha been a part of the human earch for undertanding. Mathematical dicoverie have come both from the attempt to decribe the natural world and from the deire to arrive at a form of inecapable truth from careful reaoning. Thee remain fruitful and important motivation for mathematical thinking, but in the lat century mathematic ha been uccefully applied to many other apect of the human world. Such a, voting trend in politic, the dating of ancient artifact, the analyi of automobile traffic pattern, and long-term trategie for the utainable harvet of deciduou foret, to mention a few. Today, mathematic a a mode of thought and epreion i more valuable than ever before. Due to the importance or involvement of Mathematic in other cientific domain, many paper in the open literature have tried to prove the Riemann functional equation whoe mot of thee paper have encountered difficultie to give a real proof. For eample, in page of [] the author did a great effort to prove the Riemann functional equation, but unfortunately it wa unclear and unfinihed. When arrived at below tep: ( ) () Γ ζ ω()d = (4) The author wrote we take the natural tep of plitting the integration at = then ubtituting / for in the integral from -, where the integral from

5 J. Math. & Stat., 4 (3): 8-85, 8 - ha not been indicated and no reference ha been alo made on the value of d(d = - ) and furthermore no eplanation about how the left hand ide of Eq. 4 ha been obtained. Baed on the location of nonobviou zero on the comple plane the border of Eq. 4 hould be from - intead of -. In our proof, we have focued on the clue given by Riemann in hi conjecture tate that, the zeroe of the Riemann zeta function that are inide the Critical Strip. That i, the vertical trip of the comple plane where the real part of the comple variable i in [; ], are actually located on the Critical line. That i the pecific vertical line of the comple plane with real part equal to½. COCLUSIO In thi tudy, the gamma function i ued to prove the Riemann functional equation, baed on the real part of the non-obviou zero that i eactly /; following by the ue of Poion ummation formula to how that zeta ha a imple pole at = with reidue. From thi reearch we have found that the riemann zeta function depend on propertie of the gamma function when igma i greater than zero in the comple plane and that the non-obviou zero all lie on the pecific vertical line in the comple plane. Thi new founding may help to olve ome comple problem related to zeta function. REFERECES. Kedlaya, K.S., 7. The functional equation for riemann zeta function. J. Anal. Theor., Bochner, S., 958. On riemann functional equation with multiple gamma factor. Ann. Math., 67: Sondow, J., 994. Analytic continuation of riemann zeta function and value at negative integer via euler tranformation of erie. Proc. Am. Math. Soc., : Spanier, J. and K.B. Aldham, 987. The Zeta umber and Related Function: An Atla of Function. t Edn., Taylor and Franci, Hemipher, Wahington, DC., ISB: , pp: Srivatava, H.M.,. Some imple algorithm for the evaluation and repreentation of the zeta function at poitive integer argument. J. Math. Anal. Appl., 46: Caldwell, C. and H. Dubner, 998. Prime in pi. J. Recreat. Math., 9: InPi.pdf 7. Brian Conrey, J., 3. The riemann hypothei. otic. Am. Math. Soc., 5: &ETOC=R&from=earchengine 8. Molhem, H. and R. Pourgoli, 8. A numerical algorithm for olving a one-dimenional invere heat conduction problem. Am. J. Math. Stat., 4: Ayoub, R., 974. Euler and the zeta function. Am. Math. Monthly, 8: Biane, P., J. Pitman and M. Yor,. Probability law related to the jacobi theta and riemann zeta function and brownian ecurion. Bull. Am. Math. Soc., 38: /home.html. Bloch, S., 996. Zeta value and differential operator on the circle. J. Algebra, 8: DOI:.6/jabr

Demonstration of Riemann Hypothesis

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