Generalized Riemann Hypothesis

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1 Generalized Riemann Hypothesis Léo Agélas To cite this version: Léo Agélas. Generalized Riemann Hypothesis. 9 pages.. <hal-74768v> HAL Id: hal Submitted on 7 Nov HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 Generalized Riemann Hypothesis Léo Agélas November 7, Abstract Generalized) Riemann Hypothesis that all non-trivial zeros of the Dirichlet L-function) zeta function have real part one-half) is arguably the most important unsolved problem in contemporary mathematics due to its deep relation to the fundamental building blocks of the integers, the primes. The proof of the Riemann hypothesis will immediately verify a slew of dependent theorems [BRW], [SA]). In this paper, we give a proof of Generalized Riemann Hypothesis which implies the proof of Riemann Hypothesis and Goldbach s weak conjecture also known as the odd Goldbach conjecture) one of the oldest and bestknown unsolved problems in number theory and in all of mathematics. Keywords Generalized) Riemann hypothesis;complex function; analytic function. Introduction Riemann Hypothesis has defied proof so far, and very complicated and advanced abstract mathematics is often brought to bear on it. Does it need abstract mathematics, or just a flash of elementary inspiration. The Riemann zeta function, defined [AS] p. 87) by the series, ζs) = n=, s C, ) ns is analytic in Rs) > see [BRW]). The basic connection between complex analysis and prime numbers is the following beautiful identity of Euler see [STSH],[BRW],[CO],[SA]): ζs) = ) p s, Rs) >, ) p P where P =, 3, 5, 7,, 3,...} is the set of prime positive integers p. Equation ) is used to shown that [BRW]), ζs) in Rs) >. 3) Riemann proved see [RIE], [ED]), that ζs) has an analytic continuation to the whole complex plane except for a simple pole at s =. Moreover, he showed that ζs) satisfies the functional equatioee [TIT],[ED],[BRW]), ζs) = s π s sin πs )Γ s)ζ s), 4) where Γs) is the complex gamma function. From 4), it can be deduced that, Department of Mathematics, IFPEN, & 4 avenue Bois Préau, 985 Rueil-Malmaison Cedex, France leo.agelas@ifpen.fr)

3 . ζs) is nonzero in Rs) <, except for the real zeros m} m N,. m} m N are the only real zeros of ζs), 3. ζs) possesses infinitely many non real zeros in the strip Rs), the so-called) critical strip for ζs) In 859, B. Riemann formulated the following conjecture, Conjecture. Riemann Hypothesis). All non real zeros of ζs) lies exactly on Rs) =. It was later showee [TIT] p. 45), independently in 896 by Hadamard and de la Vallée-Poussin, that ζs) has no zeros on Rs) =, which provided the first proof of the Prime Number Theorem : πx) x log x x + ), 5) where πx) = number of primes p for which p x} where x > ). From 4), it also follows that ζs) has no zeros on Rs) =, whence, ζs) possesses infinitely many non real zeros in < Rs) <. 6) It is interesting to mention that, ζs) has infinitely many non real zero in Rs) = Hardy [HA]), ζs) has at least 3 of its zeros on Rs) = Levinson [LE]). It also follows from 4) that if ζs) = where s is non real then, s, s, s} are also zeros of ζ. Thus, it suffices to search for the non real zeros of ζs) in the upper half-plane of the critical strip : S = s C : < Rs) < and Im s > }. 7) The Riemann zeta function ζs) and the Riemann Hypothesis have been the object of a lot of generalizations and there is a growing literature in this regard comparable with that of the classical zeta function itself. The most direct generalization, which is also what we will mainly deal with, concerns the Dirichlet L-functions with the corresponding Generalized Riemann Hypothesis. Dirichlet defined his L-functions in 837 as follows. A function χ : Z C is called a Dirichlet character modulo q if it satisfies the following criteria: i) χn) if n, q) = ; ii) χn) = if n, q) > ; iii) χ is periodic with period q : that is χn + q) = χn) for all n; iv) χ is completely) multiplicative : that is χmn) = χm)χn) for all integers m and n. The principal character or trivial character) is the one such that χ n) = whenever n, q) =. Then, one can define the Dirichlet series for Rs) >, Ls, χ) = n= χn) ; 8) this series is also convergent for Rs) > except in the case of a principal character see e.g [WI]) and extend it meromorphically to the complex plane. If χ : Z C is a principal character, then Ls, χ) has a simple pole at s = and is analytic everywhere, otherwise Ls, χ) is analytic everywhere see Theorem

4 .5 in [AP]). As in the case of the Riemann zeta function, by multiplicativity, there is an Euler product decomposition over the primes, for Rs) > see [DA], [EE], [SE]), Ls, χ) = χp) ) p s. 9) p P For any Dirichlet character χ mod q there is a smallest divisor q q such that χ agrees with a Dirichlet character χ mod q on integers coprime with q. The resulting χ is called primitive and has many distinguished properties. First of all, χ being induced from χ means analytically that Ls, χ) = Ls, χ ) p q χ p)p s ), whence Ls, χ) and Ls, χ ) have the same zeros in the critical strip Rs). Zeros outside this strip are well understood, indeed Ls, χ) if Rs) > and for a primitive character χ, the only zeros of Ls, χ) for Rs) < the so-called trivial zeros) are as follows s = ε k, ε, } such that χ ) = ) ε and k positive integer see e.g [MV] corollary.8, see also [COH], Corollary..5 and Definition..6), as well as s = in case χ is a non principal or non-trivial) even character see Theorem. in [AP]). Now let us assume that χ is primitive i.e. χ = χ ), then we have the following beautiful functional equation, discovered by Riemann in 86 for the case q = Riemann zeta function) and worked out for general q by Hurwitz in 88 see e.g [MV], Corollary.8): q s/ Γ R s + κ)ls, χ) = εχ)q s)/ Γ R s + κ)l s, χ). ) Here Γ R s) := π s/ Γs/), κ, } such that χ ) = ) κ, and εχ) is an explicitly computable complex number of modulus. It follows that there are infinitely many zeros with real part at least / see [BH]); in fact it seems that all zeros in the critical strip have real part equal to /. Similar to the Riemann zeta function, there is a Riemann Hypothesis, Conjecture. Generalized Riemann Hypothesis) The Generalized Riemann Hypothesis conjectures that the Dirichlet L-functions have all their non-trivial zeros on the critical line Rs) =, Or, in other words, that Ls, χ), for a primitive character χ modulo q, has no zeros with real part different from in the critical strip Rs), since we can exclude non-trivial zeros outside. Proof of Generalized Riemann Hypothesis Let us record some immediate consequences from definition of Dirichlet character modulo q. For any integer n we have χn) = χn ) = χn)χ) by iv), and since χn) for some n by i), we conclude that χ) =. Next, if n, q) = then, using iv, iii) and Euler s theorem that n ϕq) mod q) with ϕ the Euler s totient functioee e.g Theorem 5.7 in [AP]), χn) ϕq) = χn ϕq) ) = χ) =, so that χn) is a ϕq)-th root of unity. Therefore, we get, χn) = if n, q) =, χn) = if n, q) >. Let us now introduce the Mobius function µ defined for all positive integers by if n =, µn) = ) r if n is the product of r distinct primes, if n is divisible by the square of a prime. ) ) 3

5 The Mobius function has generating functions n= µn) = ζs), for Rs) > see [AP] p. 9, see also [NA] p.3, [DER] pp 45-49, [KV], [DA]). Similarly, we have for Rs) > see [AP], p. 9), We begin by the following Lemma. n= = Lemma. For all integers N N and for all integer n N, N, αe πin m)α dα. m=n Ls, χ). 3) Proof. Let N N, N N such that N N and n N, N. For all p Z, after an integration by parts, we observe, αe πipα dα = πip. 4) Then, taking p = n m in Equation 4) with m n, m N, N and summing it over m n, m N, N, we obtain, m=n,m n αe πin m)α dα = πi m=n,m n n m. Then, we deduce, Therefore, we get, m=n αe πin m)α dα = i m=n π m=n,m n αe πin m)α dα,. n m which concludes the proof. Now, we turn to the proof of Conjecture.. Theorem. [Generalized Riemann Hypothesis] All non-trivial zeros of Ls, χ) lies exactly on Rs) =. Proof. For any primitive character χ modulo q, all non-trivial zeros of Ls, χ) lies in the critical strip s C : Rs) }. We notice if ρ is a non-trivial zero of Ls, χ), then by the functional equation ), ρ is a zero of Ls, χ). Similarly, if β is a non-trivial zero of Ls, χ), then β is a zero of Ls, χ). Then, it suffices to show that for all primitive character χ modulo q, there is no non-trivial zeros of Ls, χ) 4

6 in the critical strip s C : } < Rs). Let us show that the functio n= is well-defined and analytic in s C : Rs) > }. Since each member of the sequence of functions u )} n N defined by u ) =, is analytic in s C : Rs) > }, then thanks to the result obtained iubsection.8 of [TIT], it suffices to show that the series is uniformly convergent in any open bounded subset O of s C : Rs) > }. n= Thanks to subsection.3 and subsection. in [TIT], we infer it suffices to show that for any open bounded subset O of s C : Rs) > }, for all ε >, there exists an integer M > such that for all integers M N < N, for all s O, Then, let us show 5). For this, let ε > and O an open bounded subset of ε. 5) s C : Rs) > }, then there exists θ > such that for all s O, Rs) + θ. We denote by O the value O = sup s, s O}. Let N N, N N and N < N. We introduce c n } n N+,N the sequence of complex numbers defined for all integer n N +, N by, c n = m=n + Thanks to Lemma., for all n N +, N, we have, αe πin m)α dα. 6) c n. 7) Then, we introduce a n } n,n the sequence of complex numbers defined for all n N +, N by, and for all n, N, Then, using 8) and 6), we get for all s O, a n = c n, 8) a n =. 9) = = = α c n a n N m=n + m=n + c n e πimα ) αe πin m)α dα N ) a n e πinα dα. 5

7 Then, we deduce for all s O, α m=n + e πimα a n e πinα dα. ) Thanks to Lemma in [CG], for any < α, we get the following estimate on the exponential sum, e πimα + πα, which implies for all < α, m=n + α m=n + Using ), from ), we deduce for all s O, + ) π e πimα + π. ) a n e πinα dα. ) Let us estimate, the integrand at the right hand side of Inequality ). Using Abel s summation formula integration by parts for a Riemann-Stieltjes integral, see Theorem 4. in [AP]) and thanks to 9), we notice that for all α, where for all y R, y [, N ], Eα, y) = n y a n e πinα. Then, from 3), we obtain, a n e πinα = Eα, N N ) Eα, x) N s + s N + x s+ dx, 3) a n e πinα Eα, N ) N Rs) N + s N + Eα, x) dx. 4) xrs)+ Therefore, thanks to 4), from ), after inverting the integrals, we deduce for all s O, + π ) Eα, N ) dα N Rs) N ) + s Eα, x) dα dx N + x Rs)+. 5) Thanks to Parseval identity and 9), we get for all x [N +, N ], Eα, x) dα a n n x 6) = a n N + n x Thanks to 7), from 8), we deduce for all n N +, N, a n µn), 7) 6

8 where we have used the fact that χn), thanks to ). Therefore, using 6) and 7), we obtain for all x [N +, N ], Eα, x) dα 4 µn) N + n x 4 8) µn). n x Since, we have for all x [N +, N ] see [MV] Theorem., see also [KG]), µn) = 6 π x + O x). 9) n x Then, using 9), from 8), we deduce that there exists a constant C > such that for all x [N +, N ], Eα, x) dα C x. 3) Thanks to Cauchy-Schwarz inequality and 3), we get for all x [N +, N ], Using 3), from 5), we infer for all s O, C + ) π = C + ) π C + π Eα, x) dα C x. 3) N Rs) N Rs) ) N Rs) N + s N + x Rs)+ dx ) s + Rs) N + ) Rs) ) s + Rs) ). N + ) Rs) Since N > N and Rs) + θ, then we deduce for all s O, C + ) ) O π N θ + θn + ) θ. N Rs) O Since N θ as N and θn + ) θ as N, then we deduce that there exists a integer M > depending only on ε, θ and O such that for all N > N M, we have, C + ) ) O + π θn + ) θ ε. Then, we deduce for all N > N M and for all s O, ε, which gives us 5). Therefore, we infer that the functio n= N θ 7 )) is well-defined and analytic i C : Rs) > }.

9 Thanks to 3), by means of analytic continuatioee subsections in [TIT]), we deduce that the complex functio is analytic i C : Rs) > }. Ls, χ) Hence, we infer that for all s C such that Rs) >, Ls, χ), which concludes the proof. 3 Conclusion In this article, we have proved that the Generalized Riemann Hypothesis is true which implies Riemann Hypothesis is true. Moreover, thanks to the results obtained in [DERZ], we obtain the proof of Goldbach s weak conjecture which states that Every odd number greater than 5 can be expressed as the sum of three odd primes. References [AP] T. M. Apostol : Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag, 976. [AS] M. Abramowitz and I.A. Stegun : Handbook of Mathematical Functions, Dover, 964. [BEL] G. Bhowmik, D. Essouabri and B. Lichtin : Meromorphic continuation of multivariable Euler products, Forum Math. p. -9, DOI.55, 6. [BH] E. Bombieri and D. A. Hejhal : On the distribution of zeros of linear combinations of Euler products. Duke Math. J., 83):8-86, 995. [BRW] P. Borwien, S. Choi, B. Rooney, and A. Weirathmueller : The Riemann Hypothesis: A Resource for the Afficionado and Virtuoso Alike, Springer, 8. [CG] Y. F. Cheng and S. W. Graham : Explicit estimates for the Riemann zeta Function, Rocky Mountain Journal of Mathematics, Volume 34, No 4, 4. [CO] J. B. Conrey, The Riemann Hypothesis, Notices of the American Mathematical Society, Vol. 5, pp , 3. [COH] H. Cohen : Number theory. Vol. II. Analytic and modern tools, volume 4 of Graduate Texts in Mathematics. Springer, New York, 7. [DA] H. Davenport : Multiplicative Number Theory, nd edition, revised by H. Montgomery, Graduate Texts in Mathematics, Vol. 74, Springer-Verlag, New York, 98. [MV] H. L. Montgomery and R. C. Vaughan : Multiplicative Number Theory I. Classical Theory, Cambridge University Press 6. [HA] G. H. Hardy : Sur les zéros de la fonction ζs) de Riemann, C. R. Acad. Sci. Paris 58 94), -4. [DER] J. Derbyshire : Prime Obsession: Bernhard Riemann and the Greatest Unsolved Problem in Mathematics. New York: Penguin, pp. 45-5, 4. [DERZ] Deshouillers, Effinger, Te Riele and Zinoviev : A complete Vinogradov 3-primes theorem under the Riemann hypothesis, Electronic Research Announcements of the American Mathematical Society 3 5): 99-4, 997. [ED] H.M. Edwards : Riemann s zeta function Acad. Press. New York

10 [EE] W. J. Ellison and F. Ellison, Prime Numbers, John Wiley & Sons, New York, 985. [GE] L. Gegenbauer : Asymptotische Gesetze der Zahlentheorie, Denkschriften. Akad. Wien 49 Abt. 885) [HU] M. Huxley : The Distribution of Prime Numbers, Oxford University Press, London, 97. [KG] Karl Greger : Square Divisors and Square-Free Numbers, Mathematics Magazine, Vol. 5, No 4, 978, pp. -9. [KV] A. A. Karatsuba and S. M. Voronin : The Riemann zeta-function, Berlin - New York De Gruyter 99. [LE] N. Levinson : More than one third of the zeros of Riemann s zeta-function are on σ =, Adv. Math ), [NA] T. Nagell : Introduction to Number Theory. New York: Wiley, p. 7, 95. [NAT] M. B. Nathanson : Elementary methods in number theory, volume 95 of Graduate Texts in Mathematics. Springer-Verlag, New York,. [SA] K. Sabbagh, The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics, Farrar, Straus, and Giroux,. [RIE] B. Riemann : Über die Anzahl der Prinzahlen unter einer gegebener Gröse. Mo-nastsber. Akad. Berlin, 67-68, 859). [SE] J. P. Serre : A Course in Arithmetic, Springer-Verlag, New York, 996. [STSH] E. M. Stein and R. Shakarchi, Complex Analysis, Princeton Lectures in Analysis II, Princeton University Press, Princeton and Oxford, 3. [TIT] E. C. Titchmarsh : The Theory of the Riemann Zeta-function, nd edition revised by D. R. Heath-Brown), Oxford University Press, Oxford, 986. [TIT] E. C. Titchmarsh : The Theory of Functions, nd edition, Oxford University Press, 939 [WI] A. Wintner : The fundamental lemma in Dirichlet s theory of the arithmetical progressions. Amer. J. Math., 68:85-9,