Riemann’s memoir is devoted to the function π(x) defined as the number of prime
numbers less or equal to the real and positive number x. This is really the fact, but the
“main role” in it is played by the already mentioned zeta-function

Rusev, Peter

Bulgarian Digital Mathematics Library at IMI-BAS

МАТЕМАТИКА И МАТЕМАТИЧЕСКО ОБРАЗОВАНИЕ, 2010MATHEMATICS AND EDUCATION IN MATHEMATICS, 2010Proceedings of the Thirty Ninth Spring Conference ofthe Union of Bulgarian MathematiciansAlbena, April 6–10, 2010RIEMANN’S HYPOTHESISPeter RusevIn 1859, i.e. a litle more than a centure and a half ago, appeared the selebratememoir Ueber die Anzahl der Primzahlen unter einer gegebenen Gro¨sse,Monatsb. der Ko¨nigl. Preuss. Akad. der Wissen. zu Berlin aus dem Jahr 1859(1860), 671-680 of B. Riemann (1826 - 1866). In E. Bombieri’s survey Problem ofthe Millenium: the Riemann Hypothesis, this memoir is qualified as “epoch-making” as well as “really astonished for the novelty of ideas included”. In SiebenMilleniums-Probleme. I, Internat. Math. Nachr., Nr 184 (2000) 29-36, M. Dr-mota calls it “bahnbrechend fu¨r die analytische Zahlentheorie” thus repeating Jo¨rgBru¨dern, Primzahlverteilung, Vorlesung im Wintersemester 1991/92, Mathema-tisches Institut Go¨ttingen. These “estimates” and other similar ones confirmed thewords of E. Titchmarsh at the begining of Chapter X of his The theory of Rie-mann zeta-function, Oxford 1951: “The memoir, in which Riemann consideredthe zeta-function became famous thanks to the great number of ideas included in it.Many of them has been worked afterwards, and some of them are not exausted eventill now”.Riemann’s memoir is devoted to the function pi(x) defined as the number of primenumbers less or equal to the real and positive number x. This is really the fact, but the“main role” in it is played by the already mentioned zeta-function. In order to makeclear the basic idea of Riemann, let us quote a part of the second section of his memoir:“Bei dieser Untersuchungen diente mir als Ausgangspunkt die von Euler gemachteBemerkung, dass das Product(1.1)∏ 11−1ps=∑ 1nswenn fu¨r p alle Primzahlen, fu¨r n alle ganzen Zahlen gesetzt werden. Die Functionder complexen Vera¨nderlichen s, welche durch diese beiden Ausdru¨cke, so lange sie con-vergieren, darstellt wird, bezeichne ich durch ζ(s)”.Namely Riemann was the first who considered the function denoted by him by ζ(s)as a function of a complex variable and this leaded him to a number of remarkablediscoveries.After Riemann the notation ζ(s) remained unchanged but the notion “zeta-function”got a very wide revalence. As Riemann has pointed out, by the equality(1.2) ζ(s) =∞∑n=11ns21is defined a holomorphic function of the complex variable s = σ + it provided σ > 1.An easy computation leads to an integral represenation of ζ(s) by means of the functionpi(x). Indeed, from (1.1) it follows that ζ(s) 6= 0 in the half-plane Re s > 1 and, moreover,thatlog ζ(s) = −∑plog(1−1ps)= −∞∑n=2{pi(n)− pi(n− 1)} log(1−1ns)= −∞∑n=2pi(n){log(1−1ns)− log(1−1(n+ 1)s)}=∞∑n=2pi(n)∫ n+1ns dxx(xs − 1),i.e.log ζ(s) = s∫ ∞2pi(x) dxx(xs − 1), s = σ + it, σ > 1.Much more deeper is the formula expressing pi(x) in terms of the so-called non-trivialzeros of the meromorphic function obtained by the analytical continuation of the functionζ(s) in the whole complex plane. The discovery of this formula is one of the mainachievements of Riemann included in the memoire under consideration.There are many ways to prove that ζ(s) can be analytically contunued on the left ofthe line Re s = 1. The idea of that one which we are going to sketch here belongs toRiemann. It is based on the equalities∫ ∞0exp(−pin2x)xs/2−1 dx =Γ(s/2)pis/2ns, s = σ + it, σ > 0, n = 1, 2, 3, . . . ,which are corollaries of the well-known integral representation of Γ(s) in the half-planeσ > 0, namelyΓ(s) =∫ ∞0xs−1 exp(−x) dx.If σ > 1, then(1.3) pi−s/2Γ(s/2)ζ(s) =∫ ∞0ψ(x)xs/2−1 dx,where(1.4) ψ(x) =∞∑n=1exp(−pin2x), x > 0.In order to justify the validity of (1.3), Riemann used the function(1.5) θ(x) =∞∑n=−∞exp(−pin2x), x > 0.From the inequalities exp(−pin2x) ≤ exp(−pinx), x > 0, n = 1, 2, 3, . . . it follows thatψ(x) = O(exp(−pix)) when x→ ∞. Since θ(x) = 2ψ(x) + 1, θ(x) = O(1) when x→ ∞.Then, the relations(1.6) θ(x) = x−1/2θ(1/x)and ψ(x) = (1/2)(θ(x) − 1) imply ψ(x) = O(x−1/2) when x → 0. Since σ > 1, theintegral on the right-hand side of (1.3) is absolutely convergent. The series defining the22function (1.4) is uniformly convergent on every compact subset of the ray (0,∞). Hence,∫ ∞0ψ(x)xs/2−1 dx =∫ ∞0(∞∑n=1exp(−pin2x))xs/2−1 dx=∞∑n=1∫ ∞0exp(−pin2x)x−s/2−1 dx = pi−s/2Γ(s/2)ζ(s).Further, using 2ψ(x) + 1 = x−1/2(2ψ(x−1) + 1), from (1.3) after some computationsRiemann came to the representation(1.7) pi−s/2Γ(s/2)ζ(s) =1s(s− 1)+∫ ∞1ψ(x)(xs/2−1 + x(1−s)/2−1) dx, Re s > 1.The integral in the last equality is uniformly convergent on every compact subset ofthe complex plane. That means the left-hand side of (1.7) admits an analytic continuationin the whole complex plane except the points 0 and 1. The zero point is a pole of thefunction Γ(s/2), the point s = 1 is a simple pole of ζ(s) with residu equal to one. Theother poles of Γ(s/2) are at the points −2,−4,−6, . . . and all they are simple. Theequality (1.7) shows that all they are regular points for the already contunued functionζ(s). More precisely, these points are simple zeros of ζ(s). They are called trivial.Moreover, there are no other zeros of ζ(s) in the half-plane Re s < 0. Since ζ(s) 6= 0when Re s > 1, all other possible zeros of ζ(s) are in the closed strip 0 ≤ Re s ≤ 1 namedcritical strip. The zeros of ζ(s) in the crirtical strip are called non-trivial.The right-hand side of (1.7) does not change if we replace s by 1 − s. This leadsimmediately to the relation(1.8) pi−s/2Γ(s/2)ζ(s) = pi−(1−s)/2Γ((1− s)/2)ζ(1− s)usually refered as a functional equation for ζ(s).Let the function ξ(s) be defined by the equality(1.9) ξ(s) = s(s− 1)pi−s/2Γ(s/2)ζ(s).It is clear that ξ(s) is an entire function. Further, the functional equation (1.8) yieldsthat(1.10) ξ(s) = ξ(1− s).It is clear also that all the zeros of ξ(s) coinside with the non-trivial zeros of ζ(s), i.e.all they are in the critical strip. Moreover, since (s + (1 − s))/2 = 1/2, from (1.10) itfollows that the zeros of ξ(s) are symmetrically situated with respect to the point 1/2.It is known that the entire function ξ(s) is real, i.e. ξ(s) is a real number if and only ifs is real and, hence, the euality ξ(s) = ξ(s) holds for every s ∈ C. That means: if ρ is azero of ξ(s), then so does ρ as well as 1− ρ. If ρ = σ + it, then 1− ρ = 1− σ + it, i.e. ρand 1− ρ are symmetrcally situated with respect to the line Re s = 1/2.In his memoir Riemann claimed that the function ζ(s) has infinitely many zeros inthe critical strip and moreover, that the following “explicit” formula(1.11) pi(x) = li(x) +∑ρ∈N ,Im ρ>0(li(xρ) + li(x1−ρ))+∫ ∞xdt(t2 − 1) log t− log 2, x ≥ 2.23holds for the function pi(x), where li is the integral logarithm and N is the set of zerosof ζ(s) in the critical strip. The validity of the above formula was really confirmed byH. von Mangoldt in 1894.Let, further, the function Ξ(z) be defined by(1.12) Ξ(z) = ξ(12+ iz), z ∈ C.Then (1.10) immediately yields(1.13) Ξ(z) = Ξ(−z),i.e. the functional equation for ζ(s) is equivalent to the fact that Ξ(z) is an even function.Moreover, it is clear that all the possible zeros of Ξ(z) are in the strip | Im z| ≤ 1/2.From (1.7), (1.9) and (1.12) it follows thatΞ(z) =12−(z2 +14)∫ ∞1ψ(x)x−3/4 cos(z2log x)dx.Integrating by parts and using the equality 4ψ′(1)+ψ(1) = −1/2, Riemann got that(1.14) Ξ(z) = 4∫ ∞1{x3/2ψ′(x)}′x−1/4 cos(z2log x)dx.Based on the last representation he made his remarkable comment about the zeros ofthe function Ξ(z) namely: “. . . es ist sehr wahrscheinlich, dass alle Wurzeln reell sind.Hiervon wa¨re allerdings ein strenger Bewis zu wu¨nschen, ich habe indess die Aufsuchdesselben nach einigen flu¨chtigen vergeblichen Versuche vorla¨ufig bei Seite gelassen, daer fu¨r den na¨chsten Zweck meiner Untersuchungen entberlich schien”.The above words of Riemann has born the conjecture, usually called Riemann’s hy-pothesis, that all the zeros of the function Ξ(z) are real. It is equivalent to the conjecturethat all the non-trivial zeros of the function ζ(s) are on the line Re s = 1/2. In spite ofthe efforts made in the last 150 years, this conjecture is neither proved, nor disproved.Remarks1. Euler’s identity (1.1), considered by him only when s is a real number, is equiv-alent to the fundamental theorem of Arithmetic to say that if p1 < p2 < p3 < . . . arethe prime numbers, then every natural number n ≥ 2 has a unique representation of thekind n = pα11 pα22 pα33 . . . pαkk , where αj , 1 ≤ j ≤ k are non-negative integers.2. The series in the right-hand sides of the equalitiesη(s) =∞∑n=1(−1)nns, L(s) =∞∑n=0(−1)n(2n+ 1)s,as Dirichlet series, are uniformly convergent on every compact subset of the half-planeRe s > 0. The holomorphic functions defined by them satisfy the functional equations(2s−1 − 1)η(1 − s) = −(2s − 1)pi−s cospis2Γ(s)η(s),L(1− s) = 2spi−s sinpis2Γ(s)L(s).corresondingly.The first of them is equivalent to the functional equation for ζ(s). Thesecond is contained in a paper of L. Euler published in 1749 and “checked” by him onlyfor some real values of s.24In fact, both functions η(s) and L(s) are analytically contunuable as entire func-tions. Moreover, η(s) = (1− 21−s)ζ(s) (G.H. Hardy, Divergent series, Oxford, 1949,Chapter II, 2.2).3. The function pi(x) and more precissely, its growth as x→∞ has been an object ofinvestigation of many mathematicians. About the year 1800 it was already numericallycleared that(1.15) pi(x) ∼xlog x, x→∞,i.e that pi(x)x−1 log x→ 1 when x→∞. At the same time K. Gauss proposed that(1.16) pi(x) ∼∫ x2dtlog t, x→∞,so that it is not surprising that the first member of Riemann’s explicit formula (1.11)is just li(x). Let us point out that the validity of Gauss’ asymptotic formla was provedindependently by J. Hadamard and J. de la Valle´e Poussin at the end of 19thcentery. It is well-known that (1.15) and (1.16) are equivalent, but it seems that (1.15)is much more popular. Each one of them is called asymptotic low of prime numbersdistribution or, briefly, asymptotic low (prime number theorem as well as Primzahlsatzare also used). It is well-known that its validity is equivalent to the fact that ζ(1+it) 6= 0for every real t 6= 0. Namely the last property of Riemann’s zeta-function has beenestablished by J. Hadamard and J. de la Valle´e Poissin.The asymptotics of the function pi∗(x) = pi(x) − li(x) as x → ∞ is still an openproblem. Riemann’s explicit formula shows that its behaviour is strongly connectedwith the zero-distribution of ζ(s) in the crirtical strip. Indeed, all the results known tillnow confirmed that the asymptotics of the function pi∗(x) depends on the absence ofzeros of ζ(s) in subregions of the critical strip.In 1899 J. de la Valle´e Pousin proved that ζ(s) has no zeros in the region definedby the inequality σ > 1 − A(log(|t| + 2))−1 and as a corollary he got that pi∗(x) =O(x exp(−a(log x)1/2)). In 1922 J.E. Litlewood proved that ζ(s) 6= 0 if σ > 1 −A log(log t)(log t)−1, t ≥ t0 > 0 and thus obtained thatpi∗(x) = O(x exp(−a(log x log log x)1/2))(here and below A and a denote positive constants different in different cases). A sharp-ening of Littlewood’s results is given in 1936 by N.G. Tchudakov. He proved thatζ(s) 6= 0 when σ > 1 − A(log t)−3/4(log log t)−3/4, provided t is suficiently large, and asa corollary that pi∗(x) = O(x exp(−a(log x)−4/7(log log x)−3/7)).In 1958 I.M. Vinogradov and N.M. Korobov proved (independently) that ζ(s) 6=0 when σ > 1−A(log(|t|+3))−1/3(log log(|t|+3))−2/3. A corollary of this result is thatpi∗(x) = O(x exp(−a(log x)3/5)(log log x)−1/5)as x→∞. It seems the last asymptotic estimate is the best one known till now.In 1901 H. von Koch proved that if Riemann’s hypothesis is true, then pi∗(x) =O(x1/2 log x) as x → ∞. He proved also that if pi∗(x) = O(xθ+ε) for some fixed θ ∈[1/2, 1) and arbitrary positive ε when x → ∞, then ζ(s) 6= 0 for σ > θ. Since log x =O(xε) for every positive ε when x → ∞, it follows that the validity of the estimatepi∗(x) = O(x1/2 log x), x→∞ leads to the validity of Riemann’s hypothesis.4. The denotations for the function (1.9) and (1.12) are assumed by E. Titchmarsh.25The entire function defined by the right-hand side of (1.9) appeared first in Riemann’smemoir. Replacing s by 1/2+ it, he defined in fact the function Ξ(t), denoting it by ξ(t).From (1.14) it follows thatΞ(z) = 2∫ ∞1Ψ(x) cos((z/2) logx) dx,where(1.17) Ψ(x) = {3ψ′(x) + 2xψ′′(x)}x1/4.Then, after substituting x by exp 2u,−∞ < u <∞, one gets(1.18) Ξ(z) = 2∫ ∞0Φ(u) cos zu du,where(1.19) Φ(u) = 2Ψ(exp 2u) exp 2u, −∞ < u <∞,i.e.(1.20) Φ(u) = 2∞∑n=1(2pi2n4 exp(9u/2)− 3pin2 exp(5u/2)) exp(−pin2 exp 2u).It is not quite obvious that the function Φ(u) is even, but this is really the fact.Indeed, (1.17), (1.19) and the relation x = exp 2u yieldΦ(−u) = 2{3ψ′(x−1) + x−1ψ′′(x−1)}x−5/4Further, using that 2ψ(x)+1 = x−1/2{2ψ(x−1)+1}, 0 < x <∞, after some computationone gets2{3ψ′(x−1) + x−1ψ′′(x−1)}x−5/4 = 2{sψ′(x) + xψ′′(x)}x5/4, 0 < x <∞,i.e. Φ(−u) = Φ(u), 0 < u <∞. Then, (1.18) can be written as(1.21) Ξ(z) =∫ ∞−∞Φ(u) exp(izu) du.5. It was already mentioned that Rieman’s hypothesis is neither proved nor dis-proved. But it is well-known that infinitely many zeros of the function ζ(s) are locatedon the line Re s = 1/2 which is equivalent to the existing of infinitely many real zeros ofRieman’s ξ-function. The first proof was given by H.G. Hardy Sur les ze´ros de lafonction ξ(s) de Riemann, C. R. 153 (1914), 1012-1014.Acknowledgements. This survey is performed in frames of the Project ID−02−0129“Integral TransformMethods, Special Functions and Applications” with National ScienceFund – Ministry of Education, Youth and Science, Bulgaria.Peter RusevInstitute of Mathematics and InformaticsBulgarian Academy of SciencesAcad. G. Bonchev Str., Bl. 81113 Sofia, Bulgariae-mail: pkrusev@math.bas.bg26ХИПОТЕЗАТА НА РИМАНПетър РусевТази година се навършват 150 години от публикуването на знаменития мемоарUeber die Anzahl der Primzahlen unter einer gegebenen Gro¨sse (Върхуброя на простите числа по-малки от дадена “величина”) Monatsb. der Ko¨nigl.Akad. der Wissen. zu Berlin aus dem Jahr 1959 (1960), 671–680 на Б. Риман (1826–1866).Отправна точка на изследванията на Риман е дължимото на Ойлер тъждес-твоY 11− 1ps=X 1ns, s > 1,където p “пробягва” простите, а n – естествените числа.Главното внимание в мемоара е отделено на функцията(1) ζ(s) =∞Xn=11ns,която за пръв път е третирана като функция на комплексната променливаs = σ + it именно от Риман.С (1) функцията ζ(s) е дефинирана когато σ > 1. Както установява Риман,тя е аналитично продължима в цялата комплексна равнина като мероморфнафункция с единствен (прост) полюс в точката s = 1. Това е следствие от функ-ционалното уравнениеpi−s/2Γ(s/2)ζ(s) = pi−(1−s)/2Γ((1− s)/2)ζ(1− s)получено от Риман. От него следва, че точките −2,−4,−6, . . . са прости нулина “продължената” ζ-функция, наречени тривиални. Тъй като ζ(s) 6= 0 когато|Re s| > 1, s 6= −2,−4,−6, . . . други нули може да има в ивицата 0 ≤ Re s ≤ 1.Всъщност, както е доказано по-късно, върху правите линии Re s = 0 и Re s = 1няма нули на ζ(s).Нека N е множеството на нетривиалните нули на ζ(s), т.е. тези, които са вивицата 0 < Re s < 1 и pi(x) е броят на простите числа ненадминаващи x > 0.Риман приема че ζ(s) има безбройно много нетривиални нули и освен това твърдиче за pi(x) е валидна следната формула(2) pi(x) = li(x) +Xρ∈N ,Im ρ>0(li(xρ) + li(x1−ρ))+Z ∞xdt(t2 − 1) log t− log 2, x ≥ 2,където li е интегралният логаритъм.Функцията ξ(s), дефинирана с равенствотоξ(s) = s(s− 1)pi−s/2Γ(s/2)ζ(s)е холоморфна в цялата комплексна равнина, т.е. тя е цяла функция. Освен това,множеството на нулите и` съвпада с множеството на нетривиалните нули на ζ(s),т.е. на тези за които 0 < Re s < 1. С равенството27Ξ(z) = ξ12+ izе дефинирана също цяла функция, нулите на която са в ивицата | Im z| <12.На Риман се дължи хипотезата, че функцията Ξ(z) има само реални нули, ко-ято е еквивалентна с хипотезата, че нетривиалните нули на ζ(s) са върху праваталиния Re s =12. Тази хипотеза нито е потвърдена, нито е опровергната досега.За аналитичната теория на числата от фундаментално значение е асимпто-тичното поведение на функцията pi(x) когато x→∞. Около 1800 г е установеноемпирично, че pi(x) ∼ x(log x)−1, x→∞, което е еквивалентно с(3) pi(x) ∼ li(x), x→∞.Поведението обаче на функцията pi∗(x) = pi(x)− li(x) когато x→∞ е все ощезагадка за съвременната математическа наука.В предложения обзор главното внимание върху мемоара на Риман е отде-лено на извода на функционалното уравнение за функцията ζ(s), което, кактоизглежда, е довело до хипотезата за нетривиалните и` нули.Упоменато е, че формулата (2) за pi(x), както и валидността на съотношени-ето (3), е доказана в края на 19-и век и, че безбройно много от нулите на ζ(s) савърху правата линия Re s =12е установено в началото на миналия век. Прос-ледени са усилията респ. резултатите от тях за уточняване на разпределениетона нетривиалните нули на ζ(s), както и влиянието им върху асимптотиката нафункцията pi∗(x).28

Riemann's Hypothesis

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Riemann's Hypothesis

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