828 THE THEORY OF NUMBERS 1800-1900 then F is said to be transformable into f.If further the six numbers p192-q1p2,p193-91p3,p194-91p4,P293-92P3,p294-92p4,p394-93P4 do not have a common divisor,then F is called a composite of the forms f and f'. Gauss was then able to prove an essential theorem:Iffand g belong to the same class and f and g'belong to the same class,then the form composed of f and f'will belong to the same class as the form composed of g and g'. Thus one can speak of a class of forms composed of two (or more)given classes.In this composition of classes,the principal class acts as a unit class; that is,if the class K is composed with a principal class,the resulting class will be K. Gauss now turns to a treatment of ternary quadratic forms Ax2 2Bxy Cy2+2Dxz +2Eyz +Fz2, where the coefficients are integers,and undertakes a study very much like what he has just done for binary forms.The goal as in the case of binary forms is the representation of integers.The theory of ternary forms was not carried far by Gauss. The objective of the entire work on the theory of forms was,as already noted,to produce theorems in the theory of numbers.In the course of his treatment of forms Gauss shows how the theory can be used to prove any number of theorems about the integers including many that were previously proved by such men as Euler and Lagrange.Thus Gauss proves that any prime number of the form 4n I can be represented as a sum of squares in one and only one way.Any prime number of the form 8n+1 or 8n +3 can be represented in the form 2+2y2 (for positive integral x and y)in one and only one way.He shows how to find all the representations of a given number M by the given formx provided the dis- criminant D is a positive non-square number.Further if Fis a primitive form (the values of a,b,and c are relatively prime)with discriminant D and if is a prime number dividing D,then the numbers not divisible by p which can be represented by F agree in that they are all either quadratic residues of p or non-residues of p. Among the results Gauss drew from his work on ternary quadratic forms is the first proof of the theorem that every number can be represented as the sum of three triangular numbers.These,we recall,are the numbers n2+n 1,36,10,15,.,2 He also re-proved the theorem already proved by Lagrange that any positive integer can be expressed as the sum of four squares.Apropos of the former result,it is worth noting that Cauchy read a paper to the Paris Academy in
Analytic NUMBER THEORY 829 1815 which established the general result first asserted by Fermat that every integer is the sum of k or fewer k-gonal numbers.26 (The general k-gonal number is n+(n2-n)(k -2)/2.) The algebraic theory of binary and ternary quadratic forms as presented by Gauss has an interesting geometrical analogue which Gauss himself initiated.In a review which appeared in the Gottingische Gelehrte Anzeigen of 183027 of a book on ternary quadratic forms by Ludwig August Seeber (1793-1855)Gauss sketched the geometrical representation of his forms and classes of forms.28 This work is the beginning of a development called the geometrical theory of numbers which first gained prominence when Hermann Minkowski (1864-1909),who served as professor of mathematics at several universities,published his Geometrie der Zahlen (1896) The subject of forms became a major one in the theory of numbers of the nineteenth century.Further work was done by a host of men on binary and ternary quadratic forms and on forms with more variables and of higher degree. 6.Analytic Number Theory One of the major developments in number theory is the introduction of analytic methods and of analytic results to express and prove facts about integers.Actually,Euler had used analysis in number theory (see below) and Jacobi used elliptic functions to obtain results in the theory of con- gruences and the theory of forms.30 However,Euler's uses of analysis in number theory were minor and Jacobi's number-theoretic results were almost accidental by-products of his analytic work. The first deep and deliberate use of analysis to tackle what seemed to be a clear problem of algebra was made by Peter Gustav Lejeune-Dirichlet (1805-59),a student of Gauss and Jacobi,professor at Breslau and Berlin, and then successor to Gauss at Gottingen.Dirichlet's great work,the Vorlesungen iber Zahlentheorie,3 expounded Gauss's Disquisitiones and gave his own contributions The problem that caused Dirichlet to employ analysis was to show that every arithmetic sequence a,a+b,a+2b,a+3b,.,a+nb,. 26.Mtm.de1'Acad des Sci.,Pari,(1),14,1813-15,177-220=Euvres,(2),6,320-53. 27.Wcrk,2,188-196 28.Felix Klein in his Entwicklung (see the bibliography at the end of this chapter), 29.See works af Smith and Diskson listed in the bibliggrapby for further dctails 30.Jour fuir Math,37,1848,61-94and221-54=werk,2,21988. 31.Published 1863;the second,third,and fourth editions of 1871,1879,and 1894 were supplemented extensively by Dedekind
830 THE THEORY OF NUMBERS 1800-1900 where a and b are relatively prime,contains an infinite number of primes. Euler32 and Legendre 33 made this conjecture and in 1808 Legendre34 gave a proof that was faulty.In 1837 Dirichlet35 gave a correct proof.This result generalizes Euclid's theorem on the infinitude of primes in the sequence 1,2,3,.Dirichlet's analytical proof was long and complicated.Specifically it used what are now called the Dirichlet series,an-,wherein the a and are complex.Dirichlet also proved that the sum of the reciprocals of the primes in the sequence {a nb)diverges.This extends a result of Euler on the usual primes (see below).In 184138 Dirichlet proved a theorem on the primes in progressions of complex numbers a bi. The chief problem involving the introduction of analysis concerned the function (x)which represents the number of primes not exceeding x.Thus m(8)is 4 since 2,3,5,and 7 are prime and m(11)is 5.As x increases the additional primes become scarcer and the problem was,What is the proper analytical expression for m(x)?Legendre,who had proved that no rationa expression would serve,at one time gave up hope that any expression could be found.Then Euler,Legendre,Gauss,and others surmised that (x) (5) Gauss used tables of primes (he actually studied all the primes up to 3,000,000)to make conjectures about m(x)and inferred37 that m(x)differs little fromdtllog t.He knew also that 2diflog t lim =1. 。x/logx In 1848 Pafnuti L.Tchebycheff(1821-94),a professor at the University of Petrograd,took up the question of the number of prime numbers less than or equal toand made a big step forward in this old problem.In a key paper,"Sur les nombres premiers"38 Tchebycheff proved that 41<<42, xllog x where 0.922 <A<I and 1 A2 1.105,but did not prove that the function tends to a limit.This inequality was improved by many mathe- 32.Obuscula Analutica.2.1783 33.Mem.de l'Acad.des Sci.,Paris,1785,465-559,pub.1788. 34.Theorie des nombres,2nd ed,p.404. 35.Abh.Konig.Akad.der Wiss.,Berlin,1837,45-81 and 108-10 Werke,1,307-42. 36.Abh.Konig.Akad.der Wiss.,Berlin,1841,141-61 =Werke,2,509-32. 37.weke,2,444-47. 38.Mm.Acad.Sci.SL.Peter5.,7,1854,15-33;also Jour.de Math.,(1),17,1852.366-90= (Euvres,51-70
ANALYTIC NUMBER THEORY 831 maticians including Sylvester,who among others doubted in 1881 that the function had a limit.In his work Tchebycheff used what we now call the Riemann zeta function, -S! though he used it for real values of z.(This series is a special case of Dirichlet's series.)Incidentally he also proved in the same paper that for n>3 there is always at least one prime between n and 2n-2. The zeta function for real z appears in a work of Euler3 where he introduced -0- gs)=】 Here the p's are prime numbers.Euler used the function to prove that the sum of the reciprocals of the prime numbers diverges.For s an even positive integer Euler knew the value of(s)(Chap.20,sec.4).Then in a paper read in 174940 Euler asserted that for reals t1-）=2(2m）-cosr((. He verified the equation to the point where, he said,there was s no doubt about it.This relation was established by Riemann in the 1859 paper referred to below.Riemann,using the zeta function for complex z,attempted to prove the prime number theorem,that is,(5)aboveHe pointed out that to further the investigation one would have to know the complex zeros of (z).Actually,(z)fails to converge when but the values of in the half-plane x s 1 are defined by analytic continuation.He expressed the hypothesis that all the zeros in the strip sI lie on the line =1/2.This hypothesis is still unproven. In 1896 Hadamard,43 by applying the theory of entire functions (of a complex variable),which he investigated for the purpose of proving the prime number theorem,and by proving the crucial fact that (z)0 for x=1,was able finally to prove the prime number theorem.Charles-Jean de la Vallee Poussin (1866-1962)obtained the result for the zeta function 39.Comm.Acad.Sci.Pet .9.1737.16088.Dub.1744=0b a.(1).14.216-44 40.Hist.de im,17,1761,83-106 ub.1768=0 ,(1),15,70-90. 41.Monatsber.Berliner Akad.,1859.671-80 =Werke,145-55. 42.In 1914 Godfrey H.Hardy p oved (Comp.Rend.,158,1914,1012-14=Coll.Papers,2, 6-9)that an infinit s of (z)lie on the line 43.Bull.Soc.Math de Fro a,141896,19-220=aEr,1,189-210
832 THE THEORY OF NUMBERS I800-IQ00 and proved the prime number theorem at the same time.44 This theorem is a central one in analytic number theory Bibliography Bachmann,P.:"Uber Gauss'zahlentheoretische Arbeiten,"Nachrichten Konig. Ges.der Wiss.zu Gott.,1911,455-508;also in Gauss:Werke,102,1-69. Bell,Eric T.:The Development of Mathematics,2nd ed.,McGraw-Hill,1945,Chaps. 9-10. Carmichacl,Robert D.:"Some Recent Researches in the Theory of Numbers," Amer.Math.Monthly,39,1932,139-60. Dedekind,Richard:Uber die Theorie der ganzen algebraischen Zahlen (reprint of the Gesammelte ath Sohn,1930-32 elsea (reprint),1968. "Sur la theorie des nombres entiers algebriques,"Bull.des Sci.Math.,(1), 11,1876,27888;(2),1,1877,17-41,69-92,14464,20748=Gs.math. Wrke.3.263-96. Dickson, onard E.:History of the Theory of Numbers,3 vols.,Chelsea (reprint), 1951. Studies in the Theory of Numbers (1930),Chelsea (reprint),1962. :"Fermat's Last Theorem and the Origin and Nature of the Theory of Algebraic Numbers,"Annals.of Math.,(2),18,1917,161-87. etal:Algebraic】 ers,Report of C ommittee on Algebraic Numbers,National Research Council,1923 and 1928;Chelsea(reprint),1967. Dirichlet,P.G.L.:Werke(1889-97);Chelsea (reprint),1969,2 vols. Dirichlet,P.G.Land R.Dedekind:Vorlesungen berr thed94 contains dedekind's su ment);Che eprint),1968. Gauss,C.F.,trans.A.A.Clarke,Yale University Press, 1965. Hasse,H.:"Bericht uber neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkorper,"Jahres.der Deut.Math.-Verein.,35,1926, 1-55and36.1927.233-311 Hilbert,David: Die Theorie der algebraischen Zahlkorper,"Jahres.der Deut. Math.-Verein.,4,1897,175-546 Gesammelte Abhandlungen,1,63-363. Klein,Felix:Vorlesungen iiber die Entwicklung der Mathematik im 19.Jahrhundert, Chelsea (reprint),1950,Vol.1. Kronecker, ld:Werke,5 vols.(1895-1931),Chelsea (reprint),1968.Sce especially,Vol.2,pp.1-10 on the law of quadratic reciprocity. Grundzuge einer arithmetischen Theorie der algebraischen Grossen,G.Reimer, 1882=Jou.ftir Math.,92,1881/82,1-122=Werke,2,237-388. Landau,Edmund:Handbuch der Lehre von der Verteilung der Primzahlen,B.G.Teubner 1009 Vol.1,pp.1-55. Mordell,L.J.:"An Introductory Account of the Arithmetical Theory of Algebraic 44.Ann.Soc.Sci,Bruxelle.5.(1).20 Part II.1896,183-256.281-397