416 HISTORY OF THE THEORY OF NUMBERS. prime to 0, \, . . . , w, contained in ir(k~l) consecutive terms of the progression is ^P~lQ>jr(k-V-2, where P=3-5-7-ll. . ., Q= (3-1) (5-1). . .. J. J. Sylvester29 gave a proof. V. I. Berton290 found h such that between x and xh occur at least 2g primes each of one of the 2g linear forms 2py+ri} where n, . . ., r2g are the integers <2p and prime to 2p. C. Moreau30 noted the error in Legendre's22 proof. L. Kronecker5 (pp. 442-92) gave in lectures, 1886-7, the following extension* of Dirichlet's theorem (in lectures, 1875-6, for the case m a prime): If M is any given integer, we can find a greater integer v such that, if m, r are any two relatively prime integers, there exists at least one prime of the form hm+r in the interval from ju to v (p. 11, pp. 465-6). Moreover (pp. 478-9), there is the same mean density of primes in each of the <KW) progressions m/i-fr,-, where the rt- are the integers <ra and prime to m. I. Zignago31 gave an elementary proof. H. Scheffler32 devoted 31 pages to a revision of Legendre's insufficient proof and gave a process to determine all primes under a given limit. G. Speckmann33 failed in an attempt to prove the theorem. P. Bachmann34 gave an exposition of Dirichlet's23 proof. Ch. de la Valtee-Poussin36 obtained without computations, by use of the theory of functions of a complex variable, a proof of the difficult point in Dirichlet's23 proof. He36 proved that the sum of the logarithms of the primes hk+l^x equals #/<£(&) asymptotically and concluded readily that the number of primes hk+l^x equals, asymptotically, F. Mertens37 proved the existence of an infinitude of primes in an arithmetical progression by elementary methods not using the quadratic reciprocity theorem or the number of classes of primitive binary quadratic forms. He supplemented the theorem by showing how to find a constant c such that between x and ex there lies at least one prime of the progression for every x^l [cf. Kronecker,5 pp. 480-96]. 29Proc. London Math. Soc., 4, 1871, 7; Messenger Math., (2), 1, 1872, 143-4; Coll. Math. Papers, 2, 1908, 712-3. 29aComptes Rendus Paris, 74, 1872, 1390. 3°Nouv. Ann. Math., (2), 12, 1873, 323-^. Also, A. Piltz, Diss., Jena, 1884. "Improvements in the exposition were made by the editor, Hensel (cf. p. 508). 31Annali di Mat., (2), 21, 1893, 47-55. 32Beleuchtung u. Beweis eines Satzes aus Legendre's Zahlentheorie [II, 1830, 76], Leipzig, 1893. "Archiv Math. Phys., (2), 12, 1894, 439-441. Cf. (2), 15, 1897, 326-8. "Die analytische Zahlentheorie, 1894, 51, 74-88. ĞMe*m. couronnSs. . .acad. roy. sc. Belgique, 53, 1895-6, No. 6, 24-9. 36Annales de la soc. sc. de Bruxelles, 20, 1896, II, 281-361. Cf. 183-256, 361-397; 21, 1897, I, 1-13, 60-72; II, 251-368. 37Sitzungsber. Ak. Wiss. Wien (Math.), 106, 1897, II a, 254-286. Parts published earlier, ibid., 104, 1895, Ha, 1093-1121, 1158-1166; Jour, ftir Math., 78, 1874, 46-62; 117, 1897, 169-184.