416 History of the Theory of Numbers. iChap. xviii 
prime to ^,X,. . ., co, contained in tt**"^^ consecutive terms of the progression 
is ^p-'Q7r"'-''-2, where P = 3-5-7-ll . . ., (?= (3-l)(5-l) . . .. 
J. J. Sylvester-^ gave a proof. 
V. I. Berton^^" found h such that between x and xh occur at least 2g 
primes each of one of the 2g linear forms 2py+ri, where ri, . . ., r2g are the 
integers < 2p and prime to 2p. 
C. ]Moreau^° noted the error in Legendre's-- proof. 
L. Kronecker^ (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 jjl is any given integer, we can find a greater integer v such that, 
if 7u, r are any two relatively prime integers, there exists at least one prime 
of the form h7?i+r in the interval from /x to ^' (p. 11, pp. 465-6). Moreover 
(pp. 478-9), there is the same mean density of primes in each of the <l){m) 
progressions mh-\-ri, where the r, are the integers <?n and prime to m. 
I. Zignago^^ gave an elementary proof. 
H. Scheffler^^ devoted 31 pages to a re\nsion of Legendre's insufficient 
proof and gave a process to determine all primes under a given limit. 
G. Speckmann^^ failed in an attempt to prove the theorem. 
P. Bachmann^^ gave an exposition of Dirichlet's^^ proof. 
Ch. de la Vallee-Poussin^^ obtained without computations, by use of 
the theorj' of functions of a complex variable, a proof of the difficult point 
in Dirichlet's^^ proof. He^® proved that the sum of the logarithms of the 
primes hk-\-l^x equals x/4>{k) asymptotically and concluded readily that 
the number of primes hk+l'^x equals, asjonptotically, 
4> {x) log X 
F. Mertens^^ proved the existence of an infinitude of primes in an arith- 
metical progression by elementarj^ methods not using the quadratic reci- 
procity 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,^ pp. 480-96]. 
"Proc. London Math. Soc, 4, 1871, 7; Messenger Math., (2), 1, 1872, 143-4; Coll. Math. Papers, 
2, 1908, 712-3. 
""Comptes Rendus Paris, 74, 1872, 1390. 
">Nouv. Ann. Math., (2), 12, 1873, 323-4. Also, A. Piltz, Diss., Jena, 1884. 
•Improvements in the exposition were made by the editor, Hensel (cf. p. 508). 
«Annan di Mat., (2), 21, 1893, 47-55. 
**Beleuchtung u. Beweis eines Satzes aus Legendre's Zahlentheorie [H, 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. 
**M6ra. couronnes. . .acad. roy. sc. Belgique, 53, 1895-6, No. 6, 24-9. 
*Annales de la soc. sc. de Bru.xeUes, 20, 1896, II, 281-361. Cf. 183-256, 361-397; 21, 1897, I, 
1-13, 60-72; II, 251-368. 
"Sitzungsber. Ak. Wiss. Wien (Math.), 106, 1897, II a, 254-286. Parts published earlier, 
ibid., 104, 1895, II a, 1093-1121, 1158-1166; Jour, fiir Math., 78, 1874, 46-62; 117, 1897, 
169-184. 
