Chap. XVIII] INFINITUDE OF PRIMES. 417 
F. Mertens^^ gave a proof, still simpler than his" earlier one, of the 
difficult point in Dirichlet's-^ proof. The proof is very elementary, involv- 
ing computations of finite sums. 
F. Mertens^^ gave a simplification of Dirichlet's^ proof of his general- 
ization to complex primes. 
H. Teege^° proved the difficult point in Dirichlet's^^ proof. 
E. Landau'*^ proved that the number of prime ideals of norm ^ a: of an 
algebraic field equals the integral-logarithm Li(x) asymptotically. By 
specialization to the fields defined by V — 1 or V — 3? we derive theorems'*^ 
on the number of primes 4:k^l or 6A;=fcl ^x. 
L. E. Dickson^^ asked if aifi+bi {i = l,. . ., m) represent an infinitude of 
sets of m primes, noting necessary conditions. 
H. Weber^^ proved Dirichlet's^^ theorem on complex primes. 
E. Landau^^ simplified the proofs by de la Vallee-Poussin^^ and Mertens.^* 
E. Landau*^'^^ simplified Dirichlet's^^ proof. Landau^^ proved that, 
if k, I are relatively prime, the number of primes ky-\-l^x is 
where 7 is a constant depending on k. For see Pfeiffer^° of Ch. X. 
A. Cunningham*^ noted that, of the N primes ^R, approximately 
N/c{)(n) occur in the progressions nx-\-a, a<n and prime to n, and gave a 
table showing the degree of approximation when R = 10^ or 5-10^, with 
n even and < 1928. Within these limits there are fewer primes nx-\-l than 
primes nx+a, a>l. 
Infinitude of Primes Represented by a Quadratic Form. 
G. L. Dirichlet^^ gave in sketch a proof that every properly primitive 
quadratic form {a, h, c), a, 26, c with no common factor, represents an infini- 
tude of primes. 
Dirichlet^^ announced the extension that among the primes represented 
by {a, b, c), an infinitude are representable by any given linear form Mx-\-N, 
with M, N relatively prime, provided a, h, c, M, N are such that the linear 
and quadratic forms can represent the same number. 
»8Sitzungsber. Ak. Wiss. Wien (Math.), 108, 1899, II a, 32-37. 
"/bid., 517-556. Polish transl. in Prace mat. fiz., 11, 1900, 194-222. 
"Mitt. Math. GeseU. Hamburg, 4, 1901, 1-11. 
*iMath. Amialen, 56, 1903, 665-670. 
*?Sitzungsber. Ak. Wiss. Wien (Math.), 112, 1903, II a, 502-6. 
"Messenger Math., 33, 1904, 155. 
"Jour, fur Math., 129, 1905, 35-62. Cf. p. 48. 
«Sitzungsber. Akad. Berlin, 1906, 314-320. 
«Rend. Circ. Mat. Palermo, 26, 1908, 297. 
"Handbuch . . .Verteilung der Primzahlen, I, 1909, 422-35. 
"Sitzungsber. Ak. Wiss. Wien (Math.), 117, 1908, Ila, 1095-1107. 
"Proc. London Math. Soc, (2), 10, 1911, 249-253. 
"Bericht Ak. Wiss. Berhn, 1840, 49-52; Werke, 1, 497-502. Extract in Jour, fiir Math., 21, 
1840, 98-100. 
"^Comptes Rendus Paris, 10, 1840, 285-8; Jour, de Math., 5, 1840, 72-4; Werke, 1, 619-623. 
