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CHAP, xviii]                     INFINITUDE OF PRIMES.                                417
F. Mertens38 gave a proof, still simpler than his37 earlier one, of the difficult point in Dirichlet's23 proof. The proof is very elementary, involving computations of finite sums.
F. Mertens39 gave a simplification of Dirichlet's24 proof of his generalization to complex primes.
H. Teege40 proved the difficult point in Dirichlet's23 proof.
E. Landau41 proved that the number of prime ideals of norm ^xof an algebraic field equals the integral-logarithm Li(x) asymptotically. By specialization to the fields defined by V — 1 or V— 3> w® derive theorems42 on the number of primes 4fc=±=l or 6fc=t=l ^x.
L. E. Dickson43 asked if o^n+ft,- (t~l,. . ., iri) represent an infinitude of sets of m primes, noting necessary conditions.
H. Weber44 proved Dirichlet's24 theorem on complex primes.
E. Landau45 simplified the proofs by de la Valle'e-Poussin35 and Mertens.38
E. Landau46'47 'simplified Dirichlet's23 proof. Landau48 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 0 see Pfeiffer90 of Ch. X.
A. Cunningham49 noted that, of the N primes ^E, approximately N/<j>(n) occur in the progressions nx+a, a<n and prime to n, and gave a table showing the degree of approximation when jR = 105 or 5-105, with n even and < 1928. Within these limits there are fewer primes nx+1 than primes ncc+a, a> 1.
G. L. Dirichlet65 gave in sketch a proof that every properly primitive quadratic form (a, 6, c), a, 26, c with no common factor, represents an infinitude of primes.
Dirichlet56 announced the extension that among the primes represented by (a, 6, c), an infinitude are representable by any given linear form Mx+N, with My N relatively prime, provided a, 6, 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.
"Ibid., 517-556.   Polish transl. in Prace mat. fiz., 11, 1900, 194r-222.
40Mitt. Math. Gesell. Hamburg, 4, 1901, 1-11.
"Math. Axmalen, 56, 1903, 665-670.
«Sitzungsber. Ak. Wiss. Wien (Math.), 112, 1903, II a, 502-6.
"Messenger Math., 33, 1904, 155.
"Jour, flir Math., 129, 1905, 35-62.    Cf. p. 48.
46Sitzungsber. Akad. Berlin, 1906, 314-320.
"Rend. Circ. Mat. Palermo, 26, 1908, 297.
47Handbuch . . .Verteilung der Primzahlen, I, 1909, 422-35.
"Sitzungsber. Ak. Wiss. Wien (Math.), 117, 1908, Ha, 1095-1107.
"Proc. London Math. Soc., (2), 10, 1911, 249-253.
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