136 HISTORY OF THE THEORY OF NUMBERS. [CHAP, v Then Generalizing pt(s), let n(k\s) be zero if the expansion of the product 11(1— p)*, extended over all primes py does not contain a term equal to s, but let it equal the coefficient of s if s occurs in the expansion. Then The n-rowed determinant in which the element in the rth row and sth column is Fm_!(S), where d is the g. c. d. of r, s, is proved equal to Fm(l) Fm(2) . . .FJ(n), a generalization of Smith's39 theorem. Finally, the right member being r(n), 2ty(d)/d, 2<r(d)/d for r = 0, 1, -1. G. Landsberg96a gave a simple proof of Moreau's72 formula for the number of circular permutations. L. Carlini97 proved Dirichlet's21 formula by noting that (8) ' n(s*-l)=0 7»-l has unity as an n-fold root, while a root ^ 1 of xh— 1 is a root of [n/h] factors xth — 1. Hence the <l>(h) primitive roots of 2A = 1 furnish <l>(h)[n/h] roots of (8). M. Lerch98 found the number N of positive integers ^m which have no one of the divisors a, b, . . . , fc, 2, the latter being relatively prime in pairs and having m as their product. Let F(x) = 1 or 0, according as x is fractional or integral. Let L = db...k. Then [Dirichlet33] L E. Landau99 proved that the inferior limit for x— oo of -<I X is e~c, where C is Euler^s constant. Hence 4>(x) is comprised between this inferior limit and the maximum x — 1. R. Occhipinti100 proved that, if a,- is an nth root of unity, and if dil} . . . , dat are the divisors of i, j-Ht«l w°Archiv Math. Phys., (3), 3, 1902, 152-4. "Periodico di Mat., 17, 1902, 329. "Prag Sitzungsber., 1903, II. "Archiv Math. Phys., (3), 5, 1903, 86-91. 100Periodico di Mat., 19, 1904, 93. Handbuch,113 1, 217-9.