Introduction to Number Theory. by Trygve Nagell
By Trygve Nagell
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The function c (x). - Legendre and Gauss occupied themselves with the problem of finding simple functions which give good approximations to 7r (x) for large values of x. 1 By means of prime tables ' here and in the following, log denotes the natural logarithm. L and du log u (1) 2 But he gave no proof of it. The function (1) is the so-called integral logarithm of X. The first demonstrated results are. r. i (2) < z (x) < _ log a 8 are valid for all sufficiently large values of x. He also showed that the quotient of the numbers :a (x) and log x has the limit I for increasing :r, provided that the limit exists.
T t, Hence, if ('l) is satisfied, y < b. On the other hand we have Theorem 20. If a is a real irrational numlwr, inequality (2) has an infinity of solutions in relatirely prime integers x and y. Proof. Let /1 be a natural number. Applying the result just obtained, we then determine a pair of relatively prime integers x1 and ill such that rI 711 where 1 1 u < "1 tl f' 1 rpl s t1 . Since rc is irrational, 9h 54 0. 2 > 1- and determine the relatively prime in'11 tegers T2 and y2 such that x2 '12 Y2 1 .
1 in the third sutm, etc. Proof. Let A be an object having exactly r of the properties El, E2, E2, etc. Then A contributes 1 to the number N. If r>0, A contributes 1 to r of the numbers Ni. If r> 1, A contributes to (2) = a r (r - 1) of the numbers \;, j . 3 of the numbers 1,j,x, and so on. We finally see that the object A contributes 1-(I) + (')-(3) - =l1 1)r=0 to the sum (5), if r > 0. On the other hand, if r = 0, it contributes 1. Consequently the number of objects having none of the properties is given by (5).