Contributions to the founding of the theory of transfinite by Georg Cantor

Number Theory

By Georg Cantor

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7) For a proof see A. V. Malysev's paper [19]. 4. 3,for 0 < q1 ~ q we have L exp 27Tibx' = Bq112T(q) In q(b, q)112. 8) CHAPTER II LEMMAS FROM THE ANALYTIC THEORY OF QUADRATIC FORMS 1. Fundamental Equations of the Dispersion Method for the Divisor Problem In the divisor problem (see Chapter III) the sequence {

5) Taking this into account, we shall consider only those solutions for which x < y, z < t. 4) will be 4 times smaller than the number of solutions of the original equation 38 ANALYTIC THEORY OF QUADRATIC FORMS [CHAP. 5)). 6). d. 7) will be important. Clearly b JM. First we suppose b = 1. = z. 1). Since v1 is prime, (v1 , z) = 1. 3) we find y = Mv~x'(mod v2z), where v:tx' is the inverse of v1x mod v2z. 3) with the conditions on it. 1), xy ~ 2n; zt ~ 2n follows from the fundamental equation. 6) we have < x < J1n; 0 0 < z < J1n.

In doing this all constants K1 , • . 2) (cf. 4)). Furthermore, we shall assume that K 4 > 100K1 • o 7. Comparison of the Results for Coherent Numbers Let (n1 , n 2) be one of the three pairs of coherent numbers described in § 2. 12) for each of the given pairs. , < n•&; Z1 (Xi. 3) Sec. 9). 4). 9) into consideration, we obtain the error in the total number of solutions (for given CJ). Recalling that v0 = v0/(ln n)K 1 (cf. 12). 7) This circumstance enables us to carry out the summation with respect to "'i.

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