By Chaohua Jia, Kohji Matsumoto

ISBN-10: 1402005458

ISBN-13: 9781402005459

Contains a number of survey articles on top numbers, divisor difficulties, and Diophantine equations, in addition to study papers on a number of points of analytic quantity thought difficulties.

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Then we may confirm that < so that &(n) = &(n; [O, 11) = Rd(n; 1)31) This time we set s = 1, k = 2, > + Rd(n;m). 2). nd that for every integer n with N 5 n 5 (6/5)N. To facilitate our subsequent description, we denote by N(5) the set of all the odd integers in the interval [N, (6/5)N], and put N(4) = Nl n > for all primes p and integers 1 1. 10). 4, respectively. We next discuss a lower bound for Pl(n, Y). 4, we have PI(n, Y) > n (zp2)-' n (1 - 120p-') n (I - 1 2 0 ~ - ' / ~ ) , 74 ANALYTICNUMBER THEORY Ternary problems in additive prime number theory 75 for n E M(kl ) .

Thus we have for k 5 5 (see Vaughan [30], Ch. 2). 14) that I2<< N $ + f (log N ) - ~ A . 19), we obtain which implies the conclusion of the lemma immediately. 3. 2, and let D = X! with 0 < 0 < 5/12. Then one has L~'(a)h~(a)g3(a;~~ 516 2 Proof. Write ij3 = 93(a,; ) ) d o (< N V ( l o g ~ ) - ~ ~ ~ . 22) and the last inequality that for short, and set which gives the lemma. 4. For a given sequence (Ad) satisfying [Ad[ 5 1, define and let D = X! with 0 < I3 < 113. Then one has 65 Ternary problems in additive prime number theory Next we note that G3(a) << ~ ~ ( afor) all ~ a/ E~[O, 11.

15] D. R. Heath-Brown, Three primes and an almost prime in arithmetic progression. J. London Math. Soc. (2) 23 (l98l), 396-414. [I61 C. Hooley, On a new approach to various problems of Waring 's type. Recent progress in analytic number theory (Durham, 1979), Vol. 1. Academic Press, London-New York, 1981, 127-191. [17] L. K. Hua, Additive Theory of Prime Numbers. Amer. Math. , Providence, Rhode Island, 1965. [18] H. Iwaniec, Primes of the type 4(x, y) form. Acta Arith. 21 (1972), 203-224. A + A, where 4 is a quadratic [19] H.

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