By Chaohua Jia, Kohji Matsumoto

ISBN-10: 1402005458

ISBN-13: 9781402005459

Comprises numerous survey articles on leading numbers, divisor difficulties, and Diophantine equations, in addition to study papers on a variety of elements of analytic quantity concept difficulties.

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**Topics in Analytic Number Theory**

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These functions are defined by for 0 < u getting r(5) = 15 and r(4) = 6. 26) is satisfied with r = r(kl). 27), namely, < 2, and then by the difference-differentia1 equations for u 2 2. It is known that 0 5 +o(u) 5 1 5 +l(u) for u > 0, and that 2e7 2e7 f o r 0 < u < 3 . 1) 76 Ternary problems in additive prime number theory ANALYTIC NUMBER T H E O R Y Then we refer to Iwaniec's linear sieve in the following form (see Iwaniec [19], or [20], for a proof). 1. Let Q, U, V, X be real numbers >_ 1, and suppose that D = UV is suficiently large.

36 (1980), 125-141. [28] R. C. Vaughan, A ternary additive problem. Proc. London Math. (3) 41 (1980), 516-532. [29] R. C. Vaughan, Sums of three cubes. Bull. London Math. Soc. 17 (1985), 17-20. (301 R. C. , Cambridge Univ. Press, 1997. A GENERALIZATION OF E. R. China Keywords: quotients of Euler, Bernoulli polynomials, binomial coefficients Abstract In this paper, we announce the result that for any odd n (n-1)/2 1 -292 (n) + nq:(n) > 1, (mod n2), where g,(n) = (r4(n)- l ) / n , (r,n) = 1 is Euler's quotient of n with base r , which is a generalization of E.

6. For a given sequence (Ad) satisfying IAdl 5 1, define and observe that V << ( D + x2)lI4+€&+ v~/~v:/*, whence By an application of Holder's inequality, we see and let D = X! with 0 3 5 k 5 5, one has < 0 < 17/30. Then for each integer k with < We remark that when k 4, the restriction on 0 in this lemma can be relaxed to 0 < t9 < 213, as the following proof shows. 9) for the other integrals. 19) and the simple estimate we observe that Ternary problems in additive prime number theory 5. 69 WEIGHTED SIEVES Having finished the preparation concerning the Hardy-Littlewood circle method, we may proceed to application of sieve theory, and in this section we appeal to weighted linear sieves.