Read e-book online Analytic Number Theory- Jia & Matsumoto PDF

Read e-book online Analytic Number Theory- Jia & Matsumoto PDF

By Chaohua Jia, Kohji Matsumoto

ISBN-10: 1402005458

ISBN-13: 9781402005459

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

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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. Lehmer's congruence. As applications, we mention some generalizations of Morley's congruence and Jacobstahl's Theorem to modulo arbitary positive integers.

Corollary 7. If p > 5 is prime, then Acknowledgments The author is very grateful to Prof. Andrew Granville for his constructive comments and valuable suggestions. and References for any 12 1. Corollary 8. If p, q > 5 are distinct primes, then In 1862, Wolstenholme showed that ( ) 1 (mod p3) [I] T. Cai and A. Granville, O n the residue of binomial coeficients and their products modulo prime powers, preprint . [2] J. W. L. Glaisher, Quart. J. , 32 (1901), 271-305. [3] R. Guy, Unsolved problems i n number theory, Springer-Verlag, Second Edition, 1994.

Hence, for almost all n , we conclude that < whenever z > w Now we put > 2. 4) and < (x, l-I(z)) = 1. while we have w,(p) = 1 for p z > w 2 wo we have C wlp Y. 1, to obtain a lower bound for R(n), for almost all n. 23). 1 and Mertens' formula, we consequently see that for almost all n, for almost all n , where whence with a certain sequence (Adyh) satisfying (Ad,h( 1. 5 that (E0(n)l<< NW (log N ) - , ~ , < n wSp

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Analytic Number Theory- Jia & Matsumoto by Chaohua Jia, Kohji Matsumoto

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