By Chaohua Jia, Kohji Matsumoto

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

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**Sample text**

Note that For a character x modulo p and an integer m, we write As for the Gauss sum T(X,I ) , we know that IT(x, 1)1 = p1I2, when x is non-principal. It is also easy to observe that when x is non-principal, we have T(X,m) = ~ ( m ) r ( x1). , When x is principal, on the other hand, we see that T(X,m ) = p - 1 or -1 depending on whether plm or not. In particular, we have for any character x modulo p and any integer m. 3 of Vaughan [30], we know that whenever p { a, and obviously Sf (p, a ) = Sl (p, a ) - 1.

19) we obtain Now we apply Lemma 1 to estimate the upper bound of S(C, P, 2).

1. 19531010, 19801021) 99 C. Jia and K. ), Analytic Number Theory, 99-1 19. 100 ANALYTIC NUMBER THEORY where On Chen's theorem 101 where w(d) is a multiplicative function, 0 5 w(p) < p, X is independent of d. Then The oringinal proof of Chen's was simplified by Pan Chengdong, Ding Xiaqi, Wang ~ u a n w Halberst , a m - ~ i c h e r t [ ~~]a, l b e r s t a m [ ~~]o, s s [ ~In l. 689 and a detail proof was given in (51. In page 338 of [4] it says: "It would be interesting to know whether the more elaborate weighting procedure could be adapted to the numerical improvements and could be important".