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It was shown by Linnik [10] that there is an absolute constant K such that every sufficiently large even integer can be written as a sum of two primes and at most

For many L-functions of arithmetic interest, the values on or closeto theedgeof theregion of absoluteconvergenceare of great importance, as shown for instance by the proof of the Prime Number Theorem (equivalent to non-vanishing of ζ(s) for Re(s) = 1). Other examples are the Dirichlet L-functions (e.g., because of the Dirichlet class-number formula) and the symmetric square L-functions of classical automorphic forms. For analytic purposes, in the absence of the Generalized Riemann Hypothesis, it is very useful to have an upper-bound, on average, for the number of zeros of the L-functions which are very close to 1. We prove a very general statement of this typefor forms on GL(n)/Q for any n � 1, comparableto the log-free density theorems for Dirichlet characters first proved by Linnik. 1. Introduction.

INTRODUCTION Let a 1 ; a 2 ; a 3 be any nonzero integers such that gcd(a 1 ; a 2 ; a 3 ) = 1: (1.1) Let b be any integer satisfying b j a 1 + a 2 + a 3 (mod2) and gcd(b; a i ; a j ) = 1 for 1 i ! j 3: (1.2) Write A := maxf3; ja 1 j; ja 2 j; ja 3 jg. In [11], M.C. Liu and K. M. Tsang studied a problem of A. Baker [1], namely, the solubility and the size of small prime solutions p 1 ; p 2 ; p 3 of the linear equation a 1 p 1<

Abstract not found

Let N denote the set of positive integers and let Q denote the set of rational numbers. For

Trying to decompose an integer into a product of integers, we feel irritation. There should dwell the reason why any prime appears like a real gem that one can touch and hold. We thus muse ever and again how and when ancient people discovered the way of sifting out primes and began appreciating them. Perhaps those who conceived the divisibility had already some sieves in their minds. Indeed, a wealth of evidences have been excavated supporting our view. The story to be told below must have originated more than five millennia ago 1) , while the primordial intellectual irritation has remained fresh and fundamental till today. The history of Sieve Method is rich and fascinating; we would need a volume to exhaust the story. In the present article we shall instead concentrate onto several pivotal ideas that made the progress possible; so the scope is inevitably limited. Nevertheless, you will encounter instances of precious mathematical achievements that people in the future will certainly continue to relate. Notes are to be read as essential parts, although they are in the style of personal memoranda. Mathematical symbols and definitions are introduced where they are needed for the first time, and will continue to be effective until otherwise stated. Theorems are given somewhat