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Almost All Roots of zeta(s) = a Are Arbitrarily Close to sigma = 1/2
1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass. 02139.
Abstract:
Let s = sigma + it. For any complex a, all but O(1/log log T) of the roots of zeta(s) = a in T < t < 2T lie in|sigma - 1/2| < (log log T)(2)/log T. The results extend easily to other functions satisfying a functional equation such as the Dirichlet L-functions, the Lerch functions, etc.
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