Asymptotic Formula for the Coordinates of the Zeros of Sections of the Zeta Function, zeta(N)(s), Near s = 1.
The sum, Sigma1(N)n(-8) = zetaN(s), s = sigma + it, is called a section of the Riemann zeta function, zeta(s). Here an asymptotic formula, for large N, giving the location of the zeros of zetaN(s) near s = 1 will be demonstrated. In particular these zeros will lie in sigma < 1. Relationships between the location of the zeros of zetaN(s) and the zeros of zeta(s) were discovered by Turán. The location in sigma < 1 will also be demonstrated directly, without the asymptotic formula, together with results valid also when s - 1 is not small.
Levinson
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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass. 02139.