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On the exact location of the non-trivial zeros of Riemann's zeta function

Juan Arias de Reyna, Jan van de Lune (2014)

Acta Arithmetica

We introduce the real valued real analytic function κ(t) implicitly defined by e 2 π i κ ( t ) = - e - 2 i ϑ ( t ) ( ζ ' ( 1 / 2 - i t ) ) / ( ζ ' ( 1 / 2 + i t ) ) (κ(0) = -1/2). By studying the equation κ(t) = n (without making any unproved hypotheses), we show that (and how) this function is closely related to the (exact) position of the zeros of Riemann’s ζ(s) and ζ’(s). Assuming the Riemann hypothesis and the simplicity of the zeros of ζ(s), it follows that the ordinate of the zero 1/2 + iγₙ of ζ(s) is the unique solution to the equation κ(t) = n.

On the first sign change in Mertens' theorem

Jan Büthe (2015)

Acta Arithmetica

The function p x 1 / p - l o g l o g ( x ) - M is known to change sign infinitely often, but so far all calculated values are positive. In this paper we prove that the first sign change occurs well before exp(495.702833165).

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