Displaying similar documents to “On the mean value of the remainder term of the prime number formula”

A zero density result for the Riemann zeta function

Habiba Kadiri (2013)

Acta Arithmetica

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We prove an explicit bound for N(σ,T), the number of zeros of the Riemann zeta function satisfying ℜ𝔢 s ≥ σ and 0 ≤ ℑ𝔪 s ≤ T. This result provides a significant improvement to Rosser's bound for N(T) when used for estimating prime counting functions.

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

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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.