A prime number theorem in the theory of the Riemann zeta function.
Akio Fujii (1979)
Journal für die reine und angewandte Mathematik
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Akio Fujii (1979)
Journal für die reine und angewandte Mathematik
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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.
Tsz Ho Chan (2004)
Acta Arithmetica
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Tsz Ho Chan (2004)
Acta Arithmetica
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H. M. Bui (2014)
Acta Arithmetica
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Assuming the Riemann Hypothesis we show that there exist infinitely many consecutive zeros of the Riemann zeta-function whose gaps are greater than 2.9 times the average spacing.
D. Heath-Brown (1982)
Acta Arithmetica
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Laurinčikas, Antanas, Steuding, Jörn (2004)
Publications de l'Institut Mathématique. Nouvelle Série
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V. Kumar Murty (1994)
Forum mathematicum
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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 (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.
Shaoji Feng (2005)
Acta Arithmetica
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J. Kaczorowski, W. Staś (1988)
Colloquium Mathematicae
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R. R. Hall (2006)
Acta Arithmetica
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Yuichi Kamiya, Masatoshi Suzuki (2004)
Publications de l'Institut Mathématique
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