I shall report in a series of articles my investigation on the famous 150 years old conjecture made by Bernhard Riemann (1826-1866) in 1859. The purposes of the reports are threefold: First, this writing effort will force me to study thoroughly prior arts on this subject, which should help my research towards a possible proof of the Riemann hypothesis (RH). Second, when I submit new results to mathematical journals and/or post them on arXiv (an online publication website) , I can refer to these articles on my website as for the background information, which will allow my journal paper(s) to be more concise than would be, otherwise. Third, if I am successful in compiling prior arts and my own results in a cohesive manner, this series of articles will serve as core materials for a book or monograph.
Furthermore, I hope that this series will stimulate some readers and make them interested in this fascinating and profound subject.
No. 1: Euler’s zeta function
No. 2: Riemann’s zeta function and the Riemann hypothesis
No. 3: Riemann’s xi function and the product form representation
No. 4: The Fourier transform representation of the xi function
No. 5: Some results on the ξ(s) and Ξ(t) functions associated with Riemann’s ζ(s) function
No. 6: Local Extrema of the Ξ(t) Function and The Riemann Hypothesis
No. 7: The Z(t) function, Gram’s Law, Riemann-von Mangoldt Formula, and Lehmer’s Phenomenon
No. 8: Negativity of d2/dt2 logΞ(t) and a conjecture on a sufficient condition for the Riemann hypothesis
No. 9: Application of the Euler-Maclaurin summation to log-differentials of M(t) = |ζ(1/2 + it)|
Hisashi Kobayashi
December 20, 2016
Leonhard Euler (1707-1783) Source: Wikipedia
Carl Friedrich Gauss (1777-1855)
Source: Wikipedia
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