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Associate Professor, Faculty of Mathematics, Kyushu University *Profile is at the time of the award.
2024Inamori Research GrantsScience & Engineering
Recently there has been a huge progress in the study of gaps between prime numbers. It has now been proven that we have a bounded gap between prime numbers and this breakthrough was made by my collaborator D.A. Goldston in his work with J. Pintz and C.Y. Yildirim in late 2000's. Based on that work, explicit bounds have been obtained and we got very close to the truth of the Twin Prime Conjecture, which simply states that there are infinitely many prime numbers of gap 2. This includes the work of Y. Zhang, J. Maynard (a 2022 Fields medalist), and the huge project group Polymath which is led by T. Tao (a 2006 Fields medalist). Goldston and I are digging into this problem further and we are currently working with J. Schettler. Unfortunately, not much progress has been done and we want to sit together and focus on this work. With this grant I am going to visit them in San Jose, California, to work on this problem.
In this research, we studied the spacing distribution of the non-trivial zeros of the Riemann zeta-function, called “pair correlation” of the zeros. For five decades, all related research has been conducted under the assumption of the Riemann Hypothesis (RH) due to the difficulty in controlling the horizontal distribution of these zeros. Through this research, the applicants have succeeded in removing the RH assumption from this methodology. Consequently, we have recovered the known result under RH regarding proportion of simple zeros, under an assumption weaker than the RH. Furthermore, the zeros we detected are not only simple but also satisfy RH, thus for the first time, the pair correlation method gives the proportion of zeros which are both simple and on the critical line. Based on very recent investigations, we also introduced a new concept called “horizontal multiplicity” which outlines our method better.
Science & Engineering
