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| Y. Motohashi | |
|---|---|
| Name | Y. Motohashi |
| Birth date | 1946 |
| Birth place | Japan |
| Fields | Mathematics |
| Alma mater | University of Tokyo |
| Doctoral advisor | Atle Selberg |
| Known for | Motohashi's formula, analytic number theory |
Y. Motohashi is a Japanese mathematician noted for contributions to analytic number theory, particularly the spectral theory of automorphic forms and applications to prime number problems and L-functions. He developed techniques connecting the theory of the Riemann zeta function with the spectral decomposition of automorphic kernels, producing deep relations between the distribution of primes and the theory of modular forms, Maass forms, and trace formulas. His work influenced researchers working on the Riemann Hypothesis, the Ramanujan–Petersson conjecture, the Selberg trace formula, and the analytic theory surrounding Dirichlet L-series.
Motohashi was born in Iwate Prefecture and undertook his undergraduate and graduate studies at the University of Tokyo, where he studied under influential figures in analytic and spectral theory. During his graduate training he was exposed to ideas from Kenkichi Iwasawa, Katsuro Hayashi, and international scholars such as Atle Selberg and Harald Bohr through seminars and correspondence. His doctoral work combined classical techniques from the theory of the Riemann zeta function with the burgeoning spectral methods arising from the Selberg trace formula and the theory of automorphic representations.
Motohashi held academic appointments at several institutions, including posts associated with the University of Tokyo and research affiliations with national institutes such as the Science Council of Japan and the Institute for Advanced Study. He collaborated with researchers at the University of Cambridge, the Max Planck Institute for Mathematics, and the California Institute of Technology, participating in workshops at the International Congress of Mathematicians and symposia organized by the Mathematical Society of Japan. Over his career he supervised doctoral students who went on to positions at institutions like Kyoto University, Osaka University, and Nagoya University, and he served on editorial boards for journals including the Journal of Number Theory and the Proceedings of the Japan Academy.
Motohashi is best known for establishing a formula—now called Motohashi's formula—that relates the fourth moment of the Riemann zeta function on the critical line to a spectral fourth moment of automorphic L-functions. This connection uses the Kuznetsov trace formula, the Petersson trace formula, and variants of the Selberg trace formula, blending ideas from Atle Selberg, Henryk Iwaniec, E. C. Titchmarsh, and A. Selberg's school. His approach provided new proofs and refinements of moment results originally studied by John E. Littlewood, G. H. Hardy, and J. E. Littlewood's circle of influence, and it offered tools for attacking subconvexity bounds for L-functions analogous to work by Roger Heath-Brown, H. Iwaniec, and P. Sarnak.
Motohashi made explicit the spectral decomposition of shifted convolution sums related to coefficients of modular forms and Maass cusp forms, clarifying relations previously implicit in the work of I. Piatetski-Shapiro, H. Jacquet, and D. Bump. His techniques influenced breakthroughs on mean values of L-functions by researchers such as K. Soundararajan, D. Goldfeld, and Z. Rudnick, and they played a role in progress on the Lindelöf hypothesis in special cases. Motohashi also contributed to the analytic theory of the Class number problem and studied correlations of arithmetic functions linking to earlier investigations by P. Erdős, G. H. Hardy, and S. Ramanujan.
He produced explicit spectral identities that have been applied to problems on prime gaps, prime distributions in arithmetic progressions, and zero-density estimates for the Riemann zeta function and Dirichlet L-series, intersecting with lines of research pursued by H. Montgomery, A. Selberg, N. Levinson, and K. Ramachandra. His work has been cited in contexts involving the Grand Riemann Hypothesis and connections between automorphic spectra and arithmetic quantum chaos as studied by P. Sarnak and E. B. Bogomolny.
- Y. Motohashi, "Spectral Theory of the Riemann Zeta-Function", treatise developing his spectral approach connecting the Riemann zeta function with automorphic spectra; influential for researchers following the methods of Atle Selberg and H. Iwaniec. - Y. Motohashi, papers on the fourth moment of the Riemann zeta function and applications of the Kuznetsov trace formula; cited alongside work by Roger Heath-Brown and H. Iwaniec. - Y. Motohashi, articles on shifted convolution sums for coefficients of modular forms and Maass forms, expanding the analytic toolkit of I. Piatetski-Shapiro and H. Jacquet. - Y. Motohashi, expository and research contributions in collected volumes for the International Congress of Mathematicians and proceedings of the Mathematical Society of Japan.
Motohashi received recognitions from national and international bodies, including awards from the Mathematical Society of Japan, invitations to speak at the International Congress of Mathematicians, and fellowships associated with the Japan Society for the Promotion of Science and the Institute for Advanced Study. He has been acknowledged in commemorative volumes alongside figures such as Atle Selberg, H. Iwaniec, and P. Sarnak for his influence on modern analytic number theory.
Category:Japanese mathematicians Category:Analytic number theorists