| Project/Area Number |
15K04785
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Ehime University |
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Principal Investigator |
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| Project Period (FY) |
2015-04-01 – 2018-03-31
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| Project Status |
Completed (Fiscal Year 2017)
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| Budget Amount *help |
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2017: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2016: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
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| Keywords | ゼータ関数 / ゼータ正規化積 / ラプラシアンの行列式 / ラプラシアン / 行列木定理 / ラマヌジャングラフ / 素測地線定理 / グラフの熱核 / 超幾何関数 |
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| Outline of Final Research Achievements |
The determinant of the graph Laplacian is an important example of the zeta regularized products. In this research, we study the determinants for various graphs via a harmonic analytical method established by Chinta, Jorgenson and Karlsson in 2010. Note that, in the method, the heat kernel on graphs plays a crucial rule. For example, we obtain a refinement of the prime geodesic theorem for discrete tori, which is not of the form of an asymptotic formula. In this context, we also study Ramanujan graphs, whose associated Ihara zeta function satisfies the “Riemann hypothesis”. We show that the problem finding Ramanujan graphs around the complete graph in a family of Cayley graphs of generalized quaternion group is related to the classical Hardy-Littlewood conjecture for primes represented by a quadratic polynomial.
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