Uniformity of spectra of arithmetic manifolds and the deep Riemann hypothesis

• Project/Area Number: 17K05184
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Toyo University
• Principal Investigator: Koyama Shin-ya 東洋大学, 理工学部, 教授 (50225596)
• Project Period (FY): 2017-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥4,680,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥1,080,000); Fiscal Year 2019: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2017: ¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
• Keywords: ゼータ関数 / セルバーグ・ゼータ関数 / リーマン予想 / 深リーマン予想 / オイラー積 / 素測地線定理 / 素数定理 / 数論的量子カオス

Outline of Final Research Achievements

The main results which we obtained in this research period are as follows: (1) For any Fuchsian group the Euler product of the Selberg zeta function attached with a unitary representation divided by certain explicit terms coming from exceptional zeros and poles converges in the region Re(s)>1/2. In particular, the Euler product of the Selberg zeta function converges in Re(s)>1/2 for any nontrivial irreducible unitary representation under the Selberg 1/4-eigenvalue conjecture. (2) The value of the Euler product obtained in (1) agrees to the value of the Selberg zeta function defined by its analytic continuation.

Academic Significance and Societal Importance of the Research Achievements

リーマン予想が未解決である理由の一つに，予想の定式化が不十分であるという説が長らく唱えられてきた．深リーマン予想はこれを解消するために2010年代に提唱された予想であり，リーマン予想が成り立つ背景にオイラー積の挙動があると主張している．オイラー積の挙動，特に非自明な表現に対するL関数のオイラー積の臨界領域における収束性が示されれば，リーマン予想を証明でき，さらにその成立理由も解明できる．これまで，標数正の関数体上で深リーマン予想の成立が確かめられてきた．本研究では，フックス群のセルバーグ・ゼータ関数に対し，深リーマン予想に相当する命題を証明した．これは，標数0の場合に初めて得られた結果である．

Research Products

• [Journal Article] Estimates of lattice points in the discriminant aspect over abelian extension fields (2017)
  • Author(s): Takeda Wataru、Koyama Shin-ya
  • Journal Title: Forum Mathematicum Volume: 0 Issue: 3 Pages: 1-11
  • DOI: 10.1515/forum-2017-0152
  • Peer Reviewed
• [Presentation] 多重三角関数 ～ その端緒から最近の進展まで (2020)
  • Author(s): 小山信也
  • Organizer: 第2回「多重三角関数とその一般化」
  • Invited
• [Presentation] Convergence of Euler products of the absolute tensor products of L-functions (2019)
  • Author(s): Shin-ya Koyama
  • Organizer: The Riemann-Roch Theorem over the One-Element- Field
  • Int'l Joint Research / Invited
• [Presentation] Applications of Ramanujan's method on the behavior of Euler products to Selberg zeta functions (2019)
  • Author(s): Shin-ya Koyama
  • Organizer: Analytic and Combinatorial Number Theory: The Legacy of Ramanujan
  • Int'l Joint Research
• [Presentation] Euler products of Selberg zeta functions in the critical strip (2018)
  • Author(s): Shin-ya Koyama
  • Organizer: Canadian Number Theory Association IX
  • Int'l Joint Research
• [Presentation] Euler products of Selberg zeta functions in the critical strip (2018)
  • Author(s): Shin-ya Koyama
  • Organizer: Zeta Functions in Okinawa 2018
  • Invited
• [Presentation] セルバーグ・ゼータ関数のオイラー積の収束性 (2018)
  • Author(s): 小山信也・金子生弥
  • Organizer: 日本数学会
• [Presentation] Convergence of Euler products of Selberg zeta funcitons (2017)
  • Author(s): Shin-ya Koyama and Ikuya Kaneko
  • Organizer: Zeta Functions in Okinawa 2017
  • Int'l Joint Research
• [Book] 1日1ページ 数学の教養365 (2020)
  • Author(s): クリフォード・A・ピックオーバー，小山信也（監訳）
  • Total Pages: 408
  • Publisher: ニュートンプレス
  • ISBN: 4315522198
• [Book] ゼータへの招待 (2018)
  • Author(s): 黒川信重・小山信也
  • Total Pages: 168
  • Publisher: 日本評論社