New developments in applications of exponential sums in number theory

2022 Fiscal Year Final Research Report

• Project/Area Number: 19K23402
• Research Category: Grant-in-Aid for Research Activity Start-up
• Allocation Type: Multi-year Fund
• Review Section: 0201:Algebra, geometry, analysis, applied mathematics,and related fields
• Research Institution: Rikkyo University (2021-2022); Nagoya University (2019-2020)
• Principal Investigator: SUZUKI Yuta 立教大学, 理学部, 助教 (30852199)
• Project Period (FY): 2019-08-30 – 2023-03-31
• Keywords: 解析的整数論 / 指数和 / 素数 / 篩法 / 無理数論 / 滑らかな数 / 鞍点法

Outline of Final Research Achievements

Improved or extended some preceding results on statistical results used as the basic theory of exponential sums: 1. Distribution of the product of two primes. Obtained an asymptotic formula continuously covering the results of Decker-Moree, Justus and Landau. 2. Application of the sieve method in irrationality. Extended the method of Chowla-Erdos on the irrationality of the Lambert series to the form applicable to the sum of the reciprocals of Lucas sequences. 3. Distribution of smooth numbers. Obtained a further asymptotic expansion of the uniform asymptotic formula of Hildebrand and Tenenbaum for smooth numbers.

Free Research Field

解析的整数論

Academic Significance and Societal Importance of the Research Achievements

指数和の評価に必要な, 素数の積の分布, 篩法, 滑らかな数といった主題に関する結果を, 先行結果に比べて, 制限が少なく, 適用範囲が広く, 得られる情報の多い形に改善することができたため, 指数和の評価法のより柔軟な基礎理論を提供できたと言える. 本研究には, これら新しい基礎理論により今後の指数和の評価法の発展や応用に寄与することが期待できるという学術的意義がある.