5 research outputs found

    Rationality of the zeta function of the subgroups of abelian pp-groups

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    Given a finite abelian pp-group FF, we prove an efficient recursive formula for σa(F)=∑H≤F∣H∣a\sigma_a(F)=\sum_{\substack{H\leq F}}|H|^a where HH ranges over the subgroups of FF. We infer from this formula that the pp-component of the corresponding zeta-function on groups of pp-rank bounded by some constant rr is rational with a simple denominator. We also provide two explicit examples in rank r=3r=3 and r=4r=4 as well as a closed formula for σa(F)\sigma_a(F)

    Some discussions on the Goldbach conjecture

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    According to some discussions based on syllogism, we present results on the binary Goldbach conjecture in three categories: results that are weaker than the Goldbach conjecture, sufficient conditions for the Goldbach conjecture, and results that are similar in nature to the Goldbach conjecture. Additionally, we explore the connections between the Goldbach conjecture and other well-known conjectures.Comment: Some typos corrected. Many references added. We apologize for any unintentional omissions and recognize the significant contributions made by numerous researchers to the study of the Goldbach conjecture. Any comments or suggestions are welcome

    Quaternion Algebras

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    This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout
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