Event
01_1
| 제출번호(No.) | 0115 |
|---|---|
| 분류(Section) | Contributed Talk |
| 분과(Session) | Algebra (AL) |
| 영문제목 (Title(Eng.)) |
Matrices related to the Mertens function |
| 저자(Author(s)) |
Gi-Sang Cheon1, Hana Kim1 Sungkyunkwan University1 |
| 초록본문(Abstract) | The Redheffer matrix $R_n$ is defined as an $n\times n$ matrix whose $(i,j)$-entry is 1 if either $j=1$ or $i$ divides $j$, and $0$ otherwise. Asymptotic behavior of the determinant of the Redheffer matrix $M(n)=\sum_{k=0}^n \mu(k)$, which is called the Mertens function, is known to be closely related to the Riemann hypothesis. In this talk, we introduce an infinite family of matrices using Riordan matrices in which each has the same determinant as the Redheffer matrix. A Riordan matrix is an infinite lower triangular matrix whose $(i,j)$-entry is the coefficient of $z^i$ in $g(z)f(z)^j$ for $i,j\ge0$, where $g(z)$ and $f(z)$ are formal power series satisfying $g(0)=1$, $f(0)=0$ and $f^\prime(0)=1$. We find the generating function for the characteristic polynomials of the new matrices and a sufficient condition for the Riemann hypothesis using the singular values of the new matrices. |
| 분류기호 (MSC number(s)) |
11M26, 15A18 |
| 키워드(Keyword(s)) | Redheffer matrix, Riordan matrix, equimodular matrices, singular values |
| 강연 형태 (Language of Session (Talk)) |
Korean |