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A matrix inequality for Möbius functions. (English) Zbl 1190.15024

The authors consider an integer matrix \(\Gamma_n=(\gamma_{ij})\) defined by
\[ \gamma_{ij}=\begin{cases} \text{mod}(j,2)-1,&\text{if}\;i=1\;\text{and}\;2\leq j\leq n;\\ \text{mod}(j,i+1)-\text{mod}(j,i),&\text{if}\;2\leq i\leq n-1\;\text{and}\;1\leq j\leq n;\\ 1,&\text{if}\;(i,j)\in\{(1,1),(n,1)\};\\ 0,&\text{otherwise}. \end{cases} \]
By using an LU decomposition of \(\Gamma_n\), they first prove that
\[ \text{det}\Gamma_n=n!\sum_{k=1}^n\frac{\mu(k)}{k}\quad (n\geq 2), \]
where \(\mu\) is the Möbius function, and then obtain a sufficient condition for the Prime Number Theorem and the Riemann Hypothesis in terms of the smallest singular value of the factor \(U\). At the end of this paper, an alternative proof of R. Redheffer’s theorem [ISNM 36, 213–216 (1977; Zbl 0363.65062)] is also given based on an LU decomposition of the Redheffer’s matrix.

MSC:

15A45 Miscellaneous inequalities involving matrices
15A15 Determinants, permanents, traces, other special matrix functions
15A18 Eigenvalues, singular values, and eigenvectors
11A25 Arithmetic functions; related numbers; inversion formulas
11C20 Matrices, determinants in number theory
15B36 Matrices of integers

Citations:

Zbl 0363.65062
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Online Encyclopedia of Integer Sequences:

Infinite Redheffer matrix read by upwards antidiagonals.