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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
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