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Let $h$ be a complex valued multiplicative function. For any $N\in \mathbb{N}$, we compute the value of the determinant $D_N:={det}_{i|N,j|N}\left(\frac{h\left(\left(i,j\right)\right)}{ij}\right)$ where $\left(i,j\right)$ denotes the greatest common divisor of $i$ and $j$, which appear in increasing order in rows and columns. Precisely we prove that $$D_N=\prod _{p{}^l\parallel N}{\left(\frac{1}{p^{l\left(l+1\right)}}\stackrel{l}{\prod _{i=1}}\left(h\left(p^i\right)-h\left(p^{i-1}\right)\right)\right)}^{\tau \left(N/p^l\right)}.$$ This means that $D_N^{1/\tau \left(N\right)}$ is a multiplicative function of $N$. The algebraic apparatus associated with this result allows us to prove the following two results. The first one is the characterization of real multiplicative functions $f\left(n\right)$, with $0\le f\left(p\right)<1$, as minimal values of certain...

We express $\mathrm{\Delta }\left(x,N\right)$, as defined in the title, for $x=\frac{a}{q}$ and $q$ prime in terms of values of characters modulo $q$. Using this, we show that the universal lower bound for $\mathrm{\Delta }\left(N\right)={sup}_{x\in \mathbb{R}}\left|\mathrm{\Delta }\left(x,N\right)\right|$ can, in general, be substantially improved when $N$ is composed of primes lying in a fixed residue class modulo $q$. We also prove a corresponding improvement when $N$ is the product of the first s primes for infinitely many natural numbers $s$.

Let X₁ and X₂ be complex Banach spaces, and let A₁ ∈ BL(X₁), A₂ ∈ BL(X₂), A₃ ∈ BL(X₁,X₂) and A₄ ∈ BL(X₂,X₁). We propose an iterative procedure which is a modified form of Newton's iterations for obtaining approximations for the solution R ∈ BL(X₁,X₂) of the Riccati equation A₂R - RA₁ = A₃ + RA₄R, and show that the convergence of the method is quadratic. The advantage of the present procedure is that the conditions imposed on the operators A₁, A₂, A₃, A₄ are weaker than the corresponding conditions...

Many discrepancy principles are known for choosing the parameter α in the regularized operator equation $(T*T+\alpha I)x_{\alpha }^{\delta }=T*y^{\delta }$, $|y-y^{\delta }|\le \delta$, in order to approximate the minimal norm least-squares solution of the operator equation Tx = y. We consider a class of discrepancy principles for choosing the regularization parameter when T*T and $T*y^{\delta }$ are approximated by Aₙ and $z{ₙ}^{\delta }$ respectively with Aₙ not necessarily self-adjoint. This procedure generalizes the work of Engl and Neubauer (1985), and particular cases of the results are applicable...

It is known that the nonlinear nonhomogeneous backward Cauchy problem $u_t\left(t\right)+Au\left(t\right)=f(t,u\left(t\right))$, $0\le t<\tau$ with $u\left(\tau \right)=\phi$, where $A$ is a densely defined positive self-adjoint unbounded operator on a Hilbert space, is ill-posed in the sense that small perturbations in the final value can lead to large deviations in the solution. We show, under suitable conditions on $\phi$ and $f$, that a solution of the above problem satisfies an integral equation involving the spectral representation of $A$, which is also ill-posed. Spectral truncation is used...