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Almost even arithmetical functions on semigroups. (English. Russian original) Zbl 0577.10039

Lith. Math. J. 25, No. 2, 128-136 (1985); translation from Litov. Mat. Sb. 25, No. 2, 90-101 (1985).
Starting from papers by J. Knopfmacher [Ann. Mat. Pura Appl. (4) 109, 177–201 (1976; Zbl 0337.10033)] and W. Schwarz and J. Spilker [Nieuw Arch. Wiskd. (3) 19, 198–209 (1971; Zbl 0221.10007); Colloq. Math. Soc. János Bolyai 13, 315–357 (1976; Zbl 0343.10038)] and continuing own investigations [Litov. Mat. Sb. 25, No. 1, 72–83 (1985; Zbl 0567.10039)], the author studies the space \({\mathcal B}^ u\) of almost-even functions, defined on an arithmetical semigroup \(S\); these functions are arbitrarily close to “even” functions \((f(n)=f(\gcd (n,k))\) for some \(k\)) with respect to the norm \(\| f\| =\sup_{n}| f(n)|.\)
By Gelfand’s theory, \({\mathcal B}^ u\) is isomorphic to the space \({\mathcal C}(\varepsilon S)\) of continuous functions on a compactification \(\varepsilon S\) of \(S\); the author shows that \(\varepsilon S\) is isomorphic to a product space, and he characterizes the additive resp. multiplicative functions in \({\mathcal B}^ u\) by conditions concerning the values of these functions at “prime” powers.
Next the author uses a sequence \(\{\mu_ x\}\) of measures on \(\{S, {\mathcal P}(S)\}\) to define the mean values \[ M(f;\{\mu_ x\})=\lim_{x\to \infty}\int_{S}f(m)\, d\mu_ x. \]
Extending \(\mu_ x\) to a measure \(\mu^*_ x\) on \(\varepsilon S\), it is shown that the sequence \(\mu^*_ x\) converges weakly to a product measure \(\lambda\), and the mean-value \(M(f;\{\mu_ x\})\) equals \(\int_{\epsilon S}f \,d\lambda\). Using the (semi-)norm \[ \| f\|_{q,\gamma_ x}=(\lim_{x\to \infty}\int_{S}| f|^ q \,d\gamma_ x)^{1/q}, \] the author defines the spaces \({\mathcal B}^ q(\gamma_ x)\) of almost-even functions (modulo null-functions) and shows the completeness of these spaces. Moreover these spaces are isomorphic to \(L^ q(\varepsilon S,\lambda)\). Finally for functions in \({\mathcal B}^ 1\) the existence of a limit-distribution is shown.

MSC:

11N99 Multiplicative number theory
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
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