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# A remark on a paper of Luca

Acta Mathematica Hungarica
Author: Imre Kátai

## Summary

It is proved that the set of those natural numbers which cannot be written as n-Ω(n) is of positive lower density. Here Ω(n) is the number of the prime power divisors of n. This is a refinement of a theorem of F. Luca.

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# A remark on trigonometric sums

Acta Mathematica Hungarica
Author: Imre Kátai

Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $S(x,\alpha\mid X_p):=\sum_{\substack{p_1p_2<x\\ p_1<p_2}} X_{p_1}X_{p_2}e^{2\pi i\alpha p_1p_2},\qquad \pi_2(x)=\sum_{\substack{p_1p_2<x\\ p_1<p_2}} 1,$ \end{document}$where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$p$\end{document}, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$p_1$\end{document}, \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$p_2$\end{document} run over the prime numbers. It is proved that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\max_{\substack{|X_p|\le 1\\ p}} \frac{{S(x,\alpha,X_p)}}{{\pi_2(x)}} =\Delta(x,\alpha)\to 0\qquad(x\to\infty)$$ \end{document} for almost all irrational \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$\alpha$\end{document}. Restricted access # A remark on product partition Acta Mathematica Hungarica Authors: Imre Kátai and M. V. Subbarao ## Summary The asymptotic expansion of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$\sum\limits_{n\le x}e^*(n)$\end{document} is given, where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$e^*(n)$\end{document} is defined by \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document}$\sum{\frac{{e^*(n)}}{{n^s}}} = \prod\limits_{n=2}^\infty (1-1/n^s)\$ \end{document}.

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# Arithmetical Functions Commutable with Sums of Squares II

Mathematica Pannonica
Authors: Imre Kátai and Bui Minh Phong

We give all functions ƒ , E: ℕ → ℂ which satisfy the relation

for every a, b, c ∈ ℕ, where h ≥ 0 is an integers and K is a complex number. If n cannot be written as a2 + b2 + c2 + h for suitable a, b, c ∈ ℕ, then ƒ (n) is not determined. This is more complicated if we assume that ƒ and E are multiplicative functions.

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# Influence of alternative plant nutrition methods on soil microbial characteristics in long-term experiments

Agrokémia és Talajtan
Authors: János Kátai, Thomas Döring, Magdolna Tállai, Andrea Balla-Kovács, István Henzsel, Marianna Makádi, Zsolt Sándor, and Imre Vágó

The size of the arable land is constantly decreasing all over the world due to severe anthropogenic disorders. Plant production therefore has to be adapted to changing environmental conditions along with the proper selection of crop varieties and the application of sustainable environmental technologies which also consider economic aspects. The investigations were carried out in the Westsik long-term fertilization experiment near Nyíregyháza, East Hungary, which was set up in 1929 (89 years ago). Alternative forms of nutrient supplies (A) (green manure, straw with and without fermentation, organic fertilizer with and without inorganic fertilizer supplements) were used in different crop rotations. The test plant was potato (Solanum tuberosum L.) and the soil type sand with a low humus content (Arenosols). A further long-term experiment is located on calcareous chernozem soil (Chernozems) in Debrecen (set up in 1983, 35 years ago). In one part of this experiment, organic farming (OF) has been carried out with a pea, winter wheat and maize crop rotation for over 15 years with no inorganic fertilization. In another block in this experiment, changes in soil properties as a result of the medium and high doses of fertilizers applied in intensive farming (I) were evaluated with a maize (Zea mays L.) monoculture as the test plant.

The results obtained with alternative nutrient supplies (green manure, fermented and unfermented straw, farmyard manure, fertilization) proved that the soil organic carbon content increased to varying degrees in humus-poor, acidic sand soil. The organic matter content of the soils increased in response to the treatments, contributing to a significant enhancement in soil microbial parameters (MBC, saccharase, dehydrogenase and phosphatase enzyme activities).

The carbon dioxide production and saccharase enzyme activity in organic plots (OF) were significantly lower than in intensively farmed (I) soils. At the same time, in the case of organic farming (OF) the microbial biomass carbon, phosphatase and dehydrogenase activity were significantly higher in OF plots than in I plots. Compared to the control soil, MBC was 7-8 times higher in organic plots and 1.3-3.8 times higher in intensive plots.

Organic farming on chernozem soil generally resulted in higher microbial activity (MBC, phosphatase, saccharase and dehydrogenase enzyme activity) than in either intensively farmed chernozem or in the case of alternative farming (A) on sandy soil.

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