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On the distribution function of additive arithmetical functions and on some related problems

Conferenza tenuta il 31 Gennaio 1956

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Si espongono ricerche e risultati sulla classica questione della esistenza, e delle eventuali proprietà, della funzione distribuzione di una funzione additiva o moltiplicativa.

Si accenna anche ad alcune questioni connesse, quali quella della densità dei numeri abbondanti primitivi.

L’esposizione si chiude con una elencazione di problemi tuttora aperti che si possono prospettare in questo campo di ricerche.

Summarx

In this paper researches and results about the existence and properties of the distribution function of an additive function are illustrated. Some associated questions are also considered; for example the density of primitive abundant numbers.

The paper ends with an enumeration of unsolved problems which can be encountered in these researches.

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References

  1. Much of the material discussed in this paper is also found in the excellent revue article ofM. Kac,Bull. Amer. Math. Soc. 55 (1949), 641–665, see also my paper which will soon appear in the Proc. International Math. Congress of Amsterdam.

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  2. Amer. Journal of Math. 61 (1939), 713–721.

  3. Journal London Math. Soc. 13 (1938), 119–127. The result in question is a specia case of a result of PaulLévy.

  4. Amer. Journal of Math. 61 (1939), 722–725.

  5. Amer. Journal of Math. 62 (1940), 738–742.

  6. Written communication.

  7. Dokladi Akademii Nauk SSSR, 100 (4) (1955), 623–626.

  8. Annals of Math. 47 (1946), 1–20.

  9. Math. Zeitschrift, 28 (1928), 171–200. A later paper ofSchoenberg,Trans Amer. Math. Soc. 39 (1936), 315–330 contains very much more general results.

  10. Berliner Sitzungsberichte (der Akademie) (1934) 830–834.

  11. Journal London Math. Soc. 9 (1934), 278–282.

  12. Journal London Math. Soc. 10 (1935), 49–58.

  13. The proof of this result is not published.

  14. Math. Annalen, 110 (1934), 336–341.

  15. Bull. Amer. Math. Soc. 54 (1948), 685–692.

  16. Acta Arithmetica, 2 (1936), 147–151, see alsoIndian Journal of Math. 15 (1951), 19–24.

  17. A simple argument shows that if the conditiontj ≥ai is replaced bytj ≥0 the logarithmic density in question does not always exist.

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Authors and Affiliations

  1. Technion, Haifa, Israel

    P. Erdös

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  1. P. Erdös
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Erdös, P. On the distribution function of additive arithmetical functions and on some related problems. Seminario Mat. e Fis. di Milano 27, 45–49 (1957). https://doi.org/10.1007/BF02922565

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