Erdős-Macintyre type theorem’s for multiple Dirichlet series: exceptional sets and open problems
Keywords:
Wiman's inequality; Erd\H{o}s-Macintyre theorem; maximal term; maximum modulus; minimum modulus; exceptional set; Dirichlet series
Abstract
In the paper, we formulate some open problems related to the best description of the values of the exceptional sets in Wiman's inequality for entire functions and in the Erd\H{o}s-Macintyre type theorems for entire multiple Dirichlet series. At the same time, we clarify the statement of one \v{I}.V. Ostrovskii problem on Wiman's inequality. We also prove three propositions and one theorem. On the one hand, in a rather special case, these results give the best possible description of the values of the exceptional set in the Erd\H{o}s-Macintyre-type theorem. On the second hand, they indicate the possible structure of the best possible description in the general case.
References
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12. O.B. Skaskiv, D.Yu. Zikrach, On the best possible description of an exceptional set in asymptotic estimates for Laplace–Stieltjes integrals, Mat. Stud., 35 (2011), No2, 131–141.
13. T.M. Salo, O.B. Skaskiv, Minimum modulus of lacunary power series and h-measure of exceptional sets, Ufa Math. J., 9 (2017), No4, 135–144. doi:10.13108/2017-9-4-135
14. A.O. Kuryliak, O.B. Skaskiv, Wiman’s type inequality in multiple-circular domain, Axioms, 2021, 10(4), Article ID: 348 (2021). https://doi.org/10.3390/axioms10040348
15. O.B. Skaskiv, M.R. Lutsyshyn, On the minimum modulus of multiple Dirichlet series, Ukr. Mat. Zh., 44 (1992), No9, 1296–1298 (Ukrainian). English transl.: O.B. Skaskiv, M.R. Lutsyshyn, Minimum of modulus of Dirichlet multisequence, Ukr. Math. J., 44 (1992), No9, 1186–1188. https://doi.org/10.1007/BF01058386
16. M.R. Lutsyshyn, O.B. Skaskiv, Asymptotic properties of a multiple Dirichlet series, Mat. Stud., 3 (1994), 41–48 (in Ukrainian). http://matstud.org.ua/texts/1994/3/3 041-048.pdf
17. O.B. Skaskiv, The maximum modulus and the maximal term of the entire Dirichlet series, Dop. Akad. Nauk Ukrain. SSR, Ser. A, 11 (1984), 22–24 (in Ukrainian).
18. O.B. Skaskiv, M.M. Sheremeta, On the asymptotic behavior of entire Dirichlet series, Sbornik: Mathematics, 59 (1988), No2, 379–396. https://doi.org/10.1070/SM1988v059n02ABEH003141
19. P. Erdös, A.J. Macintyre, Integral functions with gap power series, Proc. Edinburgh Math. Soc. (2), 10 (1953), 62–70.
20. P. Fenton, The minimum modulus of gap power series, Proc. Edinburgh Math. Soc., 21 (1978), No1, 49–54. https://doi.org/10.1017/S001309150001587X
21. Yu. Gal, T. Salo, O. Zhuk, Exceptional sets in Erdős-Fenton’s type theorem for Dirichlet series, International conference “Theory of approximation of functions and its applications” dedicated to the 80th anniversary of Professor Alexander Stepanets (1942-2007) (Lutsk, June 6-10, 2022), Abstracts: Lutsk, 2022, 15–16.
22. M.R. Lutsyshyn, O.B. Skaskiv, On the minimum modulus of a multiple Dirichlet series with monotone coefficients, Journal of Math. Sc., 96 (1999), No2, 2957–2960. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, 40 (1997), No4, 21–25.
23. M.R. Lutsyshyn, Entire Dirichlet series with monotonuos coefficients and complex exponents, Intern. conf. dedicated to Yu.P. Shauder (Lviv, August 23–29 1999): Abstracts: Lviv, 1999, P. 132.
2. H. Wittich, Neuere Untersuchungen über eindeutige analytische Funktionen, Berlin-Göttingen-Heidelberg: Springer-Verlag, 1955.
3. P.C. Rosenbloom, Probability and entire functions, Stud. Math. Anal. and Related Topics, Stanford: Calif. Univ. Press., 1962, 325–332.
4. O.B. Skaskiv, P.V. Filevych, On the size of an exceptional set in the Wiman theorem, Mat. Stud., 12 (1999), No1, 31–36 (in Ukrainian).
5. O.B. Skaskiv, O.V. Zrum, On an exceptional set in the Wiman inequalities for entire functions, Mat. Stud., 21 (2004), No1, 13–24 (in Ukrainian).
6. A.O. Kuryliak, O.B. Skaskiv, Wiman’s type inequality for analytic and entire functions and h-measure of an exceptional sets, Carpathian Math. Publ., 12 (2020), No2, 492–498.
7. W. Bergweiler, On meromorphic function that share three values and on the exceptional set in Wiman-Valiron theory, Codai. Math. J., 13 (1990), No1, 1–9. doi: 10.2996/kmj/1138039154
8. T.M. Salo, O.B. Skaskiv, O.M. Trakalo, On the best possible description of exceptional set in Wiman-Valiron theory for entire function, Mat.Stud., 16 (2001), No2, 131–140.
9. O.B. Skaskiv, O.M. Trakalo, On exceptional set in Borel relation for multiple entire Dirichlet series, Mat. Stud., 15 (2001), No2, 163–172 (in Ukainian).
10. P.V. Filevych An exact estimate for the measure of the exceptional set in the Borel relation for entire functions, Ukrainian Math. J., 53 (2001), No2, 328–332. https://doi.org/10.1023/A:1010489609188
11. O.B. Skaskiv, O.M. Trakalo, Sharp estimate of exceptional set in Borel’s relation for entire functions of several complex variables, Mat. Stud., 18 (2002), No1, 53–56 (in Ukainian).
12. O.B. Skaskiv, D.Yu. Zikrach, On the best possible description of an exceptional set in asymptotic estimates for Laplace–Stieltjes integrals, Mat. Stud., 35 (2011), No2, 131–141.
13. T.M. Salo, O.B. Skaskiv, Minimum modulus of lacunary power series and h-measure of exceptional sets, Ufa Math. J., 9 (2017), No4, 135–144. doi:10.13108/2017-9-4-135
14. A.O. Kuryliak, O.B. Skaskiv, Wiman’s type inequality in multiple-circular domain, Axioms, 2021, 10(4), Article ID: 348 (2021). https://doi.org/10.3390/axioms10040348
15. O.B. Skaskiv, M.R. Lutsyshyn, On the minimum modulus of multiple Dirichlet series, Ukr. Mat. Zh., 44 (1992), No9, 1296–1298 (Ukrainian). English transl.: O.B. Skaskiv, M.R. Lutsyshyn, Minimum of modulus of Dirichlet multisequence, Ukr. Math. J., 44 (1992), No9, 1186–1188. https://doi.org/10.1007/BF01058386
16. M.R. Lutsyshyn, O.B. Skaskiv, Asymptotic properties of a multiple Dirichlet series, Mat. Stud., 3 (1994), 41–48 (in Ukrainian). http://matstud.org.ua/texts/1994/3/3 041-048.pdf
17. O.B. Skaskiv, The maximum modulus and the maximal term of the entire Dirichlet series, Dop. Akad. Nauk Ukrain. SSR, Ser. A, 11 (1984), 22–24 (in Ukrainian).
18. O.B. Skaskiv, M.M. Sheremeta, On the asymptotic behavior of entire Dirichlet series, Sbornik: Mathematics, 59 (1988), No2, 379–396. https://doi.org/10.1070/SM1988v059n02ABEH003141
19. P. Erdös, A.J. Macintyre, Integral functions with gap power series, Proc. Edinburgh Math. Soc. (2), 10 (1953), 62–70.
20. P. Fenton, The minimum modulus of gap power series, Proc. Edinburgh Math. Soc., 21 (1978), No1, 49–54. https://doi.org/10.1017/S001309150001587X
21. Yu. Gal, T. Salo, O. Zhuk, Exceptional sets in Erdős-Fenton’s type theorem for Dirichlet series, International conference “Theory of approximation of functions and its applications” dedicated to the 80th anniversary of Professor Alexander Stepanets (1942-2007) (Lutsk, June 6-10, 2022), Abstracts: Lutsk, 2022, 15–16.
22. M.R. Lutsyshyn, O.B. Skaskiv, On the minimum modulus of a multiple Dirichlet series with monotone coefficients, Journal of Math. Sc., 96 (1999), No2, 2957–2960. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, 40 (1997), No4, 21–25.
23. M.R. Lutsyshyn, Entire Dirichlet series with monotonuos coefficients and complex exponents, Intern. conf. dedicated to Yu.P. Shauder (Lviv, August 23–29 1999): Abstracts: Lviv, 1999, P. 132.
Published
2023-01-23
How to Cite
Bandura, A. I., Salo, T. M., & Skaskiv, O. B. (2023). Erdős-Macintyre type theorem’s for multiple Dirichlet series: exceptional sets and open problems. Matematychni Studii, 58(2), 212-221. https://doi.org/10.30970/ms.58.2.212-221
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