Some properties of the Riemann zeta-function and cyclicity in weighted Hardy spaces

V. Dilnyi

We consider some reformulation of the Riemann Hypothesis in terms of cyclic functions in a weighted Hardy space with an exponential weight. We indicate a distinctive feature of the weighted case from the classical one. Also we consider connection of the Riemann Hypothesis with a convolution type equation in the semistrip.

1. A.N. Parshin, Zeta function, in: Matem. encyklopedia, v.2 D. – Koo. – Moscow: Soviet. encyklopedia, 1979. – P.111–122.

2. B. Nyman, On some groups and semigroups of translations, Thesis, Uppsala, 1950.

3. A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta math., 81 (1949), 239–255.

4. P. Lax, Translation invariant subspaces, Acta math., 101 (1959), 163–178.

5. N.K. Nikolski, Operatos, functions and systems: an easy reading, V.1, AMS, 2002.

6. V.I. Vasyunin, On a biorthogonal system associated with the Riemann hypothesis, St. Petersburg Math. Journ., 7 (1996), №3, 405-–419.

7. J.-F. Burnol, An adelic causality problem related to abelian L–functions, J. Number Theory, 87 (2001), 432–428.

8. H.S. Shapiro, Weakly invertible elements in certain function spaces and generators in l1, Mich Math J., 11 (1964), 161–165.

9. P. Lax, Functional analysis, Wiley Interscience, 2002.

10. A. Shields, Weighted shift operators and analytic function theory, in Topics in Operator Theory, Math Surveys, AMS, Providence, 13 (1974), 49–128.

11. B. Vinnitskii, On zeros of functions analytic in a half plane and completeness of systems of exponents, Ukr. Math. Jour. 46 (1994), №5, 484–500.

12. R.E.A.C. Paley, N. Wiener, Fourier transforms in complex domain (AMS Colloq. Publ. XIX), Providence: AMS, 1934.

13. V. Dil’nyi, On the equivalence of some conditions for weighted Hardy spaces, Ukr. Math. Jour., 58 (2006), №9, 1425–1432.

14. V. Dilnyi, On cyclic functions in weighted hardy spaces, Journ. of Math. Phys., Anal., Geom., 7 (2011), №1, 19–33.

15. B. Vinnitskii, V. Dil’nyi, A generalization of the Beurling-Lax theorem, Math. Notes, 79 (2006), №3–4, 335–341.

16. H. Dwight, Tables of integrals and other mathematical data, 4-th ed., The Macmillan Company, New York, 1961.

17. M.A. Fedorov, A.F. Grishin, Some questions of the Nevanlinna theory for the complex half-plane, Math. Physics, Anal. and Geom., 1 (1998), 223–271.

18. P. Koosis, Introduction to Hp spaces, Second edition. Cambridge Tracts in Mathematics, 115, Cambridge University Press, Cambridge, 1998.

19. A. M. Sedleckii, An equivalent definition of Hp spaces in the half-plane and some applications, Math. USSR Sb., 25 (1975), №1, 69–76.

20. B. Vinnitsky, On solutions of homogeneous convolution equation in one class of functions analytic in a semistrip, Mat. Stud., 7 (1997), №1, 41–52. (in Ukrainian)

21. B.V. Vynnytskyi, V.M. Dil’nyi, On necessary conditions for existence of solutions of convolution type equation, Mat. Stud., 16 (2001), №1, 61–70. (in Ukrainian)

22. A.A. Karatsuba, S.M. Voronin, The Riemann zeta-function, Valter de Gruyter, 1992.

23. B.V. Vynnyts’kyi, V.M. Dil’nyi, On solutions of homogeneous convolution equation generated by singularity, Mat. Stud., 19 (2003), №2, 149–155.

	Some properties of the Riemann zeta-function and cyclicity in weighted Hardy spaces(in Ukrainian)
Author	V. Dilnyi dilnyi@ukr.net Ivan Franko National University of Lviv
Abstract	We consider some reformulation of the Riemann Hypothesis in terms of cyclic functions in a weighted Hardy space with an exponential weight. We indicate a distinctive feature of the weighted case from the classical one. Also we consider connection of the Riemann Hypothesis with a convolution type equation in the semistrip.
Keywords	outer function; cyclic function; Riemann zeta-function; weighted Hardy space
Reference	1. A.N. Parshin, Zeta function, in: Matem. encyklopedia, v.2 D. – Koo. – Moscow: Soviet. encyklopedia, 1979. – P.111–122. 2. B. Nyman, On some groups and semigroups of translations, Thesis, Uppsala, 1950. 3. A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta math., 81 (1949), 239–255. 4. P. Lax, Translation invariant subspaces, Acta math., 101 (1959), 163–178. 5. N.K. Nikolski, Operatos, functions and systems: an easy reading, V.1, AMS, 2002. 6. V.I. Vasyunin, On a biorthogonal system associated with the Riemann hypothesis, St. Petersburg Math. Journ., 7 (1996), №3, 405-–419. 7. J.-F. Burnol, An adelic causality problem related to abelian L–functions, J. Number Theory, 87 (2001), 432–428. 8. H.S. Shapiro, Weakly invertible elements in certain function spaces and generators in l1, Mich Math J., 11 (1964), 161–165. 9. P. Lax, Functional analysis, Wiley Interscience, 2002. 10. A. Shields, Weighted shift operators and analytic function theory, in Topics in Operator Theory, Math Surveys, AMS, Providence, 13 (1974), 49–128. 11. B. Vinnitskii, On zeros of functions analytic in a half plane and completeness of systems of exponents, Ukr. Math. Jour. 46 (1994), №5, 484–500. 12. R.E.A.C. Paley, N. Wiener, Fourier transforms in complex domain (AMS Colloq. Publ. XIX), Providence: AMS, 1934. 13. V. Dil’nyi, On the equivalence of some conditions for weighted Hardy spaces, Ukr. Math. Jour., 58 (2006), №9, 1425–1432. 14. V. Dilnyi, On cyclic functions in weighted hardy spaces, Journ. of Math. Phys., Anal., Geom., 7 (2011), №1, 19–33. 15. B. Vinnitskii, V. Dil’nyi, A generalization of the Beurling-Lax theorem, Math. Notes, 79 (2006), №3–4, 335–341. 16. H. Dwight, Tables of integrals and other mathematical data, 4-th ed., The Macmillan Company, New York, 1961. 17. M.A. Fedorov, A.F. Grishin, Some questions of the Nevanlinna theory for the complex half-plane, Math. Physics, Anal. and Geom., 1 (1998), 223–271. 18. P. Koosis, Introduction to Hp spaces, Second edition. Cambridge Tracts in Mathematics, 115, Cambridge University Press, Cambridge, 1998. 19. A. M. Sedleckii, An equivalent definition of Hp spaces in the half-plane and some applications, Math. USSR Sb., 25 (1975), №1, 69–76. 20. B. Vinnitsky, On solutions of homogeneous convolution equation in one class of functions analytic in a semistrip, Mat. Stud., 7 (1997), №1, 41–52. (in Ukrainian) 21. B.V. Vynnytskyi, V.M. Dil’nyi, On necessary conditions for existence of solutions of convolution type equation, Mat. Stud., 16 (2001), №1, 61–70. (in Ukrainian) 22. A.A. Karatsuba, S.M. Voronin, The Riemann zeta-function, Valter de Gruyter, 1992. 23. B.V. Vynnyts’kyi, V.M. Dil’nyi, On solutions of homogeneous convolution equation generated by singularity, Mat. Stud., 19 (2003), №2, 149–155.
Pages	115-122
Volume	41
Issue	2
Year	2014
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html

Some properties of the Riemann zeta-function and cyclicity in weighted Hardy spaces(in Ukrainian)