Advertisement

The Riemann Zeta-function: Approximation of Analytic Functions

  • Antanas LaurinčikasEmail author
Conference paper
Part of the Trends in Mathematics book series (TM)

Abstract

In the paper, a short survey on the theory of the Riemann zetafunction is given. The main attention is given to universality-approximation of analytic functions by shifts of the Riemann zeta-function. This includes the effectivization problem, generalization for other zeta-functions, joint universality as well as some applications.

Keywords

Approximation of analytic functions joint universality Riemann zeta-function universality. 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [G.D. Birkhoff (1929)] G.D. Birkhoff, D´emonstration d’un th´eor`eme ´el´ementaire sur les fonctions enti`eres, Comptes Rendus Acad. Sci. Paris 189 (1929), 473–475. 100Google Scholar
  2. [K.M. Bitar, N.N. Khuri, H.C. Ren (1991)] K.M. Bitar, N.N. Khuri, H.C. Ren, Path integrals and Voronin’s theorem on the universality of the Riemann zeta-function, Ann. Phys. 211(1)( 1991), 151–175. 112Google Scholar
  3. [J.B. Conrey (1989)] J.B. Conrey, More than two fifths of the zeros of the Riemann zetafunctions are on the critical line, J. Reine Angew. Math. 399 (1989), 1–26. 98Google Scholar
  4. [R. Garunkˇstis (2003)] R. Garunkˇstis, The effective universality theorem for the Riemann zeta-function, in: Special Activity in Analytic Number Theory and Diophantine Equations, Proc. Workshop at Plank-Institut Bonn 2002, D. R. Heath-Brown and B. Moroz (eds.), Bonner Math. Schriften 360, 2003. 102Google Scholar
  5. [R. Garunkˇstis, A. Laurinˇcikas, K. Matsumoto, J. Steuding, R. Steuding (2010)] R. Garunkˇstis, A. Laurinˇcikas, K. Matsumoto, J. Steuding, R. Steuding, Effective uniform approximation by the Riemann zeta-function, Publ. Mat. 54 (2010), 209– 219. 102Google Scholar
  6. [A. Good (1981)] A. Good, On the distribution of the values of Riemann’s zeta-function, Acta Arith. 38 (1981), 347–388. 102Google Scholar
  7. [J. Hadamard (1896)] J. Hadamard, Sur les z´eros de la fonction 𝜁(𝑠)de Riemann, Comptes Rendus Acad. Sci. Paries 122 (1896), 1470–1473. 97Google Scholar
  8. [G.H. Hardy (1914)] G.H. Hardy, Sur les z´eros de la fonction de Riemann, Comptes Rendus Acad. Sci. Paries 158 (1914), 1012–1014. 98Google Scholar
  9. [O. H¨older (1887)] O. H¨older, ¨Uber die Eigenschaft der Gammafunktion keinen algebraischen Differentialgleichungen zu gen¨ugen, Math. Ann. 28 (1887), 1–13. 111Google Scholar
  10. [M.N. Huxley (2005)] M.N. Huxley, Exponential sums and the Riemann zeta-function, V, Proc. London Math. Soc. 90 (2005), 1–41. 99Google Scholar
  11. A. Iviˇc, (1985) The Riemann Zeta-function, the Theory of the Riemann Zetafunction with Applications. Wiley, New York, p 112Google Scholar
  12. [A. Javtokas, A. Laurinˇcikas (2006)] A. Javtokas, A. Laurinˇcikas, Universality of the periodic Hurwitz zeta-function, Integral Transforms Spec. Funct. 17 (10)( 2006), 711– 722. 106Google Scholar
  13. [J. Kaczorowski (2009)] J. Kaczorowski, Some remarks on the universality of periodic 𝐿- functions, in: New Directions in Value Distribution Theory of Zeta and 𝐿-Functions, W¨urzburg Conf. 2008, Shaker Verlag, Aachen, 2009. 105Google Scholar
  14. [A. Laurinˇcikas (1996)] A. Laurinˇcikas, Limit Theorems for the Riemann Zeta-Function, Kluwer, Dordrecht, 1996. 99, 101Google Scholar
  15. [A. Laurinˇcikas (2010)] A. Laurinˇcikas, Joint universality of zeta-functions with periodic coefficients, Izv. Math.74:3 (2010)515–539. 109Google Scholar
  16. [A. Laurinˇcikas, R. Garunkˇstis (2002)] A. Laurinˇcikas, R. Garunkˇstis, The Lerch Zeta- Function, Kluwer, Dordrecht, 2002. 104Google Scholar
  17. [A. Laurinˇcikas, K. Matsumoto (2001)] A. Laurinˇcikas, K. Matsumoto, The universality of zeta-functions attached to certain cusp forms, Acta Arith. 98(2001), 345–359. 107Google Scholar
  18. [A. Laurinˇcikas, D. ˇSiauˇciunas (2006)] A. Laurinˇcikas, D. ˇSiauˇciunas, Remark on the universality of the periodic zeta-function, Math. Notes, 80 (3–4)( 2006), 252–538. 105Google Scholar
  19. [N. Levinson (1974)] N. Levinson, More than one third of the zeros of Riemann’s zetafunction are on 𝜎 = 1/2, Adv. Math. 13 (1974), 383–436. 98Google Scholar
  20. [J. van de Lune, H.J.J. te Riele, D.T. Winter (1986)] J. van de Lune, H.J.J. te Riele, D.T. Winter, On the zeros of the Riemann zeta-function in the critical strip, IV, Math. Comp. 46 (1986), 667–681. 98Google Scholar
  21. [H. von Mangoldt (1895)] H. von Mangoldt, Zu Riemann’s Abhandlung ¨uber die Anzahl der Primzahlen unter einer gegebenen Gr¨osse, J. Reine Angew. Math. 114 (1895), 255–305. 98Google Scholar
  22. [J. Marcinkiewicz (1935)] J. Marcinkiewicz, Sur les nombres d´eriv´es, Fund. Math. 24 (1935), 305–308. 100Google Scholar
  23. [S.N. Mergelyan (1951)] S.N. Mergelyan, On the representation of functions by series of polynomials on closed sets, Doklady Akad. Nauk SSSR (N.S.) 78 (1951), 405–408 (in Russian). 95Google Scholar
  24. [S.N. Mergelyan (1952)] S.N. Mergelyan, Uniform approximations to functions of a complex variable, Usp. Mat. Nauk. 7 (1952), 31–122 (in Russian). 95Google Scholar
  25. [The Millennium Prize problems (2006)] The Millennium Prize problems, J. Carlson, A. Joffe and A. Wiles (eds.), Clay Math. Inst., Cambridge, Massachusetts, 2006. 98Google Scholar
  26. [B. Riemann (1859)] B. Riemann, ¨Uber die Anzahl der Primzahlen unterhalb einer gegebenen Gr¨osse, Monatsber. Preuss. Akad. Wiss. Berlin (1859), 671–680. 97Google Scholar
  27. [A. Selberg (1942)] A. Selberg, On the zeros of Riemann’s zeta-function, Skr. Norske Vid. Akad. Oslo 10 (1942), 1–59. 98Google Scholar
  28. [J. Steuding (2003)] J. Steuding, Upper bounds for the density of universality, Acta Arith. 107 (2003), 195–202. 102Google Scholar
  29. [J. Steuding (2007)] J. Steuding, Value-Distribution of 𝐿-Functions, Lectures Notes Math. 1877, Springer, Berlin, 2007. 101Google Scholar
  30. [C.J. de la Vall´ee-Poussin (1896)] C.J. de la Vall´ee-Poussin, Recherches analytiques sur la th´eorie des nombres premiers, I–III, Ann. Soc. Bruxelles 20 (1896), 183–256, 281– 362, 363–397. 97Google Scholar
  31. [S.M. Voronin (1975)] S.V. Voronin, Theorem on the “universality” of the Riemann zetafunction, Izv. Akad. Nauk SSSR, ser. matem. 39 (1975), 475–486 (in Russian) Math. USSR Izv. 9 (1975), 443–455. 100Google Scholar
  32. [S.M. Voronin (1975)] S.M. Voronin, On the functional independence of Dirichlet 𝐿- functions, Acta Aritm. 27 (1975), 493–503 (in Russian). 107, 108Google Scholar
  33. [S.M. Voronin (1973)] S.M. Voronin, On the differential independence of 𝜁-functions, Dokl. AN SSSR, 209(6) (1973), 1264–1266, (in Russian). 111Google Scholar
  34. [J.L. Walsh (1960)] J.L. Walsh, Interpolation and Approximation by Rational Functions in the Complex Domain, Amer. Math. Soc. Coll. Publications, Vol. XX, 1960. 95Google Scholar

Copyright information

© Springer Basel 2012

Authors and Affiliations

  1. 1.Faculty of Mathematics and InformaticsVilnius UniversityVilniusLithuania

Personalised recommendations