Extremal values of Dirichlet L -functions in the half-plane of absolute convergence

Jörn Steuding[1]

  • [1] Institut für Algebra und Geometrie Fachbereich Mathematik Johann Wolfgang Goethe-Universität Frankfurt Robert-Mayer-Str. 10 60 054 Frankfurt, Germany

Journal de Théorie des Nombres de Bordeaux (2004)

  • Volume: 16, Issue: 1, page 221-232
  • ISSN: 1246-7405

Abstract

top
We prove that for any real θ there are infinitely many values of s = σ + i t with σ → 1 + and t → + ∞ such that ℜ { exp ( i θ ) log L ( s , χ ) } ≥ log log log log t log log log log t + O ( 1 ) . The proof relies on an effective version of Kronecker’s approximation theorem.

How to cite

top

Steuding, Jörn. "Extremal values of Dirichlet $L$-functions in the half-plane of absolute convergence." Journal de Théorie des Nombres de Bordeaux 16.1 (2004): 221-232. <http://eudml.org/doc/249253>.

@article{Steuding2004,
abstract = {We prove that for any real $\theta $ there are infinitely many values of $s=\sigma +it$ with $\sigma \rightarrow 1+$ and $t\rightarrow +\infty $ such that\[ \Re \lbrace \exp (i\theta )\log L(s,\chi )\rbrace \ge \log \{\log \log \log t \over \log \log \log \log t\}+O(1).\]The proof relies on an effective version of Kronecker’s approximation theorem.},
affiliation = {Institut für Algebra und Geometrie Fachbereich Mathematik Johann Wolfgang Goethe-Universität Frankfurt Robert-Mayer-Str. 10 60 054 Frankfurt, Germany},
author = {Steuding, Jörn},
journal = {Journal de Théorie des Nombres de Bordeaux},
language = {eng},
number = {1},
pages = {221-232},
publisher = {Université Bordeaux 1},
title = {Extremal values of Dirichlet $L$-functions in the half-plane of absolute convergence},
url = {http://eudml.org/doc/249253},
volume = {16},
year = {2004},
}

TY - JOUR
AU - Steuding, Jörn
TI - Extremal values of Dirichlet $L$-functions in the half-plane of absolute convergence
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2004
PB - Université Bordeaux 1
VL - 16
IS - 1
SP - 221
EP - 232
AB - We prove that for any real $\theta $ there are infinitely many values of $s=\sigma +it$ with $\sigma \rightarrow 1+$ and $t\rightarrow +\infty $ such that\[ \Re \lbrace \exp (i\theta )\log L(s,\chi )\rbrace \ge \log {\log \log \log t \over \log \log \log \log t}+O(1).\]The proof relies on an effective version of Kronecker’s approximation theorem.
LA - eng
UR - http://eudml.org/doc/249253
ER -

References

top
  1. H. Bohr, E. Landau, Über das Verhalten von ζ ( s ) und ζ ( k ) ( s ) in der Nähe der Geraden σ = 1 . Nachr. Ges. Wiss. Göttingen Math. Phys. Kl. (1910), 303–330. Zbl41.0290.01
  2. H. Bohr, E. Landau, Nachtrag zu unseren Abhandlungen aus den Jahren 1910 und 1923. Nachr. Ges. Wiss. Göttingen Math. Phys. Kl. (1924), 168–172. Zbl50.0233.01
  3. H. Davenport, H. Heilbronn, On the zeros of certain Dirichlet series I, II. J. London Math. Soc. 11 (1936), 181–185, 307–312. Zbl62.0138.01MR20578
  4. R. Garunkštis, On zeros of the Lerch zeta-function II. Probability Theory and Mathematical Statistics: Proceedings of the Seventh Vilnius Conf. (1998), B.Grigelionis et al. (Eds.), TEV/Vilnius, VSP/Utrecht, 1999, 267–276. Zbl0997.11070
  5. K. Ramachandra, On the frequency of Titchmarsh’s phenomenon for ζ ( s ) - VII. Ann. Acad. Sci. Fennicae 14 (1989), 27–40. Zbl0628.10041MR997968
  6. G.J. Rieger, Effective simultaneous approximation of complex numbers by conjugate algebraic integers. Acta Arith. 63 (1993), 325–334. Zbl0788.11024MR1218460
  7. E.C. Titchmarsh, The theory of functions. Oxford University Press, 1939 2nd ed. Zbl0022.14602MR197687
  8. E.C. Titchmarsh, The theory of the Riemann zeta-function. Oxford University Press, 1986 2nd ed. Zbl0601.10026MR882550
  9. M. Waldschmidt, A lower bound for linear forms in logarithms. Acta Arith. 37 (1980), 257-283. Zbl0357.10017MR598881
  10. H. Weyl, Über ein Problem aus dem Gebiete der diophantischen Approximation. Göttinger Nachrichten (1914), 234-244. Zbl45.0325.01

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.