no. 1
Extremal values of Dirichlet L-functions in the half-plane of absolute convergence
Steuding, Jörn1
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, Tome 16 (2004) no. 1, pp. 221-232.

On démontre que, pour tout θ réel, il existe une infinité de s=σ+it avec σ→1+ et t→+∞ tel que

ℜ{exp(iθ)logL(s,χ)}≥loglogloglogt loglogloglogt+O(1).

La démonstration est basée sur une version effective du théorème de Kronecker sur les approximations diophantiennes.

We prove that for any real θ there are infinitely many values of s=σ+it with σ→1+ and t→+∞ such that

ℜ{exp(iθ)logL(s,χ)}≥loglogloglogt loglogloglogt+O(1).

The proof relies on an effective version of Kronecker’s approximation theorem.

DOI : 10.5802/jtnb.444
Steuding, Jörn 1

1 Institut für Algebra und Geometrie Fachbereich Mathematik Johann Wolfgang Goethe-Universität Frankfurt Robert-Mayer-Str. 10 60 054 Frankfurt, Germany 
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Steuding, Jörn. Extremal values of Dirichlet $L$-functions in the half-plane of absolute convergence. Journal de théorie des nombres de Bordeaux, Tome 16 (2004) no. 1, pp. 221-232. doi : 10.5802/jtnb.444. http://www.numdam.org/articles/10.5802/jtnb.444/

[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. | EuDML | JFM

[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. | EuDML | JFM

[3] H. Davenport, H. Heilbronn, On the zeros of certain Dirichlet series I, II. J. London Math. Soc. 11 (1936), 181–185, 307–312. | JFM | MR

[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. | Zbl

[5] K. Ramachandra, On the frequency of Titchmarsh’s phenomenon for ζ(s) - VII. Ann. Acad. Sci. Fennicae 14 (1989), 27–40. | MR | Zbl

[6] G.J. Rieger, Effective simultaneous approximation of complex numbers by conjugate algebraic integers. Acta Arith. 63 (1993), 325–334. | EuDML | MR | Zbl

[7] E.C. Titchmarsh, The theory of functions. Oxford University Press, 1939 2nd ed. | JFM | MR

[8] E.C. Titchmarsh, The theory of the Riemann zeta-function. Oxford University Press, 1986 2nd ed. | MR | Zbl

[9] M. Waldschmidt, A lower bound for linear forms in logarithms. Acta Arith. 37 (1980), 257-283. | EuDML | MR | Zbl

[10] H. Weyl, Über ein Problem aus dem Gebiete der diophantischen Approximation. Göttinger Nachrichten (1914), 234-244.

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