Limit theorem in the space of continuous functions for the Dirichlet polynomial related with the Riemann zeta-funtion

Antanas Laurinčikas

Journal de théorie des nombres de Bordeaux (1996)

  • Volume: 8, Issue: 2, page 315-329
  • ISSN: 1246-7405

Abstract

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A limit theorem in the space of continuous functions for the Dirichlet polynomial ∑ m ≤ T d κ T ( m ) m σ T + i t where d κ T ( m ) denote the coefficients of the Dirichlet series expansion of the function ζ κ T ( s ) in the half-plane σ > 1 κ T = ( 2 - 1 log l T ) - 1 2 , σ T = 1 2 + 1 n 2 l T l T and l T > 0 , l T ≤ 1n T and l T → ∞ as T → ∞ , is proved.

How to cite

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Laurinčikas, Antanas. "Limit theorem in the space of continuous functions for the Dirichlet polynomial related with the Riemann zeta-funtion." Journal de théorie des nombres de Bordeaux 8.2 (1996): 315-329. <http://eudml.org/doc/247829>.

@article{Laurinčikas1996,
abstract = {A limit theorem in the space of continuous functions for the Dirichlet polynomial\begin\{equation*\} \sum \_\{m \le T\} \frac\{d\_\{\kappa \_T\} (m)\}\{m^\{\sigma \_\{T\}+ it\}\} \end\{equation*\}where $d_\{\kappa _T\} (m)$ denote the coefficients of the Dirichlet series expansion of the function $\zeta ^\{\kappa _T\} (s)$ in the half-plane $\sigma &gt; 1$$\kappa _T = (2^\{-1\} \log l_T)^\{ - \frac\{1\}\{2\}\}$, $\sigma _T = \frac\{1\}\{2\} + \frac\{1n^2 l_T\}\{l_T\}$ and $l_T &gt; 0$, $l_T \le $ 1n $T$ and $l_T \rightarrow \infty $ as $T \rightarrow \infty $, is proved.},
author = {Laurinčikas, Antanas},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {Dirichlet polynomial; Riemann zeta-function; distribution; limit theorem; space of continuous functions},
language = {eng},
number = {2},
pages = {315-329},
publisher = {Université Bordeaux I},
title = {Limit theorem in the space of continuous functions for the Dirichlet polynomial related with the Riemann zeta-funtion},
url = {http://eudml.org/doc/247829},
volume = {8},
year = {1996},
}

TY - JOUR
AU - Laurinčikas, Antanas
TI - Limit theorem in the space of continuous functions for the Dirichlet polynomial related with the Riemann zeta-funtion
JO - Journal de théorie des nombres de Bordeaux
PY - 1996
PB - Université Bordeaux I
VL - 8
IS - 2
SP - 315
EP - 329
AB - A limit theorem in the space of continuous functions for the Dirichlet polynomial\begin{equation*} \sum _{m \le T} \frac{d_{\kappa _T} (m)}{m^{\sigma _{T}+ it}} \end{equation*}where $d_{\kappa _T} (m)$ denote the coefficients of the Dirichlet series expansion of the function $\zeta ^{\kappa _T} (s)$ in the half-plane $\sigma &gt; 1$$\kappa _T = (2^{-1} \log l_T)^{ - \frac{1}{2}}$, $\sigma _T = \frac{1}{2} + \frac{1n^2 l_T}{l_T}$ and $l_T &gt; 0$, $l_T \le $ 1n $T$ and $l_T \rightarrow \infty $ as $T \rightarrow \infty $, is proved.
LA - eng
KW - Dirichlet polynomial; Riemann zeta-function; distribution; limit theorem; space of continuous functions
UR - http://eudml.org/doc/247829
ER -

References

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  2. [2] H. Bohr and B. Jessen, Über die Wertverteilung der Riemannschen Zeta funktion, Zweite Mitteilung, Acta Math.58 (1932),1-55. Zbl0003.38901JFM58.0321.02
  3. [3] B. Jessen and A. Wintner, Distribution functions and the Riemann zeta-function, Trans.Amer.Math.Soc.38 (1935), 48-88. Zbl0014.15401MR1501802JFM61.0462.03
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  6. [7] A.P. Laurincikas, Distribution of values of complex-valued functions, Litovsk. Math. Sb.15 Nr.2 (1975), 25-39, (in Russian); English transl. in Lithuanian Math. J., 15, 1975. Zbl0311.10047MR384720
  7. [8] D. Joyner, Distribution Theorems for L-functions, John Wiley (986). Zbl0609.10032
  8. [9] A.P. Laurincikas, A limit theorem for the Riemann zeta-function close to the critical line. II, Mat. Sb., 180, 6 (1989), 733-+749, (in Russian); English transl. in Math. USSR Sbornik, 67, 1990. Zbl0703.11037MR1015037
  9. [10] A. Laurincikas, A limit theorem for the Riemann zeta-function in the complex space, Acta Arith.53 (1990), 421-432. Zbl0713.11057MR1075034
  10. [11] D.R. Heath-Brown, Fractional moments of the Riemann zeta-function, J.London Math. Soc.24(2) (1981), 65-78. Zbl0431.10024MR623671
  11. [12] A. Ivic, The Riemann zeta-functionJohn Wiley, 1985. Zbl0556.10026MR792089
  12. [13] P. Billingsley, Convergence of Probability Measures, John Wiley, 1968. Zbl0172.21201MR233396
  13. [14] H. Heyer, Probability measures on locally compact groups, Springer-Verlag, Berlin-Heidelberg- New York (1977). Zbl0376.60002MR501241
  14. [15] H.L. Montgomery and R.C. Vaughan, Hilbert's inequality, J. London Math. Soc.8(2) (1974), 73-82. Zbl0281.10021MR337775

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