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, Volume 8 (1996) no. 2, pp. 315-329.

A limit theorem in the space of continuous functions for the Dirichlet polynomial ∑ m≤T d κ T (m) m σ T +it 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 logl T ) -1 2 , σ T =1 2+1n 2 l T l T and l T >0, l T ≤ 1n T and l T →∞ as T→∞, is proved.

Dans cet article on prouve un théorème limite dans l’espace des fonctions continues pour le polynôme de Dirichlet ∑ m≤T d κ T (m) m σ T +it où d κ T (m) sont les coefficients du développement en série de Dirichlet de la fonction ζ κ T (s) dans le demi-plan σ>1, κ T =(2 -1 logl T ) -1 2 , σ T =1 2+log 2 l T l T , l T >0, l T ≤logT et l T →∞ lorsque T→∞.

@article{JTNB_1996__8_2_315_0,
     author = {Antanas Laurin\v{c}ikas},
     title = {Limit theorem in the space of continuous functions for the {Dirichlet} polynomial related with the {Riemann} zeta-funtion},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {315--329},
     publisher = {Universit\'e Bordeaux I},
     volume = {8},
     number = {2},
     year = {1996},
     doi = {10.5802/jtnb.171},
     zbl = {0871.11059},
     mrnumber = {1438472},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.171/}
}
TY  - JOUR
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
DA  - 1996///
SP  - 315
EP  - 329
VL  - 8
IS  - 2
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.171/
UR  - https://zbmath.org/?q=an%3A0871.11059
UR  - https://www.ams.org/mathscinet-getitem?mr=1438472
UR  - https://doi.org/10.5802/jtnb.171
DO  - 10.5802/jtnb.171
LA  - en
ID  - JTNB_1996__8_2_315_0
ER  - 
%0 Journal Article
%T Limit theorem in the space of continuous functions for the Dirichlet polynomial related with the Riemann zeta-funtion
%J Journal de Théorie des Nombres de Bordeaux
%D 1996
%P 315-329
%V 8
%N 2
%I Université Bordeaux I
%U https://doi.org/10.5802/jtnb.171
%R 10.5802/jtnb.171
%G en
%F JTNB_1996__8_2_315_0
Antanas Laurinčikas. 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, Volume 8 (1996) no. 2, pp. 315-329. doi : 10.5802/jtnb.171. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.171/

[1] H. Bohr and B. Jessen, Über die Wertverteilung der Riemannschen Zeta funktion, Ernste Mitteilung, Acta Math. 51 (1930), 1-35. | JFM: 56.0287.01

[2] H. Bohr and B. Jessen, Über die Wertverteilung der Riemannschen Zeta funktion, Zweite Mitteilung, Acta Math. 58 (1932),1-55. | JFM: 58.0321.02 | Zbl: 0003.38901

[3] B. Jessen and A. Wintner, Distribution functions and the Riemann zeta-function, Trans.Amer.Math.Soc. 38 (1935), 48-88. | JFM: 61.0462.03 | MR: 1501802 | Zbl: 0014.15401

[4] V. Borchsenius and B. Jessen, Mean motions and values of the Riemann zeta-function, Acta Math. 80 (1948), 97-166. | MR: 27796 | Zbl: 0038.23201

[5] A. Laurinčikas, Limit theorems for the Riemann zeta-function on the complex space, Prob. Theory and Math. Stat., 2, Proceedings of the Fifth Vilnius Conference, VSP/Mokslas (1990), 59-69. | MR: 1153860 | Zbl: 0733.11030

[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. | MR: 384720 | Zbl: 0311.10047

[8] D. Joyner, Distribution Theorems for L-functions, John Wiley (986). | Zbl: 0609.10032

[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. | MR: 1015037 | Zbl: 0703.11037

[10] A. Laurincikas, A limit theorem for the Riemann zeta-function in the complex space, Acta Arith. 53 (1990), 421-432. | MR: 1075034 | Zbl: 0713.11057

[11] D.R. Heath-Brown, Fractional moments of the Riemann zeta-function, J.London Math. Soc. 24(2) (1981), 65-78. | MR: 623671 | Zbl: 0431.10024

[12] A. Ivic, The Riemann zeta-function John Wiley, 1985. | MR: 792089 | Zbl: 0556.10026

[13] P. Billingsley, Convergence of Probability Measures, John Wiley, 1968. | MR: 233396 | Zbl: 0172.21201

[14] H. Heyer, Probability measures on locally compact groups, Springer-Verlag, Berlin-Heidelberg- New York (1977). | MR: 501241 | Zbl: 0376.60002

[15] H.L. Montgomery and R.C. Vaughan, Hilbert's inequality, J. London Math. Soc. 8(2) (1974), 73-82. | MR: 337775 | Zbl: 0281.10021

Cited by Sources: