Geodesic
On a Problem of Karatsuba
Matematičeskie zametki, Tome 88 (2010) no. 4, pp. 517-528.

Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain lower bounds for the fractional moments of linear combinations of analogs of the Hardy function. In addition, we apply these estimates to the Karatsuba problem of finding a lower bound for the number of zeros of the linear combination of analogs of Hardy functions on the interval $(0,T]$.
Keywords: Hardy function, Dirichlet character, Euler function, fractional moments of the Hardy function, Stirling's formula, Cauchy's theorem, Dirichlet $L$-function. 
@article{MZM_2010_88_4_a3,
     author = {S. A. Gritsenko},
     title = {On a {Problem} of {Karatsuba}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {517--528},
     publisher = {mathdoc},
     volume = {88},
     number = {4},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2010_88_4_a3/}
}
TY  - JOUR
AU  - S. A. Gritsenko
TI  - On a Problem of Karatsuba
JO  - Matematičeskie zametki
PY  - 2010
SP  - 517
EP  - 528
VL  - 88
IS  - 4
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2010_88_4_a3/
LA  - ru
ID  - MZM_2010_88_4_a3
ER  - 
%0 Journal Article
%A S. A. Gritsenko
%T On a Problem of Karatsuba
%J Matematičeskie zametki
%D 2010
%P 517-528
%V 88
%N 4
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2010_88_4_a3/
%G ru
%F MZM_2010_88_4_a3
S. A. Gritsenko. On a Problem of Karatsuba. Matematičeskie zametki, Tome 88 (2010) no. 4, pp. 517-528. http://geodesic.mathdoc.fr/item/MZM_2010_88_4_a3/

[1] G. H. Hardy, J. E. Littlewood, “The zeroes of Riemann's zeta-function on the critical line”, Math. Z., 10:3-4 (1921), 283–317 | DOI | MR | Zbl

[2] A. Selberg, “On the zeroes of Riemann's zeta-function”, Skr. Norske Vid. Akad. Oslo I, 10 (1942), 59 pp. | MR | Zbl

[3] S. M. Voronin, “O nulyakh nekotorykh ryadov Dirikhle, lezhaschikh na kriticheskoi pryamoi”, Izv. AN SSSR. Ser. matem., 44:1 (1980), 63–91 | MR | Zbl

[4] A. A. Karatsuba, “O nulyakh funktsii Devenporta–Kheilbronna, lezhaschikh na kriticheskoi pryamoi”, Izv. AN SSSR. Ser. matem., 54:2 (1990), 303–315 | MR | Zbl

[5] A. A. Karatsuba, “Novyi podkhod k probleme nulei nekotorykh ryadov Dirikhle”, Teoriya chisel i analiz, Sbornik statei. Trudy Mezhdunarodnoi konferentsii po teorii chisel, posvyaschennoi 100-letiyu so dnya rozhdeniya akademika I. M. Vinogradova, Tr. MIAN, 207, Nauka, M., 1994, 180–196 | MR | Zbl

[6] S. M. Voronin, A. A. Karatsuba, Dzeta-funktsiya Rimana, Fizmatlit, M., 1994 | MR | Zbl

[7] A. A. Karatsuba, “O nulyakh spetsialnogo vida funktsii, svyazannykh s ryadami Dirikhle”, Izv. AN SSSR. Ser. matem., 55:3 (1991), 483–514 | MR | Zbl

[8] A. A. Karatsuba, “Drobnye momenty i nuli $\zeta(s)$ na kriticheskoi pryamoi”, Matem. zametki, 72:4 (2002), 502–508 | MR | Zbl

[9] K. Prakhar, Raspredelenie prostykh chisel, Mir, M., 1967 | MR | Zbl

[10] R. M. Gabriel, “Some results concerning the integrals of moduli of regular functions along certain curves”, J. London Math. Soc., 2 (1927), 112–117 | DOI | Zbl

[11] S. Zamarys, “On fractional moments of Dirichlet $L$-functions. II”, Lithuanian Math. J., 46:4 (2006), 494–508 | DOI | MR | Zbl

[12] D. R. Heath-Brown, “Fractional moments of the Riemann zeta-function”, J. London Math. Soc. (2), 24:1 (1981), 65–78 | DOI | MR | Zbl