Geodesic
A Growth Estimate for the Mellin Transform of the Riemann Zeta Function
Matematičeskie zametki, Tome 89 (2011) no. 1, pp. 70-81.

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

In this paper, we obtain an upper bound for the modified Mellin transform of the Riemann zeta function on the critical strip.
Keywords: Riemann zeta function, Mellin transform, meromorphic continuation, Atkinson's formula, Cauchy's inequality. 
@article{MZM_2011_89_1_a6,
     author = {A. Laurin\v{c}ikas},
     title = {A {Growth} {Estimate} for the {Mellin} {Transform} of the {Riemann} {Zeta} {Function}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {70--81},
     publisher = {mathdoc},
     volume = {89},
     number = {1},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_89_1_a6/}
}
TY  - JOUR
AU  - A. Laurinčikas
TI  - A Growth Estimate for the Mellin Transform of the Riemann Zeta Function
JO  - Matematičeskie zametki
PY  - 2011
SP  - 70
EP  - 81
VL  - 89
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2011_89_1_a6/
LA  - ru
ID  - MZM_2011_89_1_a6
ER  - 
%0 Journal Article
%A A. Laurinčikas
%T A Growth Estimate for the Mellin Transform of the Riemann Zeta Function
%J Matematičeskie zametki
%D 2011
%P 70-81
%V 89
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2011_89_1_a6/
%G ru
%F MZM_2011_89_1_a6
A. Laurinčikas. A Growth Estimate for the Mellin Transform of the Riemann Zeta Function. Matematičeskie zametki, Tome 89 (2011) no. 1, pp. 70-81. http://geodesic.mathdoc.fr/item/MZM_2011_89_1_a6/

[1] A. Ivić, “The Mellin transforms and the Riemann zeta-function”, Proceedings of the Conference on Analytic and Elementary Number Theory (Vienna, 1996), Univ. of Vienna, Vienna, 1996, 112–127 | Zbl

[2] A. Ivić, “On some conjectures and results for the Riemann zeta-function and Hecke series”, Acta Arith., 99:2 (2001), 115–145 | DOI | MR | Zbl

[3] A. Ivić, “On the estimation of $\mathscr Z_2(s)$”, Analytic and probabilistic methods in number theory (Palanga, 2001), TEV, Vilnius, 2002, 83–98 | MR | Zbl

[4] A. Ivić, “The Mellin transform of the square of Riemann's zeta-function”, Int. J. Number Theory, 1:1 (2005), 65–73 | DOI | MR | Zbl

[5] A. Ivić, A modified Mellin transform of powers of the zeta-function, Preprint, 2005

[6] A. Ivić, “On the estimation of some Mellin transforms connected with the fourth moment of $|\zeta(\frac 12+it)|$”, Elementare und analytische Zahlentheorie, Schr. Wiss. Ges. Johann Wolfgang Goethe Univ. Frankfurt am Main, 20, Franz Steiner Verlag Stuttgart, Stuttgart, 2006, 77–88 | MR | Zbl

[7] A. Ivić, M. Jutila, Y. Motohashi, “The Mellin transform of powers of the zeta-function”, Acta Arith., 95:4 (2000), 305–342 | MR | Zbl

[8] M. Jutila, “The Mellin transforms of the square of Riemann's zeta-function”, Period. Math. Hungar., 42:1-2 (2001), 179–190 | DOI | MR | Zbl

[9] M. Jutila, “The Mellin transform of the power of Riemann's zeta-function”, Number Theory, Ramanujan Math. Soc. Lect. Notes Ser., 1, Ramanujan Math. Soc., Mysore, 2005, 15–29 | MR | Zbl

[10] M. Lukkarinen, The Mellin transform of the square of Riemann's zeta-function and Atkinson's formula, Dissertation, Ann. Acad. Sci. Fenn. Math. Diss. No. 140, Univ. of Turku, Turku, 2005 | MR | Zbl

[11] Y. Motohashi, “A relation between the Riemann's zeta-function and the hyperbolic Laplacian”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 22:2 (1995), 299–313 | MR | Zbl

[12] Y. Motohashi, Spectral Theory of the Riemann Zeta-Function, Cambridge Tracts in Math., 127, Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

[13] A. Laurinčikas, “Limit theorems for the Mellin transform of the square of the Riemann zeta-function. I”, Acta Arith., 122:2 (2006), 173–184 | DOI | MR | Zbl

[14] A. Laurinčikas, “Limit theorems for the Mellin transform of $|\zeta(\frac12+it)|^2$”, Probability and Number Theory – Kanazawa 2005, Adv. Stud. Pure Math., 49, Math. Soc. Japan, Tokyo, 2007, 185–198 | MR | Zbl

[15] V. Balinskait.{e}, A. Laurinčikas, “Discrete limit theorems for the Mellin transform of the Riemann zeta-function”, Acta Arith., 131:1 (2008), 29–42 | DOI | MR | Zbl

[16] V. Balinskait.{e}, A. Laurinčikas, “A two-dimensional discrete limit theorem in the space of analytic functions for Mellin transforms of the Riemann zeta-function”, Nonlinear Anal. Model. Control, 13:2 (2008), 159–167 | MR

[17] A. Laurinčikas, “The Mellin transform of the square of the Riemann zeta-function in the critical strip” (to appear)

[18] A. Laurinčikas, “The Mellin transform of the Riemann zeta-function in the critical strip. II”, Voronoi's Impact on Modern Science, Book 4, Vol. 1, NAS of Ukraine, Kyiv, 2008, 89–96

[19] K. Matsumoto, “The mean square of the Riemann zeta-function in the critical strip”, Japan. J. Math. (N.S.), 15:1 (1989), 1–13 | MR | Zbl

[20] K. Matsumoto, T. Meurman, “The mean square of the Riemann zeta-function in the critical strip. III”, Acta Arith., 64:4 (1993), 357–382 | MR | Zbl

[21] K. Matsumoto, “Recent developments in the mean square theory of the Riemann zeta and other zeta-functions”, Number Theory, Trends Math., Basel, Birkhäuser-Verlag, 2000, 241–286 | MR | Zbl

[22] A. Ivić, K. Matsumoto, “On the error term in the mean square formula for the Riemann zeta-function in the critical strip”, Monatsh. Math., 121:3 (1996), 213–229 | DOI | MR | Zbl

[23] K. Matsumoto, T. Meurman, “The mean square of the square of the Riemann zeta-function in the critical strip. II”, Acta Arith., 68:4 (1994), 369–382 | MR | Zbl

[24] K.-Y. Lam, Some results on the mean values of certain error terms in analytic number theory, Thesis, Univ. of Hong Kong, Hong Kong, 1997

[25] A. Ivić, The Riemann Zeta-Function. The Theory of the Riemann Zeta-Function with Applications, Wiley-Intersci. Publ., John Wiley Sons, New York, 1985 | MR | Zbl