Title:
|
On the modulus of the Riemann zeta function in the critical strip (English) |
Author:
|
Saidak, Filip |
Author:
|
Zvengrowski, Peter |
Language:
|
English |
Journal:
|
Mathematica Slovaca |
ISSN:
|
0139-9918 |
Volume:
|
53 |
Issue:
|
2 |
Year:
|
2003 |
Pages:
|
145-172 |
. |
Category:
|
math |
. |
MSC:
|
11M06 |
MSC:
|
11M26 |
idZBL:
|
Zbl 1048.11069 |
idMR:
|
MR1986257 |
. |
Date available:
|
2009-09-25T14:13:41Z |
Last updated:
|
2012-08-01 |
Stable URL:
|
http://hdl.handle.net/10338.dmlcz/136881 |
. |
Reference:
|
[1] AYOUB R.: Euler and the zeta function.Amer. Math. Monthlу 81 (1974), 1067-1086. Zbl 0293.10001, MR 0360116 |
Reference:
|
[2] BACKLUND R.: Sur les zéros de la fonction $\zeta(s)$ de Riemann.Comptes Rendus 158 (1914), 1979-1981. |
Reference:
|
[3] BERNOULLI J.: Ars Conjectandi.Basel, 1713. Zbl 0957.01032 |
Reference:
|
[4] BRENT R. P.-VAN DE LUNE J.-TE RIELE H. J. J.-WINTER D. T.: On the zeros of the Riemann zeta function in the critical strip. II.Math. Comp. 39 (1982), 681-688. MR 0669660 |
Reference:
|
[5] DE BRUIJN N. G.: Asymptotic Methods in Analysis.North-Holland Publ., Amsterdam, 1958. Zbl 0082.04202 |
Reference:
|
[6] CHOI J.-SRIVASTAVA H. M.: Series Associated with the Zeta and Related Functions.Kluwer Academic Pub., Dordrecht-Boston-London, 2001. Zbl 1160.11339, MR 1849375 |
Reference:
|
[7] DAVENPORT H.: Multiplicative Number Theory.(1st ed.) Markham Publ. Co., Chicago, 1967. Zbl 0159.06303, MR 0217022 |
Reference:
|
[8] DIRICHLET P. G. L.: Sur l'usage des series infinies dans la théorie des nombres.J. Reine Angew. Math. 18 (1838), 257-274. |
Reference:
|
[9] EDWARDS H. M.: Riemann's Zeta Function.Academic Press, San Diego, 1974. Zbl 0315.10035 |
Reference:
|
[10] EULER L.: De progressionibus transcendentibus.Comm. Acad. Sci. Petrop. 5 (1730), 36-57. |
Reference:
|
[11] EULER L.: Variae observationes circa series infinitas.Comm. Acad. Sci. Petrop. 9 (1737), 222-236. |
Reference:
|
[12] EULER L.: Introductio in Analysin Infinitorum.(Chapter 15), Lausanne, 1748. Zbl 0967.01027 |
Reference:
|
[13] EULER L.: .Novi Comment. Acad. Petrop. 4 (1771), 105. Zbl 1221.65025 |
Reference:
|
[14] GROSSWALD E.: Topics from the Theory of Numbers.The Macmillan Co., New York, 1966. Zbl 0158.29501, MR 0228408 |
Reference:
|
[15] HADAMARD J.: Sur la distribution des zéros de la fonction $\zeta(s)$ et ses consequences arithmétiques.Bull. Soc Math. France 24 (1896), 199-220. MR 1504264 |
Reference:
|
[16] HARDY G. H.-LITTLEWOOD J. E.: The zeros of the Riemann zeta function on the critical line.Math. Z. 10 (1921), 283-317. MR 1544477 |
Reference:
|
[17] IVIC A.: The Riemann Zeta-Function (The Theory of the Riemann Zeta-Function with Applications).A Wiley-Interscience, New York, 1985. Zbl 0556.10026, MR 0792089 |
Reference:
|
[18] KARASTUBA A. A.-VORONIN S. M.: The Riemann Zeta-Function.de Gruyter Exp. Math., de Gruyter, Berlin, 1992. MR 1183467 |
Reference:
|
[19] LANG S.: Complex Analysis.(4th ed.). Grad. Texts in Math. 103, Springer-Verlag, New York, 1999. Zbl 0933.30001, MR 1659317 |
Reference:
|
[20] LINDELÖF E.: Quelques remarques sur la croissance de la fonction $\zeta(s)$.Bull. Sci. Math. 32 (1908), 341-356. |
Reference:
|
[21] VAN DE LUNE J.-TE RIELE H. J. J.: On the zeros of the Riemann zeta function in the critical strip.HI, Math. Comp. 41 (1983), 759-767. MR 0717719 |
Reference:
|
[22] MELLIN H.: Eine Formel fÜr den Logarithmus transcendenter Funktionen von endlichem Geschlecht.Acta Math. 25 (1902), 165-183. MR 1554942 |
Reference:
|
[23] MERTENS F.: Über eine Eigenschaft der Riemann'schen $\zeta$-Function.Sitzungsberichte Akad. Wiss. Wien. Math.-Natur. Kl. 107 (1898), 1429-1434. |
Reference:
|
[24] MONTGOMERY H. L.: The pair correlation of zeros of the zeta function.In: Analytic Number Theory, (Proc Sympos. Pure Math., Vol. XXIV, St. Louis Univ., St. Louis, Mo., 1972), Amer. Math. Soc, Providence, RI, 1973, pp. 181-193. MR 0337821 |
Reference:
|
[25] MURTY M. R.: Problems in Analytic Number Theory.Grad. Texts in Math. 206, Springer, New York, 2001. Zbl 1190.11001, MR 1803093 |
Reference:
|
[26] NEWMAN D. J.: Simple analytic proof of the prime number theorem.Amer. Math. Monthly 87 (1980), 693-696. Zbl 0444.10033, MR 0602825 |
Reference:
|
[27] ODLYZKO A. M.: The $1022$-nd zero of the Riemann zeta function.In: Dynamical, Spectral, and Arithmetic Zeta Functions. AMS special session, San Antonio, TX, USA, January 15-16, 1999. Contemp. Math. 290, Amer. Math. Soc, Providence, RI, 2001, pp. 139-144. MR 1868473 |
Reference:
|
[28] ODLYZKO A. M.-SCHOENHAGE A.: Fast algorithms for multiple evaluations of the Riemann zeta function.Trans. Amer. Math. Soc. 309 (1988), 797-809. Zbl 0706.11047, MR 0961614 |
Reference:
|
[29] RIEMANN B.: Über die Anzahl der Primzahlen unter einer gegebenen Grosse (1859).Collected Works, Teubner, Leipzig, 1892. |
Reference:
|
[30] SELBERG A.: The zeta-function and the Riemann Hypothesis.In: Proc. Skand. Math. Kongr., Kobenhavn, 1947, pp. 187-200. Zbl 0030.05003, MR 0019676 |
Reference:
|
[31] STIELTJES T. J.: Sur le developpement de $\log\Gamma(a)$.J. Math. Pures Appl. (9) 5 (1889), 425-444. |
Reference:
|
[32] STIRLING J.: Methodus Differential.London, 1730. |
Reference:
|
[33] SWINNERTON-DYER H. P. F.: A Brief Guide to Algebraic Number Theory.London Math. Soc. Stud. Texts 50, Univ. Press, Cambridge, 2001. Zbl 0963.11001, MR 1826558 |
Reference:
|
[34] TITCHMARSH E.: The Theory of the Riemann Zeta-Function.(2nd ed., revised by D. R. Heath-Brown), Oxford Univ. Press, Oxford, 1986. Zbl 0601.10026, MR 0882550 |
Reference:
|
[35] DE LA VALLEE-POUSSIN, CH.: Recherches analytiques sur la theorie des nombres.Ann. Soc. Sci. Bruxelles 20 (1896), 183-256. |
Reference:
|
[36] WALFISZ A.: Weylsche Exponentialsummen in der Neueren Zahlentheorie.VEB Deutcher Verlag der Wiss., Berlin, 1963. Zbl 0146.06003, MR 0220685 |
Reference:
|
[37] WHITTAKER E. T.-WATSON G. N.: A Course of Modern Analysis.(An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions), Cambridge Univ. Press, Cambridge, 1927. MR 1424469 |
Reference:
|
[38] ZAGIER D.: Newman's short proof of the prime number theorem.Amer. Math. Monthly 104 (1997), 705-708. Zbl 0887.11039, MR 1476753 |
. |