On the modulus of the Riemann zeta function in the critical strip

Filip Saidak; Peter D. Zvengrowski

Mathematica Slovaca (2003)

  • Volume: 53, Issue: 2, page 145-172
  • ISSN: 0139-9918

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Saidak, Filip, and Zvengrowski, Peter D.. "On the modulus of the Riemann zeta function in the critical strip." Mathematica Slovaca 53.2 (2003): 145-172. <http://eudml.org/doc/34573>.

@article{Saidak2003,
author = {Saidak, Filip, Zvengrowski, Peter D.},
journal = {Mathematica Slovaca},
keywords = {Riemann zeta function; modulus of ; functional equation; Stirling's series; horizontal behaviour of },
language = {eng},
number = {2},
pages = {145-172},
publisher = {Mathematical Institute of the Slovak Academy of Sciences},
title = {On the modulus of the Riemann zeta function in the critical strip},
url = {http://eudml.org/doc/34573},
volume = {53},
year = {2003},
}

TY - JOUR
AU - Saidak, Filip
AU - Zvengrowski, Peter D.
TI - On the modulus of the Riemann zeta function in the critical strip
JO - Mathematica Slovaca
PY - 2003
PB - Mathematical Institute of the Slovak Academy of Sciences
VL - 53
IS - 2
SP - 145
EP - 172
LA - eng
KW - Riemann zeta function; modulus of ; functional equation; Stirling's series; horizontal behaviour of
UR - http://eudml.org/doc/34573
ER -

References

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  1. AYOUB R., Euler and the zeta function, Amer. Math. Monthlу 81 (1974), 1067-1086. (1974) Zbl0293.10001MR0360116
  2. BACKLUND R., Sur les zéros de la fonction ζ ( s ) de Riemann, Comptes Rendus 158 (1914), 1979-1981. (1914) 
  3. BERNOULLI J., Ars Conjectandi, Basel, 1713. Zbl0957.01032
  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. (1982) MR0669660
  5. DE BRUIJN N. G., Asymptotic Methods in Analysis, North-Holland Publ., Amsterdam, 1958. (1958) Zbl0082.04202
  6. CHOI J.-SRIVASTAVA H. M., Series Associated with the Zeta and Related Functions, Kluwer Academic Pub., Dordrecht-Boston-London, 2001. Zbl1160.11339MR1849375
  7. DAVENPORT H., Multiplicative Number Theory, (1st ed.) Markham Publ. Co., Chicago, 1967. (1967) Zbl0159.06303MR0217022
  8. DIRICHLET P. G. L., Sur l'usage des series infinies dans la théorie des nombres, J. Reine Angew. Math. 18 (1838), 257-274. 
  9. EDWARDS H. M., Riemann's Zeta Function, Academic Press, San Diego, 1974. (1974) Zbl0315.10035
  10. EULER L., De progressionibus transcendentibus, Comm. Acad. Sci. Petrop. 5 (1730), 36-57. 
  11. EULER L., Variae observationes circa series infinitas, Comm. Acad. Sci. Petrop. 9 (1737), 222-236. 
  12. EULER L., Introductio in Analysin Infinitorum, (Chapter 15), Lausanne, 1748. Zbl0967.01027
  13. EULER L., [unknown], Novi Comment. Acad. Petrop. 4 (1771), 105. Zbl1221.65025
  14. GROSSWALD E., Topics from the Theory of Numbers, The Macmillan Co., New York, 1966. (1966) Zbl0158.29501MR0228408
  15. HADAMARD J., Sur la distribution des zéros de la fonction ζ ( s ) et ses consequences arithmétiques, Bull. Soc Math. France 24 (1896), 199-220. MR1504264
  16. HARDY G. H.-LITTLEWOOD J. E., The zeros of the Riemann zeta function on the critical line, Math. Z. 10 (1921), 283-317. (1921) MR1544477
  17. IVIC A., The Riemann Zeta-Function (The Theory of the Riemann Zeta-Function with Applications), A Wiley-Interscience, New York, 1985. (1985) Zbl0556.10026MR0792089
  18. KARASTUBA A. A.-VORONIN S. M., The Riemann Zeta-Function, de Gruyter Exp. Math., de Gruyter, Berlin, 1992. (1992) MR1183467
  19. LANG S., Complex Analysis, (4th ed.). Grad. Texts in Math. 103, Springer-Verlag, New York, 1999. (1999) Zbl0933.30001MR1659317
  20. LINDELÖF E., Quelques remarques sur la croissance de la fonction ζ ( s ) , Bull. Sci. Math. 32 (1908), 341-356. (1908) 
  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. (1983) MR0717719
  22. MELLIN H., Eine Formel fÜr den Logarithmus transcendenter Funktionen von endlichem Geschlecht, Acta Math. 25 (1902), 165-183. (1902) MR1554942
  23. MERTENS F., Über eine Eigenschaft der Riemann’schen ζ -Function, Sitzungsberichte Akad. Wiss. Wien. Math.-Natur. Kl. 107 (1898), 1429-1434. 
  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. (1972) MR0337821
  25. MURTY M. R., Problems in Analytic Number Theory, Grad. Texts in Math. 206, Springer, New York, 2001. Zbl1190.11001MR1803093
  26. NEWMAN D. J., Simple analytic proof of the prime number theorem, Amer. Math. Monthly 87 (1980), 693-696. (1980) Zbl0444.10033MR0602825
  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. (1999) MR1868473
  28. ODLYZKO A. M.-SCHOENHAGE A., Fast algorithms for multiple evaluations of the Riemann zeta function, Trans. Amer. Math. Soc. 309 (1988), 797-809. (1988) Zbl0706.11047MR0961614
  29. RIEMANN B., Über die Anzahl der Primzahlen unter einer gegebenen Grosse (1859), Collected Works, Teubner, Leipzig, 1892. 
  30. SELBERG A., The zeta-function and the Riemann Hypothesis, In: Proc. Skand. Math. Kongr., Kobenhavn, 1947, pp. 187-200. (1947) Zbl0030.05003MR0019676
  31. STIELTJES T. J., Sur le developpement de log Γ ( a ) , J. Math. Pures Appl. (9) 5 (1889), 425-444. 
  32. STIRLING J., Methodus Differential, London, 1730. 
  33. SWINNERTON-DYER H. P. F., A Brief Guide to Algebraic Number Theory, London Math. Soc. Stud. Texts 50, Univ. Press, Cambridge, 2001. Zbl0963.11001MR1826558
  34. TITCHMARSH E., The Theory of the Riemann Zeta-Function, (2nd ed., revised by D. R. Heath-Brown), Oxford Univ. Press, Oxford, 1986. (1986) Zbl0601.10026MR0882550
  35. DE LA VALLEE-POUSSIN, CH., Recherches analytiques sur la theorie des nombres, Ann. Soc. Sci. Bruxelles 20 (1896), 183-256. 
  36. WALFISZ A., Weylsche Exponentialsummen in der Neueren Zahlentheorie, VEB Deutcher Verlag der Wiss., Berlin, 1963. (1963) Zbl0146.06003MR0220685
  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. (1927) MR1424469
  38. ZAGIER D., Newman's short proof of the prime number theorem, Amer. Math. Monthly 104 (1997), 705-708. (1997) Zbl0887.11039MR1476753

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