Refinement of an estimate for the Hurwitz zeta function in a neighbourhood of the line σ = 1

Mieczysław Kulas

Acta Arithmetica (1999)

  • Volume: 89, Issue: 4, page 301-309
  • ISSN: 0065-1036

Abstract

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The well-known estimate of the order of the Hurwitz zeta function      ζ ( s , α ) - α - s ≪ t c ( 1 - σ ) 3 / 2 l o g 2 / 3 t 0.    The improvement of the constant c is a consequence of some technical modifications in the method of estimating exponential sums sketched by Heath-Brown ([11], p. 136).

How to cite

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Mieczysław Kulas. "Refinement of an estimate for the Hurwitz zeta function in a neighbourhood of the line σ = 1." Acta Arithmetica 89.4 (1999): 301-309. <http://eudml.org/doc/207273>.

@article{MieczysławKulas1999,
abstract = {The well-known estimate of the order of the Hurwitz zeta function      $ζ(s,α) - α^\{-s\} ≪ t^\{c(1-σ)^\{3/2\}\} log^\{2/3\}t$ 0.    The improvement of the constant c is a consequence of some technical modifications in the method of estimating exponential sums sketched by Heath-Brown ([11], p. 136).},
author = {Mieczysław Kulas},
journal = {Acta Arithmetica},
keywords = {Hurwitz zeta-function},
language = {eng},
number = {4},
pages = {301-309},
title = {Refinement of an estimate for the Hurwitz zeta function in a neighbourhood of the line σ = 1},
url = {http://eudml.org/doc/207273},
volume = {89},
year = {1999},
}

TY - JOUR
AU - Mieczysław Kulas
TI - Refinement of an estimate for the Hurwitz zeta function in a neighbourhood of the line σ = 1
JO - Acta Arithmetica
PY - 1999
VL - 89
IS - 4
SP - 301
EP - 309
AB - The well-known estimate of the order of the Hurwitz zeta function      $ζ(s,α) - α^{-s} ≪ t^{c(1-σ)^{3/2}} log^{2/3}t$ 0.    The improvement of the constant c is a consequence of some technical modifications in the method of estimating exponential sums sketched by Heath-Brown ([11], p. 136).
LA - eng
KW - Hurwitz zeta-function
UR - http://eudml.org/doc/207273
ER -

References

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  1. [1] J. G. van der Corput et J. F. Koksma, Sur l'ordre de grandeur de la fonction ζ(s) de Riemann dans la bande critique, Ann. Fac. Sci. Univ. Toulouse (3) 22 (1930), 1-39. Zbl56.0978.03
  2. [2] W. J. Ellison et M. Mendès-France, Les nombres premiers, Hermann, Paris, 1975. 
  3. [3] D. R. Heath-Brown, private correspondence, 1992. 
  4. [4] A. Ivić, The Riemann Zeta Function, Wiley, 1985. 
  5. [5] M. Kulas, Some effective estimation in the theory of the Hurwitz-zeta function, Funct. Approx. Comment. Math. 23 (1994), 123-134. Zbl0845.11033
  6. [6] E. I. Panteleeva, On a problem of Dirichlet divisors in number fields, Mat. Zametki 44 (1988), 494-505 (in Russian). Zbl0654.10041
  7. [7] E. I. Panteleeva, On mean values of some arithmetical functions, Mat. Zametki 55 (1994), no. 2, 109-117 (in Russian). 
  8. [8] K. Prachar, Primzahlverteilung, Springer, Berlin, 1957. 
  9. [9] H. E. Richert, Zur Abschätzung der Riemannschen Zetafunktion in der Nähe der Vertikalen σ = 1, Math. Ann. 169 (1967), 97-101. Zbl0161.04802
  10. [10] W. Staś, Über das Verhalten der Riemannschen ζ-Funktion und einiger verwandter Funktionen, in der Nähe der Geraden σ = 1, Acta Arith. 7 (1962), 217-224. Zbl0099.26804
  11. [11] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Clarendon Press, Oxford, 1986. Zbl0601.10026
  12. [12] P. Turán, On some recent results in the analytical theory of numbers, in: Proc. Sympos. Pure Math. 20, Amer. Math. Soc., 1971, 339-347. 
  13. [13] O. V. Tyrina, A new estimate for a trigonometric integral of I. M. Vinogradov, Izv. Akad. Nauk SSSR Ser. Mat. 51 (1987), 363-378 (in Russian). Zbl0618.10035
  14. [14] I. M. Vinogradov, General theorems on the upper bound of the modulus of a trigonometric sum, Izv. Akad. Nauk SSSR Ser. Mat. 15 (1951), 109-130 (in Russian). Zbl0042.04205

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