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

• Autorzy: Mieczysław Kulas
• Języki publikacji: EN

Abstrakty

EN

The well-known estimate of the order of the Hurwitz zeta function
<br>     $ζ(s,α) - α^{-s} ≪ t^{c(1-σ)^{3/2}} log^{2/3}t$
<br is proved with the constant c = 18.4974 for 1/2 ≤ σ ≤ 1, t ≥ t₀ > 0.
<br>   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).

Kategorie tematyczne

• 11M35: Hurwitz and Lerch zeta functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Acta Arithmetica
• Rocznik: 1999
• Tom: 89
• Numer: 4
• Strony: 301-309

Daty

wydano

1999

otrzymano

1998-08-05

poprawiono

1999-01-13

Twórcy

autor

Mieczysław Kulas

• Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland

Bibliografia

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• [2] W. J. Ellison et M. Mendès-France, Les nombres premiers, Hermann, Paris, 1975.
• [3] D. R. Heath-Brown, private correspondence, 1992.
• [4] A. Ivić, The Riemann Zeta Function, Wiley, 1985.
• [5] M. Kulas, Some effective estimation in the theory of the Hurwitz-zeta function, Funct. Approx. Comment. Math. 23 (1994), 123-134.
• [6] E. I. Panteleeva, On a problem of Dirichlet divisors in number fields, Mat. Zametki 44 (1988), 494-505 (in Russian).
• [7] E. I. Panteleeva, On mean values of some arithmetical functions, Mat. Zametki 55 (1994), no. 2, 109-117 (in Russian).
• [8] K. Prachar, Primzahlverteilung, Springer, Berlin, 1957.
• [9] H. E. Richert, Zur Abschätzung der Riemannschen Zetafunktion in der Nähe der Vertikalen σ = 1, Math. Ann. 169 (1967), 97-101.
• [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.
• [11] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Clarendon Press, Oxford, 1986.
• [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] 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).
• [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).