## Effective uniform approximation by the Riemann zeta-function.(English)Zbl 1223.11101

The authors first discuss the celebrated universality theorem of S. M. Voronin [Izv. Akad. Nauk SSSR Ser. Mat. 39, 475–486 (1975; Zbl 0315.10037)], that essentially any non-vanishing analytic function can be approximated uniformly by certain purely imaginary shifts of the Riemann zeta-function $$\zeta(s)$$. The authors prove an effective uniform approximation of Voronin’s theorem. Let ${\mathcal K} = \Bigl\{\,s\in{\mathbb C}\;:\; |s-s_0| \leq r\,\Bigr\},$ where $$r>0$$ and $$s_0 = \sigma_0+it_0,\,1/2 <\sigma_0<1.$$ Suppose that $$g : {\mathcal K}\,\to\,\mathbb C$$ is continuous, $$g(s_0)\neq0$$ and $$g(s)$$ is analytic for $$|s-s_0| \leq r$$. Then, for any $$\varepsilon \in (0,|g(s_0)|)$$, there exist real numbers $$\tau \in [T,\,2T]$$ and $$\delta(\varepsilon,g,\tau)>0$$ such that $\max_{|s-s_0|\leq\delta r}|\zeta(s+i\tau) - g(s)| < \varepsilon.$ The quantities $$T$$ and $$\delta$$ are explicitly defined in the text. It is also shown that, under certain conditions, one may assume even that $$g(s_0) = 0$$. The authors remark that the above assertion holds also for the derivatives $$\zeta^{(j)}(s)$$ in place of $$\zeta(s)$$. They also discuss the case of effective approximations by linear combinations of $$\zeta^{(j)}(s)$$ for $$j = 0,\dots, J$$. The key ingredient in the proofs is an effective multidimensional $$\Omega$$-result of S. M. Voronin [Izv. Akad. Nauk SSSR Ser. Mat. 52, No. 2, 424–436 (1988; Zbl 0651.10026)].

### MSC:

 11M06 $$\zeta (s)$$ and $$L(s, \chi)$$ 30E10 Approximation in the complex plane

### Keywords:

Riemann zeta-function; universality; uniform approximation

### Citations:

Zbl 0315.10037; Zbl 0651.10026
Full Text:

### References:

  R. Garunkštis, The effective universality theorem for the Riemann zeta function, in: Proceedings of the Session in Analytic Number Theory and Diophantine Equations , Bonner Math. Schriften 360 , Univ. Bonn, Bonn, 2003, 21 pp. · Zbl 1070.11035  R. Garunkštis, On the Voronin’s universality theorem for the Riemann zeta-function, Fiz. Mat. Fak. Moksl. Semin. Darb. 6 (2003), 29\Ndash33. · Zbl 1132.11342  P. M. Gauthier and R. Clouâtre, Approximation by translates of Taylor polynomials of the Riemann zeta function, Comput. Methods Funct. Theory 8(1-2) (2008), 15\Ndash19. · Zbl 1221.30084  A. Good, On the distribution of the values of Riemann’s zeta function, Acta Arith. 38(4) (1980/81), 347\Ndash388. · Zbl 0372.10029  A. A. Karatsuba and S. M. Voronin, “The Riemann zeta-function” , Translated from the Russian by Neal Koblitz, de Gruyter Expositions in Mathematics 5 , Walter de Gruyter & Co., Berlin, 1992. · Zbl 0756.11022  A. Laurinčikas, A remark on the universality of the Riemann zeta-function, in: “Proceedings of XXXVIII Conf. of Lith. Math. Soc.” , Vilnius, Technika, 1997, pp. 29\Ndash32.  A. Laurinčikas, Effectivization of the universality theorem for the Lerch zeta function, (Russian), Liet. Mat. Rink. 40(2) (2000), 172\Ndash178; English translation in Lithuanian Math. J. 40(2) (2000), 135\Ndash139. · Zbl 0972.11081  A. Laurinčikas, Prehistory of the Voronin universality theorem, Šiauliai Math. Semin. 1(9) (2006), 41\Ndash53. · Zbl 1126.11041  K. Matsumoto, An introduction to the value-distribution theory of zeta-functions, Šiauliai Math. Semin. 1(9) (2006), 61\Ndash83. · Zbl 1126.11042  H. L. Montgomery, Extreme values of the Riemann zeta function, Comment. Math. Helv. 52(4) (1977), 511\Ndash518. · Zbl 0373.10024  D. V. Pečerskiĭ, Rearrangements of the terms in function series, (Russian), Dokl. Akad. Nauk SSSR 209 (1973), 1285\Ndash1287; English translation in: Soviet Math. Dokl. 14 (1973), 633\Ndash636.  J. Steuding, Upper bounds for the density of universality, Acta Arith. 107(2) (2003), 195\Ndash202. · Zbl 1167.11314  J. Steuding, Upper bounds for the density of universality. II, Acta Math. Univ. Ostrav. 13(1) (2005), 73\Ndash82. · Zbl 1251.11059  J. Steuding, “Value-distribution of $$L$$-functions” , Lecture Notes in Mathematics 1877 , Springer, Berlin, 2007. · Zbl 1130.11044  S. M. Voronin, The distribution of the nonzero values of the Riemann $$\zeta$$-function, (Russian), Collection of articles dedicated to Academician Ivan Matveevič Vinogradov on his eightieth birthday. II, Trudy Mat. Inst. Steklov. 128 (1972), 131\Ndash150, 260.  S. M. Voronin, The functional independence of Dirichlet $$L$$\guiofunctions, (Russian), Collection of articles in memory of Juriĭ Vladimirovič Linnik, Acta Arith. 27 (1975), 493\Ndash503. · Zbl 0308.10025  S. M. Voronin, A theorem on the “universality” of the Riemann zeta-function, (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 39(3) (1975), 475\Ndash486, 703; English translation in: Math. USSR-Izv. 9(3) (1975), 443\Ndash453. · Zbl 0333.30023  S. M. Voronin, $$\Omega$$-theorems of the theory of the Riemann zeta-function, (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 52(2) (1988), 424\Ndash436, 448; English translation in: Math. USSR-Izv. 32(2) (1989), 429\Ndash442. · Zbl 0651.10026
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