ON THE VORONIN'S UNIVERSALITY THEOREM FOR THE RIEMANN ZETA-FUNCTION

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1 Fizikos ir matematikos fakulteto Seminaro darbai, iauliu universitetas, 6, 2003, 2933 ON THE VORONIN'S UNIVERSALITY THEOREM FOR THE RIEMANN ZETA-FUNCTION Ram unas GARUNK TIS 1 Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania; ramunas.garunkstis@maf.vu.lt Abstract. We present a slight modication of the Voronin's proof for the universality of the Riemann zeta-function. The dierence (and simpli- cation) is that we do not use the rearrangement of terms in functional series. Key words and phrases: Riemann zeta-function, universality theorem. Mathematics Subject Classication: 11M Introduction In [8] Voronin proved the well known universality theorem. Let 0 < r < 1/4. Suppose that f(s) is an analytic function in the interior of the disc s r and is continuous up to the boundary of this disc. If f(s) 0 for s < r, then for every ε > 0 there exists T = T (ε) such that max g(s) ζ(s + 3/4 + it ) < ε. s r The Voronin's proof is ineective, that means that from it an upper bound for T = T (ε) can not be derived. One of the reasons is that Voronin uses the theorem of Pecherskii on the rearrangement of terms in functional series. In some partial cases the eective proof was obtained in Garunk²tis [3]. The mentioned paper, works of Gonek [4] and Good [5] suggest some simplication of Voronin's proof. In the next section (see Proposition 1) we give a slight modication of Main Lemma from Voronin [8] (or Lemma 1 from [6], Ÿ7.1). In the proof of Proposition 1 we avoid the rearrangements of series. The remaining proof of the universality theorem for the Riemann zeta-function is the same as in Ÿ7.1 of [6]. For another way to prove universality theorems, 1 Partially supported by Grant from Lithuanian Foundation of Studies and Science.

2 30 using the method of limit theorems, see Bagchi [1], Laurin ikas [7]. In this case the proof is also ineective.

3 R. Garunk²tis Proof For a complex s and for a real vector θ = (θ 1,..., θ m ) we dene ζ m (s, θ) := p m ( ) 1 e 2πiθ p 1 p s. Proposition 1. Let 0 < r < 1/4. Suppose that g(s) is analytic for s < r and continuous for s r. Then for any ε > 0 and y > 0 there exist an integer m y and a vector θ = (θ 1,..., θ n ) with θ j {0, 1/4, 1/2, 3/4}, j = 1,..., n, such that max s r g(s) log ζ m ( s + 3 4, θ ) < ε, where log ζ m (s, θ) = p m = ( e 2πiθ p p s p m ( log 1 e 2πiθp p s ) ) e 2 2πiθp + 2p 2s To prove the proposition we need few lemmas. Lemma 2. Let X be a linear normed space, and let D X be a convex set, closed in the norm of X. Then for any vector z X \ D there are ε > 0 and linear functional f X such that for all x D. Re f(x) Re f(z) ε For a proof of Lemma 2 see [2], Ÿ5.2. From Lemma 2 we obtain Lemma 3. Let H be a real Hilbert space and let D X be a convex set, closed in the norm of X. If D H, then there is a vector e H, e = 1, such that sup(x, e) < +. x D

4 32 On the Voronin's universality theorem Lemma 4. Let u n, n N, be vectors of a real Hilbert space H, and let the series n=1 u n satises the condition u n 2 <. n=1 { M Let for any e H, e = 1, the sequence n=1 (u n, e) : M N} be unbounded. Then for any number y, any z H, and any ε > 0 there are an integer N y and numbers α 1,..., α n equal to 1 or 1 such that N z α n u n < ε. n=1 Proof. Let z H, and let ε > 0 be arbitrary. Choose m so that m y and u n 2 < ε2 36. Let P m denote the set of all vectors x H of the form x = N λ n u n, where λ n [ 1, 1], n = m, m + 1,..., N, and N = m, m + 1, m + 2,.... The set P m is convex. Let P m be the closure of P m in the norm of H. Then P m is a closed convex set. By assumption of the lemma and Lemma 3 we see that P m = H. Consequently, there exist N = N(z) m, and λ n 1, n = m, m + 1,..., N such that N z λ n u n < ε 3. Using induction and the property x = (x, x), it is easy to construct numbers α m,..., α N equal to -1 or 1 such that N N 2 N λ n u n α n u n 4 u n 2 < ε2 9. Then z N α n u n < ε.

5 R. Garunk²tis 33 Now Lemma 4 follows if we apply the above to m 1 z n=1 instead of z. Proof of Proposition 1. By the continuity of g(s), there exists γ > 1 such that γ 2 r < 1/4 and u n max g(s) s r g(s/γ2 ) < ε. The function g(s/γ 2 ) belongs to the Hardy space H (γr) 2. We remind that the Hardy space H (R) 2 is the set of functions f(s) analytic in s < R with the norm f = lim f(s) dσdt <. r R s <r Let us dene a scalar product in H (R) 2 by the formula (ϕ 1 (s), ϕ 2 (s)) = Re ϕ 1 (s)ϕ 2 (s)dσdt. s R This makes H (R) 2 a real Hilbert space. To prove the proposition we apply Lemma 4. Since the series ( ) log 1 e 2πik/4 p s+3/4 k diers from the series η pk (s) := e 2πik/4 p s+3/4 k by an absolutely convergent series, it suces to verify the conditions of Lemma 4 for the last series. We have η p (s) <, p 3/4 R k since 0 < R < 1/4. Next, let ϕ(s) H (R) 2 with ϕ(s) = 1. Then Voronin [8], or see [6], Ÿ7.1, proved that there exists a subseries of (η k (s), ϕ(s)) diverging to. Proposition 1 is proved.

6 34 On the Voronin's universality theorem References [1] B. Bagchi, The statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet series, Ph. D. Thesis, Calcutta, Indian Statistical Institute (1981). [2] N. Dunford and J. T. Schwartz, Linear operators. II: General theory, New York, London, Interscience Publishers (1958). [3] R. Garunk²tis, The eective universality theorem for the Riemann zeta function. (to appear, in: Proc. Sesion "Analytic number theory and Diophantine equations", Max Planck Institute) [4] S. M. Gonek, Analytic properties of zeta and L-functions, Ph.D. Thesis, University of Michigan (1979). [5] A. Good, On the distribution of the values of Riemann's zeta-function, Acta Arith. 38, (1981). [6] A. A. Karatsuba and S. M. Voronin, The Riemann Zeta-Function, Berlin, Walter de Gruyter (1992). [7] A. Laurin ikas, Limit Theorems for the Riemann Zeta-Function, Dordrecht, Kluwer Academic Publishers (1996). [8] S. M. Voronin, Theorem on the 'universality' of the Riemann zeta-function, Izv. Akad. Nauk SSSR, Ser. Matem. 39, (1975) (in Russian) = Math. USSR Izv. 9, (1975). Apie Voronino universalumo teorem Rymano dzeta funkcijai R. Garunk²tis Voroninas pirmasis irode universalumo teorem Rymano dzeta funkcijai. Straipsnyje pateiktas paprastesnis ²io irodymo variantas: i²vengiama perstatu funkcinese eilutese. Rankra²tis gautas