THE RIEMANN HYPOTHESIS AND UNIVERSALITY OF THE RIEMANN ZETA-FUNCTION

Transcription

1 THE RIEMANN HYPOTHESIS AND UNIVERSALITY OF THE RIEMANN ZETA-FUNCTION Abstract. We rove that, under the Riemann hyothesis, a wide class of analytic functions can be aroximated by shifts ζ(s + iγ k ), k N, of the Riemann zeta-function, where γ k are imaginary arts of nontrivial zeros of ζ(s).. Introduction The Riemann zeta-function ζ(s), s = σ + iτ, is defined, for σ >, by the series ζ(s) = m s, m= and can be meromorhically continued to the whole comlex lane with the unique simle ole at the oint s =. The distribution of zeros occuies the central lace in the theory of ζ(s), and lays an imortant role in various alications. The zeros s = m, m N, are called trivial, they come from the functional equation π s Γ ( s ) ζ(s) = π s Γ ( s ) ζ( s), where Γ(s) is the Euler gamma-function. Moreover, the function ζ(s) has infinitely many the socalled non-trivial zeros which are comlex and are located in the critical stri s C : 0 < σ < }. The Riemann hyothesis (RH) asserts that all non-trivial zeros of ζ(s) lie in the critical line σ =. At the moment, it is known [4] that at least 4 ercent of all non-trivial zeros in the sense of density are on the critical line. The function ζ(s) is one of remarkable analytic objects and has a series of interesting roerties. One of them is universality discovered by S.M. Voronin in [7]. He roved that a wide class of analytic functions can be aroximated by shifts ζ(s+iτ), τ R. More recisely, Voronin obtained that if the function f(s) is continuous and non-vanishing in the disc s r, 0 < r < 4, and analytic in the interior of this disc, then, for every ε > 0, there exists a real number τ = τ(ε) such that ( max ζ s iτ) f(s) < ε. s r The Voronin theorem is high estimated by number theorists, it is imroved and extended for other zeta and L-functions. Let D = s C : < σ < }. Denote by K the class of comact subsets of the stri D with connected comlements, and by H 0 (K) with K K the class of continuous non-vanishing functions on K which are analytic in the interior of K. Let measa stand for the Lebesgue measure of a measurable set A R. Then the modern version of the Voronin theorem has the following form. 00 Mathematics Subject Classification. M06. Key words and hrases. Riemann hyothesis, Riemann zeta-function, universality, weak convergence. The first author is suorted by a grant No. MIP-049/04 from the Research Council of Lithuania.

2 Theorem.. Suose that K K and f(s) H 0 (K), then, for every ε > 0, } T T meas τ [0, T ] : su ζ(s + iτ) f(s) < ε > 0. The theorem shows that there are infinitely many shifts ζ(s+iτ) aroximating a given function: the set of these shifts has a ositive lower density. Different roofs of Theorem. were roosed by S.M. Gonek [8] and B. Bagchi [] (for slightly different sets). The roof can be also found in [0] and [4]. Theorem. is of continuous tye, the shifts τ in ζ(s+iτ) can take arbitrary real values. Also, a discrete universality theorem for ζ(s) is known. In this case, τ takes values from a certain discrete set, for examle, from arithmetical rogression kh : k = 0,,,... }, where h > 0 is a fixed number. Let #A denote the cardinality of the set A. Then the discrete version of Theorem. has the following form. Theorem.. Suose that K K and f(s) H 0 (K), then, for every ε > 0, } N N + # 0 k N : su ζ(s + ikh) f(s) < ε > 0. Discrete universality for zeta-functions was roosed by A. Reich. In [], he obtained a discrete universality theorem for Dedekind zeta-functions. Theorem. in a slightly different form was roved in [], and, by different way, in [3]. Discrete universality theorems for zeta-functions in some sense are more convenient for ractical alications. For examle, a discrete version of the Voronin theorem was alied [3] for estimation of comlicated integrals over analytic curves used in quantum mechanics. A roblem arises to relace the set kh} by more interesting sets. In [6], this was done for the set k α h} with a fixed 0 < α <. Let γ γ be the ordinates of non-trivial zeros of ζ(s). The aim of this aer is to rove that the shifts ζ(s + iγ k ), k =,,..., aroximate the functions from the class H 0 (K). Theorem.3. Suose that RH is true. Let K K and f(s) H 0 (K). Then, for every ε > 0, } N N + # k N : su ζ(s + iγ k ) f(s) < ε > 0. Proof of Theorem.3 is robabilistic, it is based on limit theorems for weakly convergent robability measures in the sace of analytic functions. For this, discrete moments of ζ(s) are needed.. Discrete moments of ζ(s) Lemma.. Suose that RH is true. Then, for fixed < σ < and all t R, ζ(σ + iγ k + it) N( + t ). Proof. It is known [6] that By [7] we have that, under RH, γ T k= Therefore, the lemma follows from (.). We note that RH is used only in Lemma.. γ n πn, n. (.) log n ζ(σ + iγ + it) T log T ( + t ).

3 THE RIEMANN HYPOTHESIS AND UNIVERSALITY OF THE RIEMANN ZETA-FUNCTION 3 Let ˆγ be the unit circle s C : s = }, and 3. A limit theorem Ω = where ˆγ = ˆγ for all rimes. With the roduct toology and ointwise multilication, the infinitedimensional torus Ω is a comact toological Abelian grou. Therefore, on (Ω, B(Ω)) (B(X) stands for the Borel σ-field of the sace X), the robability Haar measure m H exists, and we obtain the robability sace (Ω, B(Ω), m H ). Let ω() be the rojection of an element ω Ω to the circle ˆγ. Denote by H(D) the sace of analytic functions on D endowed with the toology of uniform convergence on comacta, and, on the robability sace (Ω, B(Ω), m H ), define the H(D)-valued random element ζ(s, ω) by the formula ζ(s, ω) = ( ω() ) s. We observe that the above roduct, for almost all ω Ω, converges uniformly on comact subsets of the stri D, and thus define the H(D)-valued random element [0]. Extend the function ω() to the set N by ω(m) = l m l+ m ω l (), m N. Denote by P ζ the distribution of the random element ζ(s, ω), i.e., ˆγ, P ζ (A) = m H (ω Ω : ζ(s, ω) A), Theorem 3.. Suose that RH is true. Then A B(H(D)). P N (A) def = N # k N : ζ(s + iγ k) A}, A B(H(D)), converges weakly to the measure P ζ as N. Moreover, the suort of P ζ is the set S def = g H(D) : g(s) 0 or g(s) 0}. We start the roof of Theorem 3. with some lemmas. First we remind that the sequence x k : k N} R is called uniformly distributed modulo if, for every interval I = [a, b) [0, ) of length I, n lim χ I (x k }) = I, n n k= where χ I is the indicator function of the interval I, and u} denotes the fractional art of a real u. For sequences uniformly distributed modulo, the Weyl criterion is true, see, for examle, [9]. Lemma 3.. A sequence x k : k N} R is uniformly distributed modulo if and only if, for all m Z \ 0}, n lim e πixkm = 0. n n k= Lemma 3.. For every real a 0, the sequence aγ k : k N} is uniformly distributed modulo. Proof. The lemma follows from Theorem. of [5] and Lemma 3.. Denote by P the set of all rime numbers.

4 4 Lemma 3.3. Let, for A B(Ω), Q N (A) = N # k N : ( iγ k : P) A}. Then Q N converges weakly to the Haar measure m H as N. Proof. Consider the Fourier transform g N (k), k = (k, k 3,... ), of Q N which is defined by g N (k) = ω k () dq N, Ω where only a finite number of integers k are distinct from zero. By the definition of Q N, we have that } g N (k) = ikγ k = ex iγ k k log. (3.) N N Obviously, k= k= g N (0) =. (3.) Since the logarithms log are linearly indeendent over the field of rational numbers, k log 0 for k 0. Hence, in view of Lemmas 3. and 3., and (3.), for k 0. This together with (3.) shows that lim g N(k) = N lim g N(k) = 0 N if k = 0, 0 if k 0. Since the right-hand side of the later equality is the Fourier transform of the measure m H, the lemma follows from a continuity theorem for robability measures on comact toological grous. Let θ > be a fixed number, v n(m) = ex ζ n (s) = ζ n (s, ω) = ( m n m= m= ) } θ with m, n N, and v n (m) m s, ω(m)v n (m) m s with ω Ω. Then the above series are absolutely convergent for σ > [0]. Lemma 3.4. Let, for A B(H(D)), P N,n (A) = N # k N : ζ n(s + iγ k ) A}. Then P N,n, as N, converges weakly to the measure ˆP n = m H u n, where u n : Ω H(D) is given by u n (ω) = ζ n (s, ω), and m H u n (A) = m H (u n A), A B(H(D)). Proof. Since the series for ζ n (s, ω) is absolutely convergent, the function u n is continuous one. Therefore, u n is (B(Ω), B(H(D))))-measurable, and ˆP n is a robability measure on H((D), B(H(D))). Thus, the assertion of the lemma follows from Lemma 3.3 and Theorem 5. of [].

5 THE RIEMANN HYPOTHESIS AND UNIVERSALITY OF THE RIEMANN ZETA-FUNCTION 5 Let K l : l N} be a sequence of comact subsets of the stri D such that D = l= K l, K l K l+ for all l N, and if K D is a comact set, then K K l for some l N. The existence of such a sequence is roved, for examle, in [5]. For g, g H(D), define ρ(g, g ) = l su l g (s) g (s) + su l g (s) g (s). l= Then ρ is a metric in H(D) inducing its toology of uniform convergence on comacta. The next lemma is devoted to the aroximation of the function ζ(s) by ζ n (s). Lemma 3.5. Suose that RH is true. Then lim lim su ρ (ζ(s + iγ k ), ζ n (s + iγ k )) = 0. n N N Proof. Let k= l n (s) = s θ Γ ( s θ ) n s, where the number θ is from the definition of v n (m). Then it is known [0] that ζ n (s) = πi θ+i θ i Then, for θ < σ <, the residue theorem gives ζ n (s) ζ(s) = πi θ σ+i θ σ i ζ(s + z)l n (z) dz z. ζ(s + z)l n (z) dz z + R(s), (3.3) where ( R(s) = Res ζ(s + z) l ) n(z). z= s z Now let K be an arbitrary comact subset of D. Then, using (3.3), we find that, for sufficiently large N, su ζ(s + iγ k ) ζ n (s + iγ k ) N k= + N l n (σ σ + iτ) N k= ζ(σ + iγ k + iτ + it) dτ R(σ + iγ k + it), (3.4) k= where < σ < σ <, and t is bounded by a constant deending on K. Clearly R(s) = l n( s) s. The estimate Γ(σ + it) e c t with c > 0 and the convergence of the series γ k= k

6 6 show that the second term in the right-hand side of (3.4) is estimated as o() for N. Therefore, in view of Lemma. and (3.4), we have that lim su su ζ(s + iγ k ) ζ n (s + iγ k ) l n (σ σ + iτ) ( + τ ) dτ. N N k= Hence, by the definition of l n (s), lim lim su n N N su ζ(s + iγ k ) ζ n (s + iγ k ) = 0. k= This together with the definition of the metric ρ roves the lemma. Proof of Theorem 3.. We aly quite standard arguments, therefore, we omit the details. First, Lemma 3.4 and the absolute convergence of the series for ζ n (s) in the half lane σ > imly that the family of robability measures ˆP n : n N} is tight, i.e., for every ε > 0 there exists a comact set K = K(ε) H(D) such that, for all n N, ˆP n (K) > ε. Hence, in view of the Prokhorov theorem, Theorem 6. of [], we have that the family ˆP n } is relatively comact. Thus, every sequence of ˆP n } contains a subsequence ˆP nr which weakly converges to a certain robability measure P on (H(D), B(H(D))) as r. Let θ N be a random variable defined on a certain robability sace (ˆΩ, A, µ) and having the distribution µ(θ N = γ k ) =, k =,..., N. N On the above robability sace, define the H(D)-valued random element X N,n = X N,n (s) by X N,n (s) = ζ n (s + iθ N ). Moreover, let ˆX n = ˆX n (s) be the H(D)-valued random element having the distribution ˆP n. Then, denoting by D the convergence in distribution, by Lemma 3.4 and the above remark, we obtain the relations D X N,n ˆX n (3.5) N and D ˆX nr P. (3.6) r Define one more random element X N = X N (s) by X N (s) = ζ(s + iθ N ). Then the relations (3.5) and (3.6), Lemma 3.5 and Theorem 4. of [] show that X N D P. (3.7) N This means that P N converges weakly to P as N. The relation (3.7) also shows that the measure P is indeendent of the sequence ˆP nr. Therefore, the relation D ˆX n P. n is true. Consequently, we have that P N, as N, converges weakly to the measure P, where P is a limit measure of ˆP n as n. However, it is known [0] that meas τ [0, T ] : ζ(s + iτ) A}, T A B(H(D)),

7 THE RIEMANN HYPOTHESIS AND UNIVERSALITY OF THE RIEMANN ZETA-FUNCTION 7 also, as T, converges weakly to the limit measure P of ˆP n as n, and P = P ζ. Therefore, P N also converges weakly to P ζ. Moreover, in [0], it is obtained that the suort of P ζ is the set S. The theorem is roved. 4. Proof of universality Theorem.3 is a consequence of Theorem 3. and the Mergelyan theorem on aroximation of analytic functions by olynomials []. Thus, by the Mergelyan theorem, there exists a olynomial (s) such that su f(s) e (s) ε <. (4.) Let } G = g H(D) : su g(s) e (s) ε <. Then G is an oen set in H(D), and using the equivalent of weak convergence in terms of oen sets, and Theorem 3., we find that Since, e (s) S, again by Theorem 3., we have that N P N (G) P ζ (G). (4.) P ζ (G) > 0. (4.3) Therefore, the definition of G and P N, and (4.) and (4.3) give the inequality } N N # k N : su ζ(s + iγ k ) e (s) ε < > 0. Combining this with inequality (4.) gives the assertion of Theorem.3. References [] BAGCHI, B.: The statistical behaviour and universality roerties of the Riemann zeta-function and other allied Dirichlet series, Ph. D. Thesis, Calcutta, Indian Stat. Institute, 98. [] BILLINGSLEY, P.: Convergence of Probability Measures, Wiley, New York, 968. [3] BITAR, K. M. KHURI, N. N. REN, H. C.: Paths integrals and Voronin s theorem on the universality of the Riemann zeta-function, Ann. Phys. (994), [4] BUI, H. M. CONREY, B. YOUNG, M. P.: More than 4% of zeros of zeta function are on the critical line, Acta Arith. 50 (0), [5] CONWAY, J. B.: Functions of One Comlex Variable, Sringer, Berlin, Heidelberg, New York, 978. [6] DUBICKAS, A. LAURINČIKAS, A.: Distribution modulo and the discrete universality of the Riemann zeta-function, Abh. Math. Semin. Univ. Hambg. 86 (06) No., [7] GARUNKŠTIS, R. LAURINČIKAS, A.: Discrete mean square of the Riemann zeta-function over imaginary arts of its zeros, Period. Math. Hungar. 76 (08), 7 8. [8] GONEK, S. M.: Analytic roerties of zeta and L-functions, Ph. D. Thesis, University of Michigan, 979. [9] KUIPERS, L. NIEDERREITER, H.: Uniform Distribution of Sequences, Pure and Alied Math., Wiley- Interscience, New York, London, Sydney, 974. [0] LAURINČIKAS, A.: Limit Theorems for the Riemann Zeta-Function, Kluwer, Dordrecht, Boston, London, 996. [] MERGELYAN, S. N.: Uniform aroximation to functions of a comlex variable, Us. Matem. Nauk, 7 (95) No., 3 (in Russian). [] REICH, A.: Werteverteilung von Zetafunktionen, Arch. Math. 34 (980), [3] SANDER, J. STEUDING, J.: Joint universality for sums and roducts of Dirichlet L-functions, Analysis 6 (006), [4] STEUDING, J.: Value-Distribution of L-Functions, Lecture Notes Math. 877, Sringer, Berlin, Heidelberg, New York, 007.

8 8 [5] STEUDING, J.: The roots of the equation ζ(s) = a are uniformly distributed modulo one, in: Anal. Probab. Methods Number Theory, A. Laurinčikas et al. (ed), TEV, Vilnius, 0, [6] TITCHMARSH, E. C.: The Theory of the Riemann Zeta-Function, Second edition revised by D. R. Heath- Brown, Clarendon Press, Oxford, 986. [7] VORONIN, S. M.: Theorem on the universality of the Riemann zeta-function, Izv. Akad. Nauk SSSR 39 (975), (in Russian) = Math. USSR Izv. 9 (975), Institute of Mathematics Faculty of Mathematics and Informatics Vilnius University Naugarduko 4 LT-035 Vilnius LITHUANIA address: ramunas.garunkstis@mif.vu.lt, antanas.laurincikas@mif.vu.lt