On the Periodic Hurwitz Zeta-Function. a Javtokas, a Laurinčikas

On the periodic Hurwitz zeta-function. A Javtokas, A Laurinčikas

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A Javtokas, A Laurinčikas. On the periodic Hurwitz zeta-function.. Hardy-Ramanujan Journal, Hardy-Ramanujan Society, 2006, 29, pp.18 - 36. hal-01111465

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On the periodic Hurwitz zeta-function

A. Javtokas, A. Laurinˇcikas

Abstract.In the paper an universality theorem in the Voronin sense for the periodic Hurwitz zeta-function is proved.

1. Introduction

Let N, N0, R and C denote the sets of all positive integers, non-negative integers, real and complex numbers, respectively, and let a = a , m Z { m ∈ } be a periodic sequence of complex numbers with period k 1. Denote by s = σ + it a complex variable. The periodic zeta-function ζ(s≥; a), for σ > 1, is deﬁned by

∞ a ζ(s; a) = m , ms m=1 X and by analytic continuation elsewhere. Deﬁne

1 k−1 a = a . k m m=0 X A. Javtokas, A. Laurinˇcikas 19

If a = 0, then ζ(s; a) is an entire function, while if a = 0, then the point s = 1 6 is a simple pole of ζ(s; a) with residue a. The function ζ(s; a) satisﬁes the functional equation [22]

k s Γ(s) πis πis ζ(1 s; a ) = exp ζ(s; a ) + exp ζ(s; a ) , − 2π √ 2 ∓ − 2 k where Γ(s) is the Euler gamma-function, and

1 k 2πilm a = al exp : m Z . (√k k ∈ ) Xl=1 The function ζ(s; a) was studied by many authors. In [3] the function ζ(s; a) appears as a special case of the periodic Lerch zeta-function with its functional equation. The papers [7] and [22] are devoted to Hamburger-type theorems for ζ(s; a). In [6] the Kronecker limit formula for ζ(s; a) is obtained. The mean square of ζ(s; a) is studied in [11], [18] and [19]. Probabilistic limit theorems in various spaces are proven in [10] and [12]. The paper [9] contains some expansions for ζ(s; a) and its derivatives as well as a Voronoi type formula for

dl(m) = ad1 . . . adl , d1...Xdl=m which is obtained by using the properties of ζ(s; a). The zero-distribution of ζ(s; a) is investigated in [24]. The zero-free regions and formulas for the number of non-trivial zeroes of ζ(s; a) are established. Moreover, in [25] an important result on the universality of ζ(s; a) is obtained. Let meas A denote the Lebesgue measure of the set A R, and let, for T > 0, { } ⊂

1 ν (. . .) = meas τ [0, T ] : . . . , T T { ∈ } where in place of dots a condition satisﬁed by τ is to be written. Suppose that k is an odd prime, am is not a multiple of a character modulo k, and ak = 0. 20 On the periodic Hurwitz zeta-function

Let K be a compact subset of the strip s C : 1/2 < σ < 1 with connected { ∈ } complement, and let f(s) be a continuous function on K which is analytic in the interior of K. Then in [25] it is proved that, for any ε > 0,

lim inf νT sup ζ(s + iτ; a) f(s) < ε > 0. T →∞ | − | s∈K

Note that the universality of the Riemann zeta-function was discovered by S.M. Voronin [26]. Later, the Voronin theorem was improved and generalized by many authors, see, the survey papers [8], [16] and [21].

In this paper we consider a generalization of the function ζ(s; a). Let 0 < α 1, and, for σ > 1, ≤

∞ a ζ(s, α; a) = m . (m + α)s m=0 X

For σ > 1, we have

k−1 ∞ 1 k−1 ∞ 1 ζ(s, α; a) = a = a l (m + α)s l (l + rk + α)s l=0 r=0 l=0 r=0 X mX=l+rk X X (1) 1 k−1 ∞ 1 1 k−1 l + α = a = a ζ s, , ks l l+α s ks l k r=0 r + k Xl=0 X Xl=0 where ζ(s, α) is the Hurwitz zeta-function. Therefore, the function ζ(s, α; a) is a linear combination of the Hurwitz zeta-functions, and equality (1) gives analytic continuation of the function ζ(s, α; a) to the whole complex plane, where it is regular, except, maybe, for a simple pole at s = 1 with residue a (if a = 0). 6

If am = 1 and k = 1, the function ζ(a, α; a) reduces to the Hurwitz zeta-function{ } ζ{(s,}α). If additionally α = 1, then ζ(s, α; a) becomes the Rie- 2πi lm mann zeta-function. The sequence a = e k , m N , (l, k) = 1, clearly, l ∈ 0 n o A. Javtokas, A. Laurinˇcikas 21

is periodic with period k. Therefore, in the case a = al the function ζ(s, α; a) reduces to the Lerch zeta-function

∞ e2πiλm L(λ, α, s) = , σ > 1, (m + α)s m=0 X with rational parameter λ. Thus, the function ζ(s, α; a) is a generalization of classical zeta-functions, and it is reasonable to call ζ(s, α; a) either the periodic Hurwitz zeta-function, or the Hurwitz zeta-function with periodic coeﬃcients.

The function ζ(s, α; a) has been investigated in [2], true with a small diﬀer- ence in the deﬁnition of ζ(s, α, a). Indeed, in our notation, in [2] the function

a F (s) = ζ(s, α; a) 0 − αs has been considered. Applying their original approximate functional equation for the function ζ(s, α) α−s, the authors obtained in [2] the following inter- esting estimate. Suppose− that a = 0, max a A and H = T 1/3. Then, for j | j| ≤ T 2, ≥

1 T +H F (1/2 + it) 2 A2k log3 T + A2k log k H | | ZT uniformly in α. Also, in [2] a mean-square estimate for twists of F (s) with some sequence has been obtained.

The aim of this note is to prove the universality of the function ζ(s, α; a) for some values of α and certain sequence a.

Theorem 1. Suppose that α is a transcendental number and min0≤m≤k−1 am > 0. Let K be a compact subset of the strip s C : 1/2 < σ < 1 | with| connected complement, and let f(s) be a continuous{ ∈ function on K which} is analytic in the interior of K. Then, for any ε > 0, 22 On the periodic Hurwitz zeta-function

lim inf sup ζ(s + iτ, α; a) f(s) < ε > 0. T →∞ | − | s∈K

Note that the universality theorem for the Hurwitz zeta-function with transcendental parameter was ﬁrst proved in the Ph.D. thesis of B. Bagchi [1]. He proposed a new method for the proof of universality for Dirichlet series based on limit theorems in the sense of weak convergence of probability measures in the space of analytic functions. We will apply this method of limit theorems as well as other tools used in [1] for the proof of Theorem 1.

2. The mean square of ζ(s, α; a)

As it was mentioned above, for the proof of Theorem 1 we need a limit theorem in the sense of week convergence of probability measures on the space of analytic functions for the function ζ(s, α; a). The proof of theorem of such a kind is based on the mean square estimate for ζ(s, α; a) in the region σ > 1/2.

Theorem 2. Let σ > 1/2. Then

1 T ζ(σ + it, α; a) 2dt 1. T | | σ,α,a Z0

Proof. Suppose that max a C. Then in view of (1) we ﬁnd that 0≤j≤k−1 | j| ≤

k−1 l + α 2 ζ(s, α; a) 2 k−2σ2k−1 a 2 ζ s, | | ≤ | l| k l=0 X (2) k−1 l + α 2 k−2σ2k−1C2 ζ s, . ≤ k l=0 X

By Theorems 3.3.1 and 3.3.2 from [16], for σ > 1/2, A. Javtokas, A. Laurinˇcikas 23

T ζ(σ + it, α) 2dt T. | | σ,α Z1 Therefore, taking into account (2), we obtain the theorem.

3. A limit theorem

Let D = s C : 1/2 < σ < 1 . Denote by H(D) the space of analytic on D functions equipp{ ∈ ed with the top}ology of uniform convergence on compacta. Let (S) stand for the class of Borel sets of the space S. B Deﬁne

∞

Ω = γm, m=0 Y where γm is the unit circle γ = s C : s = 1 for every m N0. With the product topology and pointwise{m∈ultiplication| | }the inﬁnite-dimensional∈ torus Ω is a compact topological Abelian group. Therefore, on (Ω, (Ω)) the proba- B bility Haar measure mH can be deﬁned, and this leads to a probability space (Ω, (Ω), m ). Denote by ω(m) the projection of ω Ω to the coordinate B H ∈ space γm.

For σ > 1/2, deﬁne

∞ a ω(m) ζ(s, α, ω; a) = m . (m + α)s m=0 X Since ω(m), m N is a sequence of pairwise orthogonal random variables { ∈ 0} and

∞ a 2 log2 m | m| < , (m + α)2σ ∞ m=1 X 24 On the periodic Hurwitz zeta-function we obtain by Rademacher’s theorem [19] on series of pairwise orthogonal random variables that the series

∞ amω(m) (m + α)s m=1 X for almost all ω Ω with respect to the measure mH converges uniformly on compact subsets∈ of the half-plane s C : σ > 1/2 . This is obtained similarly to the proof of Lemma 5.2.1 [17].{ ∈Therefore, ζ(s,}α, ω; a) is an H(D)- valued random element deﬁned on the probability space (Ω, (Ω), mH ). Let V be an arbitrary positive number, and D = s C : 1/2

P (A) = m (ω Ω : ζ(s, α, ω; a) A) , A (H(D )). ζ H ∈ ∈ ∈ B V

Theorem 3. The probability measure

P (A) = ν (ζ(s + iτ, α; a) A) , A (H(D )), T T ∈ ∈ B V converges weakly to P as T . ζ → ∞

We begin the proof of Theorem 3 with the following statement.

Lemma 4. The probability measure

Q (A) = ν (m + α)−iτ , m N A , A (Ω), T T ∈ 0 ∈ ∈ B converges weakly to the Haar measure m as T . H → ∞ A. Javtokas, A. Laurinˇcikas 25

Proof. The dual group of Ω is

∞

Zm, m=0 M where Z = Z for all m N . m ∈ 0

∞ k = (k , k , . . .) Z , 1 2 ∈ m m=0 M where only the ﬁnite number of integers kj are distinct from zero, acts on Ω by

k x x = xkm , x = (x , x , . . .) Ω. → m 1 2 ∈ m=0 Y

Therefore, the Fourier transform gT (k) of the measure QT is

∞ 1 T ∞ g (k) = xkm dQ = (m + α)−iτkm dτ T m T T Ω m=0 0 m=0 Z Y Z Y (3) 1 T ∞ = exp iτ k log(m + α) dτ. T − m 0 ( m=0 ) Z X Since α is transcendental, the system log(m + α), m N is linearly inde- { ∈ 0} pendent over the ﬁeld of rational numbers, and in view of (3)

1, if k = 0, ∞ gT (k) = 1−exp −iT Pm=0 km log(m+α)  { ∞ }, if k = 0, iT exp −iT Pm=0 km log(m+α) 6  { }  where in the second case only a ﬁnite number of kj are non zero. Thus, 26 On the periodic Hurwitz zeta-function

1, if k = 0, lim gT (k) = T →∞ 0, if k = 0, ( 6 and therefore the measure Q converges weakly to m as T . T H → ∞

Proof of Theorem 3. We will give only a sketch of the proof, since it only in some places diﬀers from the case of the Lerch zeta-function [17].

Let σ1 > 0, and

m + α σ1 v(m, n) = exp . − n + α 

Deﬁne the Dirichlet polynomials

N a v(m, n) ζ (s, α; a) = m n,N (m + α)s m=1 X and

N a ω(m)v(m, n) ζ (s, α, ω; a) = m , ω Ω. n,N (m + α)s ∈ m=1 X

Since the function h : Ω H(D ) given by the formula h(ω) = ζ (s, α, ω; a) → V n,N is continuous and ζn,N (s + iτ, α, ω; a) = h(ωτ ) where −iτ −iτ −iτ ωτ = (α , (1 + α) , (2 + α) , . . .), Lemma 4 and Theorem 5.1 of [4] show that the probability measure

P (A) = ν (ζ (s + iτ, α, ω; a) A) , A (H(D )), T,n,N T n,N ∈ ∈ B V converges weakly to the measure P = m h−1 as T . In view of n,N H → ∞ A. Javtokas, A. Laurinˇcikas 27

the invariance property of the Haar measure mH , this is also true for the probability measure

Pˆ (A) = ν (ζ (s + iτ, α, ω; a) A) , A (H(D )). T,n,N T n,N ∈ ∈ B V

Thus, both the measures P and Pˆ converge weakly to P as T . T,n,N T,n,N n,N → ∞ Now let

∞ a v(m, n) ζ (s, α; a) = m n (m + α)s m=1 X and

∞ a ω(m)v(m, n) ζ (s, α, ω; a) = m , ω Ω. n (m + α)s ∈ m=1 X

It is not diﬃcult to see that the series for ζn(s, α; a) as well as for ζn(s, α, ω; a) converge absolutely and, therefore, uniformly in t for σ > 1/2. Deﬁne on (H(D ), (H(D ))) two probability measures V B V

PT,n(A) = νT (ζn(s + iτ, α; a)) and

PˆT,n(A) = νT (ζn(s + iτ, α, ω; a)) .

Then, using the weak convergence of the probability measures PT,n,N and Pˆ to P as T , we can prove that on (H(D ), (H(D ))) there T,n,N n,N → ∞ V B V exists a probability measure Pn such that both the measures PT,n and PˆT,n converge weakly to P as T . n → ∞ 28 On the periodic Hurwitz zeta-function

In the next step we approximate ζ(s, α; a) and ζ(s, α, ω; a) in the mean by ζn(s, α; a) and ζn(s, α, ω; a) respectively. Let K be a compact subset of DV . Then, applying Theorem 2, we ﬁnd that

1 T lim lim sup sup ζ(s + iτ, α; a) ζn(s + iτ, α; a) dt = 0. (4) n→∞ T | − | T →∞ Z0 s∈K

To obtain a similar relation for ζ(s+iτ, α, ω; a), we use the Birkhoff-Khinchine theorem, see, for example, [4]. Let a = ((m + α)−iτ , m N ), τ R. Then τ ∈ 0 ∈ aτ : τ R is a one-parameter group. Define a one-parameter group hτ : τ{ R of∈ measurable} transformations of Ω by h (ω) = a ω, ω Ω. We recall{ ∈ } τ τ ∈ that a set A B(Ω) is called invariant with respect to h : τ R if for ∈ { τ ∈ } each τ the sets A and Aτ = hτ (A) differ at most by a set of zero mH -measure. All invariant sets form a σ-field. A one-parameter group h : τ R is { τ ∈ } called ergodic if its σ-field of invariant sets consists only of sets having mH - measure equal to 0 or 1. In [10] it was proved that the one-parameter group 2 hτ : τ R is ergodic. Hence we deduce that the process ζ(σ + it, α, ω; a) is{ ergodic.∈ Therefore,} the Birkhoff-Khinchine theorem shows|that, for σ > 1/2,|

T ζ(σ + it, α, ω; a) 2dt T | | σ,α,a Z0 for almost all ω Ω. Now, from this we ﬁnd the analogue of relation (4): ∈

1 T lim lim sup sup ζ(s + iτ, α, ω; a) ζn(s + iτ, α, ω; a) dt = 0 (5) n→∞ T | − | T →∞ Z0 s∈K for almost all ω. Now we are ready to prove limit theorems for ζ(s, α; a) and ζ(s, α, ω; a). Deﬁne

Pˆ (A) = ν (ζ(s + iτ, α, ω; a) A) , A (H(D )). T T ∈ ∈ B V

Since the probability measures PT,n and PˆT,n both converge weakly to the measure P as T , we deduce from (4) and (5) that on (H(D ), (H(D ))) n → ∞ V B V A. Javtokas, A. Laurinˇcikas 29

there exists a probability measure P such that the measures PT and PˆT both converge weakly to P as T . For this, as well as for the measures P → ∞ T,n and PˆT,n, the Prokhorov theorems, see [3], are applying. Thus it remains to identify the limit measure P . For this, we ﬁx a continuity set A (H(D)) of the measure P , and deﬁne a random variable θ on (Ω, (Ω)) by∈ B B

1, if ζ(s, α, ω; a) A, θ(ω) = ∈ 0, if ζ(s, α, ω; a) / A. ( ∈

Then the expectation E(θ) is

E(θ) = θdm = m (ω : ζ(s, α, ω; a) A) = P (A). (6) H H ∈ ζ ZΩ

Since the one-parameter group h : τ R is ergodic, the process θ (h (ω)) { τ ∈ } τ is also ergodic. Therefore, the Birkhoﬀ-Khinchine theorem yields

1 T lim θ (hτ (ω)) dτ = E(θ) (7) T →∞ T Z0 for almost all ω Ω. On the other hand, ∈

1 T θ (h (ω)) dτ = ν (ζ(s + iτ, α, ω; a) A) . T τ T ∈ Z0

This, (6) and (7) show that

lim νT (ζ(s + iτ, α, ω; a) A) = Pζ(A). T →∞ ∈

However,

lim νT (ζ(s + iτ, α, ω; a) A) = P (A). T →∞ ∈ 30 On the periodic Hurwitz zeta-function

Therefore, P (A) = Pζ (A) for all continuity sets A of the measure P . Since the continuity sets constitute a determining class, hence P (A) = Pζ (A) for all A (H(D)). The theorem is proved. ∈ B

4. Proof of Theorem 1

We begin with the support of the probability measure Pζ. We recall that the support of P is a minimal closed set A H(D ) such that P (A) = 1. ζ ⊂ V ζ Since Pζ is the distribution of the random element ζ(s, α, ω; a), its support coincides with the support of ζ(s, α, ω; a).

By the deﬁnition, ω(m) : m N is a sequence of independent complex- { ∈ 0} valued random variable, deﬁned on the probability space (Ω, (Ω), m ), and B H the support of each ω(m) is the unit circle γ. Hence the support of amω(m)/(m+ α)s, m = 0, 1, 2, . . . is the set

a a g H(D ) : g(s) = m , a = 1 . ∈ V (m + α)s | | 

s amω(m)/(m + α) , m N0 is a sequence of independent H(DV )-valued random{ elements, therefore∈by Theorem} 1.7.10 of [13], which was ﬁrst used in [1] as Lemma 5.2.11, the support of the random element ζ(s, α, ω; a) is the closure of the set of all convergent series

∞ a a(m) m , a(m) γ. (8) (m + α)s ∈ m=0 X

Lemma 5. The support of the measure Pζ is the whole of H(DV ).

Proof. We will apply Lemma 5.2.9 of [1], see also Theorem 6.3.10 of [13]. Let K be a compact subset of D . Then, clearly, for b(m) = 1, V | | A. Javtokas, A. Laurinˇcikas 31

∞ a b(m) 2 sup m < . (m + α)s ∞ m=0 s∈K X

Moreover, in Section 3 we have seen that the series

∞ amω(m) (m + α)s m=0 X converges uniformly on compact subsets of DV for almost all ω Ω. Therefore, there exists b(m), b(m) γ, such that ∈ ∈

∞ amb(m) (m + α)s m=0 X

◦ ◦ converges in H(DV ). Thus we have that conditions 2 and 3 of Theorem 6.3.10 from [13] for the sequence a b(m)/(m + α)s, m N are satisﬁed. It { m ∈ 0} remains to verify its condition 1◦.

Let µ be a complex Borel measure with compact support contained in DV such that

∞ amb(m) s dµ(s) < . (9) C (m + α) ∞ m=0 Z X

Since the sequence a is periodic and min a > 0, (9) shows that { m} 0≤m≤k−1 | m|

∞ dµ(s) s < . C (m + α) ∞ m=0 Z X

Hence we have that 32 On the periodic Hurwitz zeta-function

∞ %(log(m + α) < , (10) | | ∞ m=0 X where

%(z) = e−szdµ(s) = eszdµˆ(s), C C Z Z and µˆ(A) = µh−1(a) = µ(h−1A), A B(C), h(s) = s. It is clear that the support of the measure µˆ is contained∈ in s C : 1−< σ < 1/2, t < V . { ∈ − − | | } Since

eszdµˆ(s) eV , C Z the function %(z) is of exponential type. Therefore, in view of Lemma 5.2.2 from [1], see also Lemma 6.4.10 of [13], either %(z) 0, or ≡

log %(r) lim sup | | > 1. r→∞ r −

This and Lemma 5.2.5 of [1], see also Theorem 6.4.14 of [13], imply

%(log p) = , (11) | | ∞ p X where the summing runs over all primes p. Clearly, for m 2, ≥

log(m + α) log m m−1, − and consequently, A. Javtokas, A. Laurinˇcikas 33

%(log(m + α)) %(log m) m−3/2. − 

Thus, taking into account (10), we have that

∞ %(log m) < . | | ∞ m=2 X

However, this contradicts (11). Therefore, the case %(z) 0 takes place, and ≡ the diﬀerentiation of %(z) 0 at the point z = 0 yields ≡

srdµ(s) = 0 C Z

◦ for r N0. This is condition 1 of Theorem 6.3.10 from [12], and we have that the set∈ of all convergent series

∞ a b(m)a(m) m , a(m) γ, (m + α)s ∈ m=0 X is dense in H(DV ). Obviously, this shows that the set of all convergent series (8), has the same property. Since the support of the random element is the closure of the latter set, the lemma is proved.

Proof of Theorem 1. Clearly, there exists V > 0 such that K D . Suppose ⊂ V that the function f(s) is analytically continuable to the region DV . Deﬁne an open set G by

ε G = g H(D ) : sup g(s) f(s) < . ∈ V | − | 4 s∈K 

Since by Lemma 5 the function f(s) belongs to the support of the measure Pζ , we have by Theorem 2.1 of [3] that 34 On the periodic Hurwitz zeta-function

lim inf νT sup ζ(s + iτ, α; a) f(s) < ε Pζ (G) > 0. T →∞ | − | ≥ s∈K 

Now let f(s) satisfy the hypotheses of the theorem. Then by the Mergelyan theorem, see, for example, [23], there exists a polynomial pn(s) such that

ε sup f(s) pn(s) < . s∈K | − | 2

Moreover, since pn(s) is an entire function, by the beginning of the proof we have that

ε lim sup ν sup ζ(s + iτ, α; a) p (s) < > 0. T | − n | 2 T →∞ s∈K 

This proves the theorem.

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Address of the authors

Department of Math. and Informatics Vilnius University Naugarduko 24 03225 Vilnius Lithuania e-mail: [email protected]

Received on 18-10-2005, Accepted on 10-04-2006