ON THE LINEAR INDEPENDENCE OF THE SET OF DIRICHLET EXPONENTS

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1 ON THE LINEAR INDEPENDENCE OF THE SET OF DIRICHLET EXPONENTS Abstract. Given k 2 let α 1,..., α k be transcendental numbers such that α 1,..., α k 1 are algebraically independent over Q and α k Q(α 1,..., α k 1 ), but α k (aα i + c)/b for some i {1,..., k 1} and some a, b N, c Z satisfying gcd(a, b) = 1. We prove that then there exists a nonnegative integer q such that the set of so-called Dirichlet exponents log(n+α j ), where n runs through the set of all nonnegative integers for j = 1,..., k 1 and n = q, q+1, q+2,... for j = k, is linearly independent over Q. As an application we obtain a joint universality theorem for corresponding Hurwitz zeta functions ζ(s, α 1 ),..., ζ(s, α k ) in the strip {s C : 1/2 < R(s) < 1}. In our approach we follow a recent result of Mishou who analyzed the case k = Introduction For any given complex number α / {0, 1, 2, 3,... } we consider the set D(α) := {log α, log(1 + α), log(2 + α),... }, where log stands for the principal branch of the natural logarithm. The set D(α) is known as the set of Dirichlet exponents of the Hurwitz zeta function 1 ζ(s, α) := (n + α) = e s log(n+α), s n=0 where α is a real number in the interval (0, 1). More generally, for each integer q 0 let us denote so that D 0 (α) = D(α). n=0 D q (α) := {log(q + α), log(q α), log(q α),... }, Recall that a (finite or infinite) set of complex numbers V is linearly dependent over Q if there exist some m N, distinct v 1,..., v m V and nonzero r 1,..., r m Q such that m j=1 r jv j = 0 and linearly independent otherwise. Obviously, if α is a transcendental number then the set D(α) is linearly independent over Q. The set D(α) for algebraic α 2000 Mathematics Subject Classification. 11M35, 11J72. Key words and phrases. Hurwitz zeta function, Dirichlet exponents, universality, linear independence. 1

2 2 have been studied by Cassels [3] (see also [4] and [5]). The question if there is an algebraic number α for which the set D(α) is linearly independent over Q is still open (see [4], [5] and also [7], [12]). A finite set of distinct complex numbers v 1,..., v m is algebraically dependent over Q if there is a nonzero polynomial P (z 1,..., z m ) Q[z 1,..., z m ] such that P (v 1,..., v m ) = 0 and algebraically independent otherwise. The main result of this note is the following: Theorem 1. Let k 2 be an integer and let α 1,..., α k 1, α k be some transcendental numbers. Suppose that the numbers α 1,..., α k 1 are algebraically independent over Q and α k Q(α 1,..., α k 1 ), and suppose for each i = 1,..., k 1 we have α k (aα i + c)/b for a, b N, c Z satisfying gcd(a, b) = 1. Then there is an integer q 0 such that set of Dirichlet exponents is linearly independent over Q. D(α 1 ) D(α k 1 ) D q (α k ) Following the result of Nesterenko [15], the numbers π and e π are algebraically independent over Q, so Theorem 1 can be applied to the numbers α 1 := π = , α 2 := e π = , α 3 := α α 2 = π 2 + e π = Note that the condition α k (aα i + c)/b for integers a > 0, b > 0 and c satisfying gcd(a, b) = 1 cannot be removed from Theorem 1. Indeed, if α k = (aα i + c)/b with some i {1,..., k 1} and a, b, c as above then there exists d N for which u := (bd + c)/a is a positive integer. Thus for each N N we have the identity α k + d + an α k + d + a(n 1) = aα i + c + b(d + an) aα i + c + b(d + an a) = α i + u + bn α i + u + b(n 1). Consequently, the four logarithms log(α k +d+an), log(α k +d+a(n 1)), log(α i +u+bn), log(α i + u + b(n 1)) are linearly dependent over Q, and hence the set D q (α i ) D q (α k ) is linearly dependent over Q for any q N. As an application of Theorem 1 we shall prove the following joint universality theorem for Hurwitz zeta functions. (Throughout, µ(a) stands for the Lebesgue measure of the set A R.) Theorem 2. Let α 1, α 2,..., α k, k 2, be real transcendental numbers in the interval (0, 1) such that for some integers q 1, q 2,..., q k 0 the set of Dirichlet exponents D q1 (α 1 ) D q2 (α 2 ) D qk (α k )

3 ON THE LINEAR INDEPENDENCE OF THE SET OF DIRICHLET EXPONENTS 3 is linearly independent over Q. For each j in the range 1 j k let K j be a compact subset of the strip {s C : 1/2 < R(s) < 1} with connected complement and let f j (s) be a continuous function on K j which is analytic in the interior of K j. Then for any ε > 0 we have lim inf T 1 µ{τ [0, T ] : max T max ζ(s + iτ, α j ) f j (s) < ε} > 0. (1) 1 j k s K j The subject of universality for Dirichlet L-functions started with the paper of Voronin [16], where he proved that for every positive number ε and every continuous non-vanishing function f(s) in the disc s r, where 0 < r < 1/4, which is analytic in s < r there exists a number τ = τ(ε) for which max s r ζ(s+3/4+iτ) f(s) < ε. So certain shifts of zeta function are arbitrarily close to every analytic function. Later, this result have been extended to other L-functions and it was shown that the set of those τ for which the shift of the L-function by iτ approximates f(s) has positive density; see, e.g., [8], [9] for some references on this. In particular, for the Hurwitz zeta function ζ(s, α) it was shown that if α (0, 1/2) (1/2, 1) is either rational or transcendental number then for any function f(s) which is continuous in a compact set K {s C : 1/2 < R(s) < 1} with connected complement and analytic in the interior of K we have lim inf T 1 T for any given ε > 0 (see [1], [6]). µ{τ [0, T ] : max ζ(s + iτ, α) f(s) < ε} > 0 s K Later, certain joint universality theorems when instead of one function f we have several analytic functions f 1,..., f k and approximate them with some shifts of ζ(s, α j ), j = 1,..., k, were obtained in [2], [11], etc. In particular, the joint universality theorem which asserts the conclusion (1) of Theorem 1 under assumption that all k transcendental numbers α 1,..., α k are algebraically independent follows from the results of Nakamura in [14]. Laurinčikas proved the same statement under weaker assumption that the set of Dirichlet exponents D(α 1 ) D(α k ) is linearly independent over Q (see [10]). This corresponds to the case q 1 = = q k = 0 in Theorem 2. The above mentioned result of Nakamura covers the case when the transcendence degree trdeg(q(α 1,..., α k )/Q) of the field extension Q(α 1,..., α k )/Q (i.e., the largest cardinality of an algebraically independent subset of Q(α 1,..., α k ) over Q) is equal to k. On the other hand, when k 3 and 2 r k 1 the next simple example α 1 := α, α 2 := α/r, α 3 := (α + 1)/r,..., α r+1 := (α + r 1)/r,

4 4 where α and α j, j = r + 1,..., k, are algebraically independent transcendental numbers in the interval (0, 1) (so that trdeg(q(α 1,..., α k )/Q) = k r k 2), shows that the first r + 1 Hurwitz zeta functions are linearly dependent r s ζ(s, α 1 ) = ζ(s, α 2 ) + + ζ(s, α r+1 ). Therefore, no joint universality theorem holds for these k Hurwitz zeta functions ζ(s, α j ), j = 1,..., k. Theorem 2 deals with the remaining case when the transcendence degree of the field extension Q(α 1,..., α k )/Q is equal to k 1. The case k = 2 was recently analyzed by Mishou [13]. We will follow his approach. It seems likely that the conclusion (1) is true for any distinct transcendental numbers α 1,..., α k (0, 1) for which trdeg(q(α 1,..., α k )/Q) = k Proof of Theorem 1 Recall that if P (z 1,..., z m ) = i p iz i z im m C[z 1,..., z m ], where i = (i 1,..., i m ) and p i C \ {0}, is a nonzero polynomial then its leading coefficient is the coefficient p j for z j z j m m such that the vector j = (j 1,..., j m ) is the largest lexicographically among all vectors i = (i 1,..., i m ) with maximal sum i i m = deg P. For instance, the leading coefficient of the polynomial P (z 1, z 2 ) = z z 1 z z 5 2 z 1 z 2 is equal to 2. Lemma 3. Suppose that for m N two nonzero polynomials with integer coefficients P (z 1,..., z m ) with positive leading coefficient and Q(z 1,..., z m ), not both constants, are relatively prime. Then there exist infinitely many positive integers t for which P (z 1,..., z m ) + tq(z 1,..., z m ) = A (z i + a ij ), (2) i I j where I is a nonempty subset of the set {1,..., m}, A is a nonzero integer and a ij N {0} (where a ij are not necessarily distinct), if and only if there are i {1,..., m}, a, b N, c Z, gcd(a, b) = 1 for which P (z 1,..., z m ) = az i + c and Q(z 1,..., z m ) = b. Proof. For m = 1 the lemma was proved by Mishou in [13]. Our proof is different from that given in [13] and works for any m N. The lemma is trivial in one direction. If P (z 1,..., z m ) = az i + c and Q(z 1,..., z m ) = b with a, b, c as above then there are infinitely many t N for which c+bt 0 and a (c+bt). For each of those t the representation (2) for the polynomial P (z 1,..., z m ) + tq(z 1,..., z m ) = az i + c + bt = a(z i + (c + bt)/a) holds with A = a, I = {i} and j (z i + a ij ) = z i + (c + bt)/a.

5 ON THE LINEAR INDEPENDENCE OF THE SET OF DIRICHLET EXPONENTS 5 Assume now that P, Q Z[z 1,..., z m ], not both constants, are relatively prime, and the leading coefficient of P is positive. Assume that there exist infinitely many positive integers t for which (2) holds with A = A(t) Z \ {0} and a ij = a ij (t) N {0}. It is clear that the coefficients of the polynomial R t (z 1,..., z m ) := A(t) i I (z i + a ij (t)) (3) on the right hand side of (2) all have the form ut + v with some integers u, v lying in a finite set V. By the condition of the lemma, the nonzero coefficients of R t /A(t) are all positive. So if two nonzero coefficients, say r i (t) for z i z i m m and r j (t) for z j z j m m, of the polynomial R t are unbounded then r i (t) = ut+v and r j (t) = u t+v with some integers u, u 0. It follows that the modulus of their quotient r i (t)/r j (t) is bounded in terms of t. The fact that the quotient of two unbounded coefficients of R t must be bounded will be used below several times. Now we shall prove that all the zeros a ij (t) of the polynomial R t given in (3) are unbounded in terms of t. For a contradiction assume that a ij (t) for some fixed pair i, j is bounded and assume without restriction of generality that i = m. Then 0 a mj (t) K for certain K N. Since a mj (t) can only take K + 1 values, we must have a mj (t) = a for some fixed a {0, 1,..., K} and infinitely many t N. Thus the factor z m + a occurs in all those polynomials R t = P + tq defined in (2) corresponding to those t. Then the polynomial R t (z 1,..., z m 1, a ) = P (z 1,..., z m 1, a ) + tq(z 1,..., z m 1, a ) is zero identically. Thus Q(z 1,..., z m 1, a ) must be the zero polynomial. It follows that P (z 1,..., z m 1, a ) is also the zero polynomial. Hence Q(z 1,..., z m ) and P (z 1,..., z m ) are both divisible by the same factor z m + a, a contradiction. This proves that all the zeros a ij (t) of R t in (3) are unbounded, i.e. a ij (t) as t. Since A(t) Z \ {0}, in view of (3) it follows that all the nonzero coefficients of R t are also unbounded except possibly for the leading coefficient A(t). Next, if the leading coefficient A(t) is unbounded then A(t) and A(t) i I j a ij(t) are two unbounded coefficients of R t, which is impossible, because their quotient i I j a ij(t) tends to infinity as t. (Recall that, by the fact established above, the quotient of two unbounded coefficients of R t must be bounded.) So A(t) is bounded. Hence the leading coefficient A(t) of R t = P + tq must be that of P. This yields A(t) = a, where a > 0 is the leading coefficient of P. j

6 6 Suppose next that for infinitely many t N the product R t (z 1,..., z m ) = a (z i + a ij (t)) i I j contains exactly r 2 not necessarily distinct factors with the same i, say z i + a i1 (t),..., z i +a ir (t). Put B = B(t) for the constant term of the polynomial R t (z 1,..., z r )/ r j=1 (z i + a ij (t)). Then both B(t) r j=1 a ij(t) and B(t) r j=1 a ij(t) are the coefficients of the polynomial R t corresponding to its constant term and the term for z r 1 1, respectively. They are both unbounded, so their quotient r j=1 a ij(t)/ r j=1 a ij(t) must be bounded. This is not the case, because all a ij (t) are unbounded, so the product of r 2 terms r j=1 a ij(t) divided by their sum r j=1 a ij(t) tends to infinity as t. The only remaining possibility is that R t (z 1,..., z m ) = a i I (z i + a i (t)) for infinitely many t N. In case I 2 we see that the constant coefficient of R t is equal to a i I a i(t) and the coefficient for z l, where l I, is equal to a i I\{l} a i(t). They both are unbounded, because I 2. But their quotient a l (t) is also unbounded, a contradiction. It follows that I = 1 and thus R t (z 1,..., z m ) = a(z i +a i (t)) for some i {1,..., m} and infinitely many t N. From P + tq = R t = az i + aa i (t) we conclude that P (z 1,..., z m ) = az i + c, where a N, c Z and Q(z 1,..., z m ) = b 0. Then P + tq = az i + c + tb = a(z i + (c + tb)/a) has the required form only when b > 0 and a divides c + bt for infinitely many t N. From the equality at 1 bt = c, where t 1 Z, we see that such positive integers t exist if and only if gcd(a, b) divides c. However, if gcd(a, b) > 1 and gcd(a, b) c then the polynomials P = az i + c and Q = b are divisible by gcd(a, b) > 1, and so they are not relatively prime. Consequently, we must have gcd(a, b) = 1. Hence P (z 1,..., z m ) = az i + c for some i {1,..., m} and Q(z 1,..., z m ) = b with a, b N, c Z and gcd(a, b) = 1, as claimed in the statement of the lemma. Now we can give the proof of Theorem 1. Assume that the set D(α 1 ) D(α k 1 ) D q (α k ) is linearly dependent over Q. Since the sets D(α 1 ) D(α k 1 ) and D q (α k ) are both linearly independent over Q, writing α k = P (α 1,..., α k 1 )/Q(α 1,..., α k 1 ) with two relatively prime polynomials P, Q in Z[z 1,..., z k 1 ] we must have (α i + n ij ) u ij = (P (α 1,..., α k 1 )/Q(α 1,..., α k 1 ) + n j ) u j (4) j j i I

7 ON THE LINEAR INDEPENDENCE OF THE SET OF DIRICHLET EXPONENTS 7 for some I {1,..., k 1}, n ij, n j N {0}, n j q and u ij, u j Z \ {0}. Of course, P and Q are not both constants, because α k is transcendental. Also, without restriction of generality, by multiplying both P and Q by 1 if necessary, we may assume that the leading coefficient of P is positive. Since the numbers α 1,..., α k 1 are algebraically independent, the equality (4) must be the identity, namely, (z i + n ij ) u ij j (P (z 1,..., z k 1 )/Q(z 1,..., z k 1 ) + n j ) u j. j (5) i I Note that the polynomials P + n j Q and P + n l Q with n j n l can have only constant common factor, since P and Q are relatively prime. Hence selecting any n j q on the right hand side of (5) we see that the corresponding polynomial P (z 1,..., z k 1 ) + n j Q(z 1,..., z k 1 ) must be a constant multiplied by certain product i I 1 (z i +n is ) v is, where I 1 I, n is N {0} and v is N. However, by Lemma 3, this is impossible for q large enough whenever (P, Q) (az i + c, b) with a, b, c as in Lemma 3. This completes the proof of Theorem 1, since the condition of the theorem and that of the lemma which exclude the case P (z 1,..., z k 1 ) = az i + c, Q(z 1,..., z k 1 ) = b, where i {1,..., k 1}, a, b N, c Z and gcd(a, b) = 1, are the same. 3. Proof of Theorem 2 Assume that the set of Dirichlet exponents D q1 (α 1 ) D qk (α k ) is linearly independent over Q. Evidently, its subset D q (α 1 ) D q (α k ), where q := max 1 j k q j, is linearly independent over Q too. Take a maximal subset M 1 of the finite set k j=1(d(α j ) \ D q (α j )) for which the set D 1 := M 1 D q (α 1 ) D q (α k ) is linearly independent over Q. This means that each of the qk M 1 remaining logarithms log(n + α j ) / D 1, where 0 n q 1 and 1 j k, is a linear combination with rational coefficients of some elements of D 1. (Of course, the choice of the set M 1 is not necessarily unique.)

8 8 Fix an integer m q such that each of the logarithms log(n + α j ) / D 1 is expressible in the form log(n + α j ) = k r=1 m 1 i=0 c j,n,r,i log(i + α r ) with c j,n,r,i Q. (Some of the coefficients c j,n,r,i can be zeros.) Therefore, by increasing q to m if necessary and adding more logarithms to the set M 1 we may assume that each log(n + α j ) which is not in the set where D := M D m (α 1 ) D m (α k ), M := M 1 {log(q + α 1 ),..., log(m 1 + α 1 )} {log(q + α k ),..., log(m 1 + α k )}, is a linear combination of at most km logarithms of the set M. Obviously, there exists a positive integer l such that for each log(n + α j ) / D we have the representation l log(n + α j ) = c i,r log(i + α r ) (6) with c i,r Z. log(i+α r ) M Let K j be the sets and let f j (s) be the functions described in Theorem 2. Fix ε > 0. Let K be a simply connected compact subset of the strip {s C : 1/2 < R(s) < 1} such that the union k j=1k j is included in the interior of K. By Mergelyan s theorem (see Lemma 5 in [13]), there exist polynomials with complex coefficients p j (s), j = 1,..., k, such that max max f j (s) p j (s) < ε. (7) 1 j k s K j By Gonek s lemma (see Lemma 7, (29) and (30) in [13]), there is a large positive integer ν > m such that for each sufficiently large integer t and each j = 1,..., k we have max p j (s) 1 s K (n + α j ) exp(2πiθ n,j ) < ε s (n + α j ) s 0 n<ν ν n t with some θ n,j R. Selecting θ n,j = 0 for n = m,..., ν, we can rewrite the above inequality in the form max s K p j (s) 0 n<m 1 (n + α j ) s m n t For δ > 0 let B T (δ) be a set of those τ [T, 2T ] for which exp(2πiθ n,j ) (n + α j ) s < ε. (8) (τ/2π) log(n + α j ) θ n,j δ when m n t, 1 j k

9 and ON THE LINEAR INDEPENDENCE OF THE SET OF DIRICHLET EXPONENTS 9 (τ/2π) log(n + α j ) δ when log(n + α j ) M. Observe that in view of (6) the second inequality implies that for each sufficiently small δ there is a positive constant c 0 (which depends on l, M and the coefficients c i,r in qm M equalities (6)) such that (τ/2π) log(n + α j ) c 0 δ (9) for each n = 0, 1,..., m 1 and each j = 1,..., k. Since the logarithms involved in the definition of B T (δ) are linearly independent over Q, by Kronecker s theorem (see Lemma 6 in [13]), the Lebesgue measure of the set B T (δ) satisfies µ(b T (δ)) ε 1 T as T, where ε 1 := (2δ) k(t m+1)+ M. (10) For each j = 1,..., k and each τ B T (δ) we have max exp(2πiθ n,j ) 1 < ε s K (n + α j ) s (n + α j ) s+iτ m n t 0 n<m m n t whenever δ is small enough. Similarly, by (9), we obtain 1 max s K (n + α j ) 1 < ε s (n + α j ) s+iτ 0 n<m when δ is small enough. Combined with (8) this gives max max p j (s) 1 < 3ε. (11) 1 j k s K (n + α j ) s+iτ 0 n t The next two inequalities are standard and can be obtained by considering the second moments of the involved functions. Firstly, for any pair of positive numbers ε, ε 2 and a set ζ(s A T (ε, z) := {τ [T, 2T ] : max max + iτ, αj ) 1 < ε} (12) 1 j k s K j (n + α j ) s+iτ 0 n z we have µ(a T (ε, z)) lim inf > 1 ε 2 (13) T T for each sufficiently large z (see Lemma 9 in [13]). Secondly, let C T (δ) be a subset of B T (δ) for which the inequality max max 1 j k s K t<n z 1 (n + α j ) s+iτ < ε (14)

10 10 holds uniformly for z > t. Then (see Lemma 11 in [13] and (10)) for each sufficiently large t we have µ(c T (δ)) lim inf > 1 T T 2 lim µ(b T (δ)) T T Hence selecting ε 2 = ε 1 /4 in (13) we obtain lim inf T µ(a T (ε, z) C T (δ)) T > ε 1 4 = ε 1 2. for each sufficiently large z. Finally, for τ A T (ε, z) C T (δ) combining (7), (11), (12), (14) we find that This completes the proof of Theorem 2. max max ζ(s + iτ, α j ) f j (s) < 7ε. 1 j k s K j References [1] B. Bagchi, The statistical behaviour and universality properties of the Riemann zeta-function and other allied Dirichlet series, Ph. D. Thesis, Indian Statistical Institute, Calcutta, [2] B. Bagchi, A joint universality theorem for Dirichlet L-functions, Math. Z. 181 (1982), [3] J.W.S. Cassels, Footnote to a note of Davenport and Heilbronn, J. Lond. Math. Soc. 36 (1961), [4] P. Drungilas and A. Dubickas, Multiplicative dependence of shifted algebraic numbers, Colloq. Math. 96 (2003), [5] A. Dubickas, Multiplicative dependence of quadratic polynomials, Lith. Math. J. 38 (1998), [6] S.M. Gonek, Analytic properties of zeta and L-functions, PhD thesis, Univ. of Michigan, [7] R. Kačinskaitė, Discrete value distribution of the Hurwitz zeta-function with algebraic irrational parameter. I, Lith. Math. J. 48 (2008), [8] A.A. Karatsuba and S.M. Voronin, The Riemann zeta-function, de Gruyter, New York, [9] A. Laurinčikas, Limit theorems for the Riemann zeta-function, Kluwer, Dordrecht, Boston, London, [10] A. Laurinčikas, The joint universality of Hurwitz zeta-functions, Šiauliai Math. Semin. 3 (11) (2008), [11] A. Laurinčikas and K. Matsumoto, The joint universality and the functional independence for Lerch zeta-functions, Nagoya Math. J. 157 (2000), [12] A. Laurinčikas and J. Steuding, Limit theorems in the space of analytic functions for the Hurwitz zeta-function with an algebraic irrational parameter, Central European J. Math. 9 (2011), [13] H. Mishou, The joint universality theorem for a pair of Hurwitz zeta functions, J. Number Theory 131 (2011), [14] T. Nakamura, The existence and non-existence of the joint t-universality for Lerch zeta functions, J. Number Theory 125 (2007),

11 ON THE LINEAR INDEPENDENCE OF THE SET OF DIRICHLET EXPONENTS 11 [15] Yu.V. Nesterenko, Modular functions and transcendence questions, Sb. Math. 187 (1996), [16] S.M. Voronin, Theorem on the universality of the Riemann zeta-function, Math. USSR, Izv. 9 (1975), Department of Mathematics and Informatics, Vilnius University, Naugarduko 24, Vilnius LT-03225, Lithuania address: arturas.dubickas@mif.vu.lt