A rigidity theorem for translates of uniformly convergent Dirichlet series
aa r X i v : . [ m a t h . N T ] F e b A RIGIDITY THEOREM FOR TRANSLATES OF UNIFORMLYCONVERGENT DIRICHLET SERIES
A.PERELLI and M.RIGHETTI
Abstract.
It is well known that the Riemann zeta function, as well as several other L -functions, is universal in the strip 1 / < σ <
1; this is certainly not true for σ >
1. Answeringa question of Bombieri and Ghosh, we give a simple characterization of the analytic functionsapproximable by translates of L -functions in the half-plane of absolute convergence. Actually,this is a special case of a general rigidity theorem for translates of Dirichlet series in the half-plane of uniform convergence. Our results are closely related to Bohr’s equivalence theorem. Mathematics Subject Classification (2010):
Keywords:
Dirichlet series, universality of L -functions, Bohr equivalence theorem1. Introduction
In 1975, Voronin [12] discovered the following universality property of the Riemann zetafunction ζ ( s ). Let f ( s ) be holomorphic and non-vanishing on a closed disk K inside the strip1 / < σ <
1, and let ε >
0; thenlim inf T →∞ T |{ τ ∈ [ − T, T ] : max s ∈ K | ζ ( s + iτ ) − f ( s ) | < ε }| > . Voronin’s universality theorem has been extended in several directions, in particular involvingother L -functions in place of ζ ( s ), other compact sets in place of disks, and vectors of L -functions in place of a single L -function; see the survey by Matsumoto [9] and Chapter VII ofKaratsuba-Voronin [7]. On the other hand, it is well known that every Dirichlet series F ( s ) isBohr almost periodic and bounded on any vertical strip whose closure lies inside the half-plane σ > σ u ( F ) of uniform convergence, hence F ( s ) cannot be universal in the above sense for σ > σ u ( F ); in particular, ζ ( s ) is not universal for σ > L -functions in the domain of absolute convergence. Here we answer this question in a rathergeneral framework; it turns out that the answer is closely related to Bohr’s theory of equivalentDirichlet series, see Bohr [2] and Chapter 8 of Apostol [1].We recall that a general Dirichlet series (D-series for short) is of the form F ( s ) = ∞ X n =1 a ( n ) e − λ n s (1)with coefficients a ( n ) ∈ C and a strictly increasing sequence of real exponents Λ = ( λ n )satisfying λ n → ∞ . Clearly, the case λ n = log n recovers the ordinary D-series. Accordingto Bohr, a (possibly finite) sequence of real numbers B = ( β ℓ ) is a basis of Λ if it satisfiesthe following conditions: the elements of B are Q -linearly independent, every λ n is a Q -linearcombination of elements of B and, viceversa, every β ℓ is a Q -linear combination of elements of Λ. This can be expressed in matrix notation by considering Λ and B as column vectors, andwriting Λ = RB and B = T Λ for some (infinite) Bohr matrices R and T , whose row entries arerational and almost always 0; clearly, R is uniquely determined by Λ and B . Moreover, twogeneral D-series, say F ( s ) as in (1) and G ( s ) with coefficients b ( n ) and the same exponents Λ,are equivalent if there exist a basis B of Λ and a real column vector Y = ( y ℓ ) such that b ( n ) = a ( n ) e i ( RY ) n , (2)where R is the above Bohr matrix. In the case of ordinary D-series with coefficients a ( n ) and b ( n ), equivalence reduces to the existence of a completely multiplicative function ρ ( n ) such that b ( n ) = a ( n ) ρ ( n ) for all n ≥
1, and | ρ ( n ) | = 1 whenever a ( p ) = 0 and p is a prime divisor of n .We refer to Chapter 8 of [1] for an introduction to Bohr’s theory.We extend the above notion of equivalence to vectors ( F ( s ) , . . . , F N ( s )) of D-series in thefollowing way. Let N ≥ F j ( s ), G j ( s ), j = 1 , . . . , N , be as in (1) with coefficients a j ( n ) and b j ( n ), respectively, and the same exponents Λ. We say that ( F ( s ) , . . . , F N ( s )) and( G ( s ) , . . . , G N ( s )) are vector-equivalent if there exist a basis B of Λ and a real vector Y = ( y ℓ )such that for j = 1 , . . . , N we have b j ( n ) = a j ( n ) e i ( RY ) n , (3) R being as above. We stress that in (3) we require the same vector Y for every j , hence F j ( s )and G j ( s ) are equivalent via the same twist by e i ( RY ) n . Note that for N = 1, vector-equivalencereduces to the standard Bohr equivalence. We also point out that we assume all the F j ( s ) tohave the same exponents Λ just for convenience, since otherwise we may take as Λ the unionof the exponents Λ j and express all the F j ( s )’s in terms of Λ. Moreover, as in Righetti [10],we say that a D-series F ( s ) as in (1), or a sequence of exponents Λ, has an integral basis ifthere exists a basis B of Λ such that the associated Bohr matrix R has integer entries. Such abasis B is called an integral basis of F ( s ), or of Λ. Clearly, Λ = (log n ) has the integral basis B = (log p ), so the important class of ordinary D-series falls in this case.Vectors of D-series with an integral basis provide a general framework where the abovementioned problem by Bombieri and Ghosh can be settled in the following sharp form. Let N ≥ j = 1 , . . . , N , let F j ( s ) be general D-series with coefficients a j ( n ) and the sameexponents Λ, with an integral basis and with finite σ u ( F j ). Further, let K j be compact setsinside the half-planes σ > σ u ( F j ) containing at least one accumulation point, and let f j ( s ) beholomorphic on K j . Theorem 1.
Under the above assumptions, the following assertions are equivalent.(i) For every ε > there exists τ ∈ R such that max j =1 ,...,N max s ∈ K j | F j ( s + iτ ) − f j ( s ) | < ε ; (ii) f ( s ) , . . . , f N ( s ) are general Dirichlet series with exponents Λ , and ( f ( s ) , . . . , f N ( s )) isvector-equivalent to ( F ( s ) , . . . , F N ( s )) ;(iii) for every ε > we have lim inf T →∞ T |{ τ ∈ [ − T, T ] : max j =1 ,...,N max s ∈ K j | F j ( s + iτ ) − f j ( s ) | < ε }| > (iv) f j ( s ) has analytic continuation to σ > σ u ( F j ) and there exists a sequence τ k such that F j ( s + iτ k ) converges uniformly to f j ( s ) on every closed vertical strip in σ > σ u ( F j ) , j = 1 , . . . , N . Corollary.
Theorem holds for ordinary Dirichlet series. Our result may therefore be regarded as a general rigidity theorem for translates of D-seriesin the half-plane of uniform convergence, and represents the counterpart of the universalitytheorems for L -functions in the critical strip. Indeed, Theorem 1 gives a complete characteriza-tion of the analytic functions f j ( s ), called the target functions , approximable by such translatesas in (i), and the target functions are quite special. For example, thanks to Bohr’s equiva-lence theorem (see Theorem 8.16 of [1]) and its converse for D-series with an integral basis(see Righetti [10]), the functions f j ( s ) are those assuming the same set of values of the F j ( s )’son any vertical strip inside the domain of absolute convergence. Moreover, if f j ( s ) is a targetfunction on a compact set K j as in Theorem 1, then by (iv) it has continuation to σ > σ u ( F j )and is a target function on any compact set in such half-plane. We further note that the roleof F j ( s ) and f j ( s ) in (iv), and essentially in Theorem 1, may be interchanged.Note also that comparison with universality theorems for vectors of L -functions is moretransparent using (iii) of Theorem 1, which embodies the effect of the Kronecker-Weyl the-orem. Moreover, somehow unexpectedly, contrary to the case of such universality theorems,no independence relation among the F j ( s )’s is required in our result. Indeed, in the specialcase of vectors of orthogonal L -functions one obtains exactly the same result as for generalD-series with an integral basis. We further remark that one cannot expect Theorem 1 to holdin a larger half-plane, at least in such a general framework, since, for example, the abscissa ofuniform convergence of the Dirichlet L -functions with primitive character equals 1, and such L -functions are universal in 1 / < σ <
1. We refer to Kaczorowski-Perelli [6] for a discussionof the convergence abscissae of L -functions.The interest of Bombieri and Ghosh in the above problem was related to the expectationthat the real parts β of the zeros of linear combinations of L -functions are dense in the interval(1 , σ ∗ ), where σ ∗ is the supremum of the β ’s. However, such expectation has been shown to beincorrect by Righetti [11], by means of counterexamples of rather general nature. The rigidityproperty of the translates proved in Theorem 1, and in particular the fact that the vector Y in(3) is the same for all j ’s, may possibly provide a more conceptual explanation for the existenceof “holes” in the distribution of such real parts. However, at present we cannot make precisethis assertion.In the next section we add some remarks on the relevance of integral bases in Theorem 1;these remarks are summarized in Theorem 2 at the end of the paper. Here we finally note thatfor simplicity we stated the equivalence between (i)-(iv) above under the assumption that Λhas an integral basis, although some of the implications hold in full generality; this will be clearfrom the proof. 2. Proofs and remarks
We need the following result about uniformly convergent D-series, which we couldn’t find inthe literature.
Lemma 1.
Equivalent general Dirichlet series have the same abscissa of uniform convergence.
Proof.
Let F ( s ) be as in (1); we use the following formula for σ u ( F ) due to Kuniyeda [8].For x ∈ R let T x = sup t ∈ R (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X [ x ] ≤ λ n Lemma 2. Let F j ( s ) , j = 1 , . . . , N , be as in Theorem and let τ m be a sequence of realnumbers. Then there exists a subsequence τ m k such that, as k → ∞ and for j = 1 , . . . , N , F j ( s + iτ m k ) converges uniformly on any closed vertical strip inside σ > σ u ( F j ) to a gen-eral Dirichlet series G j ( s ) with exponents Λ , and ( G ( s ) , . . . , G N ( s )) is vector-equivalent to ( F ( s ) , . . . , F N ( s )) . Proof. Let B = ( β ℓ ) be an integral basis of the exponents Λ of the F j ( s ), and let θ m,ℓ = (cid:26) − τ m β ℓ π (cid:27) , m, ℓ = 1 , , . . . , where { x } denotes the fractional part of x . Since 0 ≤ θ m,ℓ < 1, by Helly’s selection principle,see Lemma 1 of Section 8.12 of [1], there exist a subsequence m k and a sequence of real numbers θ ℓ such that lim k →∞ θ m k ,ℓ = θ ℓ (4)for every ℓ ≥ 1. Next we define Y = (2 πθ ℓ ) and, for j = 1 , . . . , N , G j ( s ) = ∞ X n =1 a j ( n ) e i ( RY ) n e − λ n s , (5)where R = ( r n,ℓ ) is the Bohr matrix such that Λ = RB . Clearly, ( G ( s ) , . . . , G N ( s )) is vector-equivalent to ( F ( s ) , . . . , F N ( s )) by definition, and now we show that every F j ( s + iτ m k ) con-verges to G j ( s ) uniformly over any closed vertical strip inside σ > σ u ( F j ).We first note that since B is an integral basis of Λ we have e − iλ n τ mk = e πi P ℓ r n,ℓ ( − τmk βℓ π ) = e πi ( P ℓ r n,ℓ θ mk,ℓ ) , hence e − iλ n τ mk − e i ( RY ) n = e πi ( P ℓ r n,ℓ θ ℓ ) (cid:0) e πi P ℓ r n,ℓ ( θ mk,ℓ − θ ℓ ) − (cid:1) . (6)Moreover, recalling that the row entries of R are almost always 0, for every n ≥ c n ≥ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)X ℓ r n,ℓ ( θ m k ,ℓ − θ ℓ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c n max ℓ with r n,ℓ =0 | θ m k ,ℓ − θ ℓ | . (7) Let now W j be a closed vertical strip inside σ > σ u ( F j ), and let ε > M = M j ( ε ) such thatsup s ∈ W j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n>M a j ( n ) e − λ n ( s + iτ mk ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X n>M a j ( n ) e i ( RY ) n e − λ n s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)! < ε. (8)Next, writing C = C j ( ε ) = max n ≤ M c n and H = H j ( ε ) = max s ∈ W j X n ≤ M | a j ( n ) | e − λ n σ , in view of (4) there exists k = k j ( ε ) such that for k ≥ k max ℓ with r n,ℓ =0 | θ m k ,ℓ − θ ℓ | < ε/CH (9)for every n ≤ M . Hence, from (6)-(9), for k ≥ k we have thatsup s ∈ W j | F j ( s + iτ m k ) − G j ( s ) | < ε + max s ∈ W j X n ≤ M | a j ( n ) | (cid:12)(cid:12) e − iλ n τ mk − e i ( RY ) n (cid:12)(cid:12) e − λ n σ < ε, (10)and the lemma follows. (cid:3) Proof of Theorem 1. From (i) applied with ε = 1 /m , m = 1 , , . . . , we obtain a sequence τ m such that F j ( s + iτ m ) converges uniformly to f j ( s ) over K j , for j = 1 , . . . , N . Thanks toLemma 2 there exists a subsequence τ m k such that F j ( s + iτ m k ) converges uniformly over K j to G j ( s ). Hence f j ( s ) = G j ( s ) by the uniqueness of the limit and of the analytic continuation,and (ii) follows from the properties of the G j ( s )’s in Lemma 2.Suppose now that the f j ( s )’s are as in (ii), hence their coefficients b j ( n ) are as in (3) with thesame Y = ( y ℓ ), and let R = ( r n,ℓ ) be the Bohr matrix of a basis B = ( β ℓ ) of Λ. Note that herewe do not assume that Λ has an integral basis and that the K j ’s have an accumulation point. Given ε > τ ∈ R , thanks to Lemma 1 let, as in the proof of Lemma 2, M = M ( ε ) > j =1 ,...,N max s ∈ K j | F j ( s + iτ ) − f j ( s ) | < ε + max j =1 ,...,N max s ∈ K j X n ≤ M | a j ( n ) | (cid:12)(cid:12) e − iλ n τ − e i ( RY ) n (cid:12)(cid:12) e − λ n σ . (11)Recalling the properties of the Bohr matrices, we express the exponents λ n by means of thebasis B , write r n,ℓ = a n,ℓ /q n,ℓ and finally denote by Q = Q ( ε ) the least common multiple of allthe q n,ℓ ’s, with n ≤ M and ℓ ≥ 1, such that r n,ℓ = 0. We thus obtain, for n ≤ M , that e − iλ n τ − e i ( RY ) n = e πi P ℓ m n,ℓ ( yℓ πQ ) (cid:0) e πi P ℓ m n,ℓ ( − βℓτ πQ − yℓ πQ ) − (cid:1) (12)with certain m n,ℓ ∈ Z . Since the β ℓ are Q -linearly independent, by Kronecker’s approximationtheorem (see e.g. Chapter 8 of Chandrasekharan [4]) for every δ > τ ∈ R suchthat (cid:13)(cid:13)(cid:13)(cid:13) − β ℓ τ πQ − y ℓ πQ (cid:13)(cid:13)(cid:13)(cid:13) < δ (13)for all ℓ involved in (12) with n ≤ M , where k x k denotes the distance of x from the nearestinteger. As in Lemma 2, by an obvious choice of δ in terms of ε , of F j ( s ) and K j for j = 1 , . . . , N and of max n ≤ M P ℓ | m n,ℓ | , from (11)-(13) we obtain that there exists τ ∈ R such thatmax j =1 ,...,N max s ∈ K j | F j ( s + iτ ) − f j ( s ) | ≪ ε, and (i) follows.Finally, clearly (iii) implies (i), and replacing Kronecker’s approximation theorem by theKronecker-Weyl theorem (see Appendix 8 of [7] or Remark 1.1 on p.96-97 in [9]) in the aboveproof that (ii) implies (i), we can show that (ii) implies (iii) as well. Moreover, clearly (iv)implies (i), while (i) implies (iv) thanks to Lemma 2 exactly as in the above proof that (i)implies (ii), choosing τ k = τ m k . The proof of Theorem 1 is now complete. (cid:3) We conclude with some remarks about the relevance of integral bases in Theorem 1. Wealready remarked that the D-series with an integral basis contain the ordinary D-series. Asimple but interesting example of non-ordinary D-series with an integral basis is the Hurwitzzeta function ∞ X n =0 n + α ) s with a transcendental 0 < α < 1. Indeed, in this case the exponents λ n = log( n + α ) are all Q -linearly independent, see Davenport-Heilbronn [5], therefore Λ is already a basis and hence R is the identity matrix.Even if Λ does not have an integral basis, it is still possible to say something on the targetfunctions f j ( s ) by a variant of the above arguments, although such a set may be larger in thiscase since we have seen that (ii) implies (i) in full generality. From now on we assume (i) as inTheorem 1, but not anymore that Λ has an integral basis. We first note that by a variant ofthe first steps of Lemma 2, namely considering the double sequence θ m,n = (cid:26) − τ m λ n π (cid:27) , m, n = 1 , , . . . and the sequence θ n obtained as in (4), we are led to the D-series G j ( s ) = ∞ X n =1 a j ( n ) e πiθ n e − λ n s , j = 1 , . . . , N, (14)instead of those in (5). Next, we observe that a (simpler) variant of Lemma 1 shows that σ u ( G j ) = σ u ( F j ), for j = 1 , . . . , N . Indeed, for every ε > k = k ( x ) such that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X [ x ] ≤ λ n Under the assumptions of Theorem , with Λ not necessarily having an integralbasis, suppose that (i) holds. Then the f j ( s ) ’s are general Dirichlet series with coefficients b j ( n ) and the same exponents Λ , and satisfy the following properties. For j = 1 , . . . , N | b j ( n ) | = | a j ( n ) | , σ u ( f j ) = σ u ( F j ) and S f j ( V j ) = S F j ( V j ) , where V j is any open vertical strip inside σ > σ u ( F j ) . Moreover, (i) holds for the f j ( s ) ’sdescribed in (ii) of Theorem . Similar remarks and variants, namely without assuming the existence of an integral basis,apply also to the equivalence of (i) with (iii) and (iv) in Theorem 1. However, f j ( s ) may notbe equivalent to F j ( s ), as shown by the following example by Bohr [2, pp.151–153]. Let λ n = 2 n − n − , F ( s ) = ∞ X n =1 e − λ n s , f ( s ) = − F ( s ) . In this case all bases B of Λ consist of a single rational number, and since the least commonmultiple of the denominators of the λ n is ∞ , no one is an integral basis. Moreover, the Bohrmatrix R such that Λ = RB reduces to an infinite column vector, hence the vectors Y in (2)reduce to a single real number; thus the set of D-series equivalent to F ( s ) consists of its verticalshifts. Further, as shown by Bohr, f ( s ) is not equivalent to F ( s ). On the other hand, f ( s )satisfies (i) in Theorem 2 with τ = 2 π Q n ≤ m (2 n − m = m ( ε ). Acknowledgements. This research was partially supported by PRIN2015 Number Theoryand Arithmetic Geometry . A.P. is member of the GNAMPA group of INdAM, and M.R. waspartially supported by a research scholarship of the Department of Mathematics, University ofGenova. References [1] T.M.Apostol - Modular Functions and Dirichlet Series in Number Theory - Springer Verlag1976.[2] H.Bohr - Zur Theorie der allgemeinen Dirichletschen Reihen - Math. Ann. (1918),136–156.[3] E.Bombieri, A.Ghosh - Around the Davenport-Heilbronn function - Russian Math. Surveys (2011), 221–270.[4] K.Chandrasekharan - Introduction to Analytic Number Theory - Springer Verlag 1968.[5] H.Davenport, H.Heilbronn - On the zeros of certain Dirichlet series - J. London Math.Soc. (1936), 181–185.[6] J.Kaczorowski, A.Perelli - Some remarks on the convergence of the Dirichlet series of L -functions and related questions - To appear in Math. Z.[7] A.A.Karatsuba, S.M.Voronin - The Riemann Zeta-Function - de Gruyter 1992.[8] M.Kuniyeda - Uniform convergence-abscissa of general Dirichlet’s series - Tˆohoku Math.J. (1916), 7–27.[9] K.Matsumoto - A survey on the theory of universality for zeta and L -functions - In NumberTheory: Plowing and Starring Through High Wave Forms , ed. by M.Kaneko et al. , p.95–144, World Scientific 2015.[10] M.Righetti - On Bohr’s equivalence theorem - J. Math. An. Appl. (2017), 650–654;corrigendum ibid. (2017), 939–940.[11] M.Righetti - On the density of zeros of linear combinations of Euler products for σ > A theorem on the “universality”’ of the Riemann zeta-function (Russian) -Izv. Akad. Nauk SSSR Ser. Mat. (1975), 475–486. English transl. Math. USSR-Izv.9