Entire functions defined by Dirichlet series

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1 J. Math. Anal. Appl Entire functions defined by Dirichlet series Lina Shang, Zongsheng Gao LMIB and Department of Mathematics, Beihang University, Beijing 83, China Received 2 August 26 Available online 2 July 27 Submitted by Richard M. Aron Abstract The present paper has the object of showing that entire functions defined by Dirichlet series behave in many respects like lacunary entire functions. Both have the property of being large except in very small neighbourhoods of their zeros. 27 Elsevier Inc. All rights reserved. Keywords: Pits property; Maximum modulus; Maximum term; Index of maximum term. Introduction and main results It is well known that many entire functions have the property that they are large except in very small areas of the complex plane [4]. These small areas are known as pits which are defined by Littlewood and Offord. Broadly, a pit is a simply connected domain of exponentially small diameter, in the interior of which the function f has one or more zeros and on the boundary f is a constant. Pits are reckoned with a multiplicity equal to the number of zeros they contain. By Rouché s theorem, if the complex number a is not too large in modulus, f takes the value a in a pit as many times as it has zeros [2]. In [3] Offord studied the family of non-zero order entire functions fz= A n z λ n, where the λ n are integers such that ln n lim = α<..2 n ln λ n Writing Mr for the maximum modulus of f on z =r, mr for the maximum term and Nr for the index of maximum term, Offord proved all entire functions of the form. have the pits property.. The work is supported by the National Basic Research Program of China 973 Program, Grant No. 25CB3292. * Corresponding author. addresses: sln-88@63.com L. Shang, zshgao@buaa.edu.cn Z. Gao X/$ see front matter 27 Elsevier Inc. All rights reserved. doi:.6/j.jmaa

2 854 L. Shang, Z. Gao / J. Math. Anal. Appl Theorem A. See [3]. If fz= A n z λ n is an entire function such that.2 and ln Nr ln ln r are satisfied, then outside a set of finite logarithmic measure the following properties hold. i All the zeros of f are in pits and to all ε> there exists r ε such that for all r>r fz Mr ε outside the pits of f. ii Each pit is contained in a disk of diameter not exceeding /Nr 3. iii The sum of the diameters of all pits in z R does not exceed /NR. From Theorem A we can get a picture of a typical entire function of the family.. Indeed the surface defined by Z = ln fz, where z goes over the complex plane, is like a bowl about the vertical ordinate Z, and ln fz increases to infinity with z. The bowl is studded with deep pits which go right down to the z-plane and occupy only an exponentially small area [4]. This paper deals with the Dirichlet series b n e λns,.3 where {b n } C, <λ n < and s = σ + it σ,t are real variables. Furthermore, we assume that the series.3 satisfies ln n lim = α<.4 n ln λ n and ln b n lim =..5 n λ n In this paper we do not require λ n must be integers. By.4 and.5, it follows from Valiron formula [5] that the abscissa of uniform convergence of.3 are and so the series.3 defines an entire function fs in the complex plane. We show all entire functions defined by Dirichlet series.3 which satisfies.4.5 have similar pits property as lacunary entire functions.. Put { } Mσ = sup fσ + it, t R { mσ = max bn e λ nσ }, n { Nσ= max λn ; b n e λnσ = mσ }. n Mσ,mσ and Nσ are respectively called the maximum modulus, the maximum term, and the index of maximum term. Nσ is a non-decreasing step function and it plays a key role for pits property. Irregularities in the growth of Nσ introduce difficulties especially for entire functions of infinite order. It is therefore necessary to replace this function by an order type and this we proceed to do. Definition. With θ positive and σ R, we write σ,θ for the greater of the two numbers Nlne σ + θ and φe σ /θ and then with σ fixed define σ := inf σ,θ, θ> where φt is strictly increasing continuous function of t tending to infinity with t.

3 L. Shang, Z. Gao / J. Math. Anal. Appl The properties of σ are given in Lemma below. It is a continous non-decreasing function of σ. To simplify the notation, by itself will always mean σ. Theorem. If Dirichlet series.3 satisfies.4.5, associated with any number β satisfying α<β<, then for s such that Rs = σ>σ, there is a circumference with center s on which f > exp σ β mσ, where f is the entire function defined by.3 and σ depends only on β and f. The radius of this circumference does not exceed exp δ, where δ satisfies <δ<β.fors outside the pits of f and Rs = σ.6 holds. Theorem 2. If Dirichlet series.3 satisfies.4.5, then for σ outside a set of finite logarithmic measure, there exists σ such that for Rs σ ε, we have ln fs εln Mσ outside the pits of f. Theorem 3. If Dirichlet series.3 satisfies.4.5, then for sufficiently large σ and positive number a, the sum of the radius of the pits of f in the horizontal half-strip S σ ={s Rs σ, Is <a} does not exceed 2a exp 2 σδ, where δ has the same meaning as in Theorem. From the above we can describe the pits property of entire functions defined by Dirichlet series.3 by another picture. The surface defined by S = ln fs, where s goes over the half-plane {s Rs σ } σ is large enough, is like a slope, and ln fs increases to infinity with the real part of s. The slope is studded with deep pits which go right down to the s-plane and the area which the pits occupy does not exceed 4πa 2 exp σ δ in the horizontal half-stripe S σ ={s Rs σ, Is <a}. 2. Preliminary lemmas In the following lemmas we use C for a numerical constant. It will not be the same at each occurrence but it is always independent of all variables. Lemma. σ is a continuous non-decreasing function of σ, Nσ σ N σ + ln + 2. ψnσ and σ + ln + φ { ψ 2 } ψσ 3 2 +, where ψ is the inverse of φ. In particular if φt= expt 2, then σ + ln + ln 3 eσ. 2.3 Proof. The inequation 2. follows on putting θ = e σ /ψnσ in the expression for σ,θ in Definition. Fix σ and consider the functions Nlne σ + θ and φe σ /θ as functions of θ. Since Nlne σ + θ is nondecreasing while φe σ /θ is strictly decreasing it follows that if θ increases from small values we come to a value θ where either Nlne σ + θ = φe σ /θ or we meet a discontinuity of Nlne σ + θ so in either case there is a unique θ such that N ln e σ e σ + θ φ N ln e σ + θ θ

4 856 L. Shang, Z. Gao / J. Math. Anal. Appl and so e σ σ = φ θ. With θ defined as in 2.4 and for δ<θ, ln e σ + δ ln e σ + δ,θ δ ε = N ln e σ + θ ε e σ + δ φ θ δ ε = N ln e σ + θ φ e σ + δ θ δ ε where a b denotes the larger of a and b. So the continuity of follows from that of φ. As regards 2.2, writing ψ for ψ,wehave σ + ln + = inf N ln e σ + e σ + + θ φ ψ 3 2 θ> ψ 3 2 θ Again taking θ = θ e σ /ψ 3 2, we get σ + ln + 3 ψ 2 + φ φ { ψ 2 } ψ 3 2 ψ , 2.5 provided φθ>. In particular, if φt= expt 2, then ψt= ln t 2 and 2.3 follows. In all that follows, unless specifically stated otherwise, we shall use or σ to mean σ defined by taking φt= expt 2. Like what Offord did in [3], we now rewrite condition.4 in a slightly different form. First define ψ 3 2. kσ = [ eσ ]. Let qσ be the number of λ n which does not exceed kσ. Condition.4 becomes lim σ ln qσ ln kσ = α. By α<, we can find β> satisfying α<β< so that qσ < σ β for large σ. β may be redefined from time to time but always so that α<β<. 2.8 Lemma 2. If the Dirichlet series.3 satisfies.4.5, and if k and q are as in 2.6 and 2.7, for sufficiently large σ, we have and λ n 2k+ b n e λnσ exp fs Cσ mσ, ln 3 m σ + ln + Cmσ. mσ,

5 L. Shang, Z. Gao / J. Math. Anal. Appl Proof. According to the arguments of Valiron [6], taking the axes of coordinate OX and OY, if we plot the points A n with the coordinates λ n,g n, where g n = ln b n, we can construct a Newton polygon having some of the points A n as its vertices while the remainder lie either on it or on one side of it. If G λn is the ordinate of the point of the abscissa λ n on the curve Πf, then G λn g n. Define G λ2 G λ λ 2 λ n λ + G λ if n =[λ ], G n = G λj if λ j is integer and n = λ j, G λj+ G λj λ j+ λ j n λ j + G λj if λ j <n<λ j+, then the function n=[λ ] expns G n has the same maximum term as fs. Therefore Write λ n 2k+ b n e λ nσ R n = e G n G n λ n 2k+ e λ nσ G λn e σv G v. 2k+ so that R n is non-decreasing. Then e σv G v = e σk G k e σv k e σ k e σ mσ R k+ R v R k+ R k+ e σ. 2k+ Let e σ = e σ + 2k+ eσ, then σ eσ, and therefore k Nσ. Hence ln 3 e kσ G k e k+σ G k+ or R k+ e σ. Then we have e σ k e σ R k+ R k+ e σ + k ln 3 ln 3 exp ln 3. So 2.9 follows. 2. follows from [ fs b n e λnσ 2k + exp ] ln 3 mσ. As regard 2., if we denote Nσ + ln + = λ v, then m σ + ln + = b v e σ +ln+ Nσ +ln+ mσ + e Cmσ. Definition 2. If fsis any entire function and λ is any positive number, we define the operation D λ fs= e λs d [ e λs fs ] ds and if I is a finite increasing set of positive numbers, say λ,λ 2,...,λ q, then D I fs= D λv fs. λ v I

6 858 L. Shang, Z. Gao / J. Math. Anal. Appl Lemma 3. For any entire function fsand any increasing set I, D I fs q B q q v f v s, where the B q q v are positive numbers given by q B q q v tv = q λ v + t. Proof. It is easy to prove that where B q q D I fs= q q v B q q v f v s, = q λ v,b q =, B q q v = λ qb q q v + Bq q v and B q = Bq q =. From 2.2 we can see 2.2 q B q q v tv = q λ v + t. Hence the lemma Lemma 4. If the Dirichlet series.3 satisfies.4.5, and if α<β<, then for sufficiently large σ, we can find ξ so that ξ s =/ 4 and Pξ exp β mσ, 2.3 where Ps= λ n 2k b ne λ ns, σ =Rs. Proof. By Lemma 2, we can see that the maximum term in Ps is the same as that in fs. We denote by I the index set of all <λ n 2k except for the λ n corresponding to the maximum term which we denote by N. Letq be the number of terms in I so that q satisfies 2.7. By Definition 2, we have D I Ps = [ ] Nσ λ n 2 mσ! 2 q mσ. 2.4 λn I Take the contour C: ξ s =/ 4 and choose ξ such that Pξ is the maximum of Pξ on this contour, by Lemma 3, then D I Ps q B q q v v! 2πi q! Pξ λ n I C Pξ ξ s λn + 4 v+ dξ q! Pξ + 4q. Combining 2.4, the desired results follows.

7 L. Shang, Z. Gao / J. Math. Anal. Appl Lemma 5. There exists a complex number ξ such that s ξ =/ 4 and fξ exp β mσ, 2.5 where α<β<. Furthermore, the number of zeros of f in the disk ξ s 7 8 ln + does not exceed σ β. Proof. 2.5 follows from 2.9 and 2.3. Let nξ,t be the number of the zeros of f in the disk ξ ξ t and let us write ρ = 7 8 ln + + /2 and ρ 2 = 5 6 ln +. Then by Jensen s theorem, nξ,ρ ln ρ 2 ρ ρ 2 ρ nξ,t t dt C 2π 2π ln fξ + ρ 2 e iθ fξ dθ. By 2.3, 2. and 2., f ξ + ρ 2 e iθ C σ + ln + m σ + ln + Cσ mσ. Since the disk ξ s 7 8 ln + lies in ξ ξ ρ, the desired results follows. In the following lemma we make use of the Blaschke for the zeros α v v =, 2,...,n of f in the disk center s and radius ρ. This is defined as ξ; αv = n ρξ α v ξ,s,ρ,α v = ρ v= 2 α v sξ s. It has the property that ξ; α v =if ξ s =ρ and ξ; α v < for ξ s <ρ. Lemma 6. If the Dirichlet series.3 satisfies.4.5 and if α,α 2,...,α n are the zeros of f in the disk ξ s 7 8 ρ, then for all ξ in the disk ξ s 2 ρ, fξ exp β n mσ 2ρ n ξ α v, where ρ = min ln +,. 2 v= 2.6 Proof. Write fξ= χξ ξ; α v, then on ξ s = 7 8 ρ, χξ = fξ C Rξ m Rξ Cσ mσ. By Lemma 5 we can choose ξ so that s ξ =/ 4 8ρ, then χξ fξ exp β mσ. Since χξ/χξ has no zeros in the disk ξ ξ 6 χξ 8ρ,soqξ = ln χξ has a branch which is analytic in this disk and R [ qξ ] = ln χξ χξ β.

8 86 L. Shang, Z. Gao / J. Math. Anal. Appl Hence the function Qξ = qξ is analytic and satisfies Qξ in this disk. By Schwarz s lemma, 2 β qξ Qξ ξ ξ 6 8 ρ in this disk. This dedicates that qξ 2β ξ ξ 6 8 ρ ξ ξ. Taking ξ s 2 ρ, we have ξ ξ 5 8 ρ, and so qξ Cβ, for ξ s 2 ρ, whence R[qξ] Cβ, and so, on redefining β, χξ exp β mσ for ξ s 2ρ. It is easy to see that ξ; α v n 2ρ n ξ α v and the result follows. 3. Proof of theorems v= Proof of Theorem. To complete the proof of Theorem we appeal to Cartan s lemma []. This states that the points ξ for which n ξ αv h n can be enclosed in at most n disks the sum of whose diameters is equal to 4eh. If therefore we choose h = exp δ, then the disk center s and radius 8eh must contain a circumference of radius d at least 4eh, on which n ξ α v expn ln h exp β+δ. By Lemma 6, on this circumference fξ exp β β+δ mσ, we can now choose δ so that δ<β and redefine β, then.6 follows. But by Rouché s theorem this means that either.6 holds throughout the disk ξ s d or that in this disk f has a zero and takes every value not exceeding exp σ β mσ. If the latter occurs we say that it is a pit of f. We have therefore proved that for s outside the pits of f and Rs = σ.6 holds. Proof of Theorem 2. This will follow from the following lemma [3]. Suppose Nσ is a positive non-decreasing function which is continuous on the right. If hσ =ln Nσ 2, then e Nσ N σe h enσ for all σ outside a set of finite logarithmic measure. and From 3., outside a set of finite logarithmic measure, σ Ntdt σ σe h Ntdt N σe h σ σe h σnσ 2eln Nσ 2 3.

9 σ N σ + ln + ln Nσ 2 enσ L. Shang, Z. Gao / J. Math. Anal. Appl with φt= expt 2. Noticing that [7] ln mσ = σ Ntdt + C, we have ln σ ln ln mσ ln Nσ+ ln Nσ+ ln σ 2lnlnNσ+ C and so σ, Mσ Cσ mσ mσ +ε. By Theorem, for σ outside a set of finite logarithmic measure, there exists σ such that for Rs σ ε, we have fs Mσ ε outside the pits of f. Proof of Theorem 3. Let S σ ={s Rs < σ, Is <a} and Φ : s z = e π a s. Since ΦS σ {z z <e π a σ },we consider the number of zeros of gz = b n z a π λ n in z e π a σ. If nr denotes the number of zeros of g in z r, then n e π a σ Caσ By 2. and we have Caσ ln mσ = C + e π a σ +ln+ 2π Caσln M σ n e π a σ a 3 σ. e π a σ nt t dt ln g e π a σ +ln+ +iθ dθ σ + ln +. Ntdt < σ +ε, Hence the number of zeros and so the number of pits of f in S σ cannot exceed a 3 σ. By Lemma, The function σ is a continuous non-decreasing function of σ and so starting with σ = σ we can define a sequence σ n such that σ n+ = 2σ n = 2 n. If now we estimate the sum of the radius of all pits in the rectangle S σn+ S σn, this sum does not exceed a 3 σ n+ exp σ n δ a exp 2 σ n δ and we have

10 862 L. Shang, Z. Gao / J. Math. Anal. Appl n= as desired. References a exp 2 σ n δ 2a exp 2 σδ [] P. Erdős, A.J. Macintyre, Integral functions with gap power series, Proc. Edinb. Math. Soc [2] A.C. Offord, The distribution of the values of an entire function whose coefficients are independent random variables I, Proc. London Math. Soc [3] A.C. Offord, Lacunary entire functions, Math. Proc. Cambridge Philos. Soc [4] A.C. Offord, The pits property of entire functions, J. London Math. Soc [5] G. Valiron, Entire functions and Borel s directions, Proc. Natl. Acad. Sci. USA [6] G. Valiron, Fonctions Analytique, Paris, 954. [7] J.R. Yu, Sur les droites de Borel de certaines fonctions entières, Ann. Sci. École Norm. Sup