ON THE OVERCONVERGENCE OF CERTAIN SERIES

In this work, we consider certain class of exponential series with polynomial coefficients and study the properties of convergence of such series. Then we consider a subclass of this class and prove certain theorems on the overconvergence of such a series, which allow us to determine the conditions under which the boundary of the region of convergence of this series is a natural boundary for the function f defined by this series.

If] " Pn(x) exp-knS (i.I) 1 mn where Pn (s) anjS an..j are complex constants with an,mnO s=g+ir, j=O ((,I) E I and (k n) is a sequence of complex numbers such that (l nl) a D-sequence.That is to say (Iknl) is a sequence of positive real numbers Max Ilanl/6 (0, mn) (1.4) mn lim sup n /n6  (1.5) Let gn be the set of points of ( which are zeros of P (s) and n gd LJ n Let us denote by the derived set of and n g =I s6 Pn (s) 01 where (nj) is an infinite subsequence of  (ni) depea4/ng on s let dug is a closed set.Let us suppose thatis non empty.We put Iog l(s)exp(-s) 6(n,s) for sufficiently large n In his paper, using a technique similar to that used by M. B lamber[ and J. Simeon [2], we prove two lemmas for a LC-dichleian elemen which enable us o discuss Zhe proies of absolute convergence and uniform conver- gence for (I.I) inexclusively.Then we prove Jentzsch's heorem for a L-dichletian elemen tha is for element of he type (I.I) where k n are positive real numbers satisfying (1.2) [(kn) is a D-sequence) and a heorem on he overconvergence for a L-dichletian element.

DEFINITION.
It is said that a function is sub-lipschitzian on an open set, if it is lipschitzian on each compact subset of that open set.
LEMMA 1.Let be any compact subset of Then the following assertions are true. (1) If <(R) and If there exists a s o 6 _w such that [6w(So) < then the function 6 is sub-llpschitzlan on (- PROOF.Let w e dist (],) Then it is easy to see that v 3 V {J (l'2""'mn) = nj ds, s ]0,[ n' nn' where ds, is the open disc centred at s and of radius and (Cnj) [1,2,..., mn] is the sequence of zeros of Pn(S) (with its order of mul- plicity is taken into account).More precisely let us show that, v B v [j6(1,2,...,mn) =Znj'ds,e e]0,eX[ n' nn' s Let G e ds, e It is evident that G the closure of G e is a compact s6 subset of -Let e'9 ]0,e-e[ where e]0,e  ]m 1 lo(l+lS-S') s-s'l mn Is-s' .(So)<(R) SoEG-g which completes the proof of the lemma.
Under the condition (2) of Lemma 1 6 is continuous on (13-g which implies that cz is an open subset of -ge but /wz can have several connected components.LEMMA 2. When 8" < then / = v v 3 n(S)exp(-knS)l < exp(-nl(a-8')) Kc-uc 8'>8 n' n>n' s6K PROOF.Let a 6 R such that /*c @ # (otherwise the lemma is trivial) and let be a compact subset of /.a.
we have from lemma i, 3 6(n,s)-6(n,s') p,nS-S' '6]0,dist( ,)[ n'(=n ,) nan' (s,s')6 m n I + In particular, where Pc', n e' k n v v 6(n,s)-6(n,s') Pe',nS-S' s6 s' 6ds, n, and hence (n,s') , (n,s) ',n s-s' ) sh '>8 e'] 0,dist(e, )[ n' n>n' s'6d s,, we have Using (2.1) for the particular pair (sj,,s) Max[nsj,8,, j6(l...k)] and as [s-5 < e we have 3 6(n s) > -s(l+) , 8'>8 e'] 0,dist(ee, )[ n" nan" where s is any arbitrary point of g and n" does not deend B 6(n,s) > --8' 8'>8e cE]0,h[-n" nan" sE] As 8' is arbitrary and strictly greater than 8 e we have When 8 e < L < the LC-dirichletian element If] converges absolutely on 2,L+Se and uniformly on any compact subset of ,L+8 PROOF Let us suppose that 2),L+8 is non empty Let o be a compact subset of if] diverges on (-ge-/)eo 6e(s) < 0 and 3 6 (s) < -Hence 3 3 6(nj s) <-0 06R + e iii) ]>0 the sequence of functions (8 ) where n n 8n pO 9 s =I exp(-S(kn-k)) is bounded on ] EXAMPLE.-If (k n) is a D-sequence and 3 Inf(k-'l-k,n-,, =P then it p>0 is easy to see that (k n) is of the type (h) If the D-sequence (kn) is of the type (A) then we can easily show that L 0 Now let us prove Jentzsch's theorem for L-dirichletian element.This theorem for Dirichlet series with complex exponents was proved by JP-'(s) exp (-sk) n, =n let E denote the union of all sets En, n, corresponding to all pairs (n,n') and E be the set formed by the points which are zeros for an infinity of polynomials S (s) Let us put E EduE where E d is the derived n,n set of E E is a closed subset of ( It is evident that E g and E D g and hence E w D g vVe suppose in what follows that -E @ (which implies -g @) Then we have THEOREM 2.
Vhen the D-sequence (k) is of the type (79 and w < then we have (Fr(2>o) N (-g) c E PROOF.Let us suppose that the theorem is not true.Then there exists point b 6 (Fr(.o)N-g) and a disc d(b,p) centred at b of radius Ve have IPn (s)exp(-kn s) An(l+Is I) lexp(-knS) and From any extract'ed subsequence of (T,,** n we can extract a subsequence i, which converges uniformly on E and the limit function is holomorphlc on the Int Let E 1 be a compact subset of d(b,p) N.,o such that InthNIntE 1 .
Then we have Then the subsequence extracted from the arbitrarily extracted sub- sequence of (T,, 'n converges uniformly on UE to a limit function holo- morphic in Int(EiLJl) and continuous on the boundary of EUE 1 and takes the value one at each point of E l Hence the limit function takes the value one at each point of K UE 1 This results that the sequence (Tn l,n) conver- ges to the same limit function on Finally, let us prove a theorem on the overconvergence of If] defined by (7).Before proving the theorem let us note that REMARK '3.Let A be any compact subset of -6 and (k n) be a D-sequence of the type (A) We have IPn(s)exp(-k s) An(l+Is l)mnexp(-ok ).-When (k n) is a D-sequence of the type (h) 8" < and 8" # if there exist an infinite subsequence (n) ;6 IN, of  and a sequence of strictly positive numbers (8) such that Inf[5(s)IS6l) which implies that a i is a finite number.Hence ) is a D-sequence of the type (A) and a A -8' > 0 there exists  When the polynomial P (s) reduces to a complex number a we n n,o get the famous Ostrowaski's theorem [I] for Dirichlet series.Our theorem mn contains G.L. Lunt'z theorem [5] as a particular case when P (s) anS n COROLLARY.In theorem 3 if we replace (2.3) by the condition that there exists a sequence (8 n) of strictly positive numbers such that lim e n and 3 v kn+1 > (l+en)k n then each point of (Fr/o)-g n-n' nan' is a singular point for f defined by (2.2).In particular if (Fr 2o)c_ (-g then Fr /o is a natural boundary for f PROOF.Let us suppose that the corollary is false.Then there exists a point b 6 (Fr 9 )[ (E-g and a disc d(b,p) centred at b and of radius wo p > 0 on which f is holomorphic.As a result of theorem 3 the sequence (S n) converges on d(b,p) From remark 2 [f] diverges on ;-g-o There exists necessarily points common to (E-e-/eo and d(b,p) For these points there is a contradiction which establishes the corollary.GALLIE T.M.
in ]0,[ The set of discs ds, indexed by s ) is an infinite subsequence of Therefore [Pnj(S)exp(-ln.S)l] > exp([knj[) > and which shows that [f] diverges on -ge-eo When L 0 we have convergence of the series (1 .I) in /)eo c (B-g e and divergence in We do not discuss the property of convergence of the series in eFrom here onwards we consider a L-dirichletian element, If] , P (s) exp(-k s) is a D-sequence (here k are positive real numbers).nn DEFINITION. It s said that a D-sequence (n) is of the type(A)    if the following conditions are satisfied i) the Ditichlet series exp(-k s) converges on P s(kj-kn )) converges on Po Let n6IN-[0] j=n 8 (s) be its sum at the point s); n ii) v > 0 the sequence of functions (8) where 8 "P 9 kn) < 8' Let us take a certain ' > 8 and put 8'>* n(=ns,) n>n' o m 8' Log[l+sup[ Isl/s6d(b,p)]] -Inf[o/s6d(b,p)] and hence fA IPn(s)exp(-k s) < A exp(k ).From the definition of o we n>n' s6d(b,p) < exp(o'k n) Hence putting n Max(no,n'o,n"n(S)] is defined to be equal to exp((I/kn) iOg S n (s)) where i' l'n Im Log S n(S) 6 ]-T,r] For each integer n >n the function n n) is of the type (A) which implies L 0 we have limexp(-lIn) i. n-== n n nnl is bounded and hence nor- Hence the sequence of functions (Tnl mal on d(b,p) Let be a compact subset of d(b,p) such that Int D2) # wo

1
As is any arbitrary compact subset of d(b,p) and E 1 is any ar- bitrary compact subset of d(b,p) N ,o such that Int K [ Int E is arbitrary.Hence we arrive at a contradiction that o s o 6 /).oI-* which establishes the result.
IPn(s)exp(-knS) An(l+mA )mnexp(mAkn) where mh sup[ sl/sE] As A is a compact set mh ciently large sequence [Snv(S)} vEIN, where Snv(S) j=l P.(s)exp(-kjs),j conver- ges at each point s of any open simply connected subset (whose intersection with )*,8* in non empty) of an open set included in (I;-g* in which the function f defined by If] is holomo[phic.PROOF.Let us choose 3 bounded domains ,i 2 and A 3 in the following manner-A1 A2 A2 A3 A3 ([;-g* I * and A 3 is included in an open subset of -g* in which the function f defined by If] is holomorphic.Furthe let Fr(A I) Fr(A 2) and Fr(A 3) satisfy a condition of Hadamard's type, namely 3 Log.M 2 g b Log M 1 + (1-b) Log M 3 bE] 0,1[ where V M Max[ If(s) I/sEFr(Ai)] Log B(')+Cl-b)Log B' ()+I-b(l+%)(cA -')+l-b) Laboratoire de Mathmatiques Pures Institut Fourier dpendant de l'Universlt Sclentifique et Mdicale de Grenoble associ C.N.R.S. B.P. 116 38402 ST MARTIN D'HERES (France) By the following method, we obtain a bigger set of absolute convergence for If] Let /)eL be supposed to be non-empty and L <