1 N N(θ;d 1...d l ;N) 1 q l = o(1)

Transcription

1 NORMALITY OF NUMBERS GENERATED BY THE VALUES OF ENTIRE FUNCTIONS MANFRED G. MADRITSCH, JÖRG M. THUSWALDNER, AND ROBERT F. TICHY Abstract. W show that th numbr gnratd by th q-ary intgr part of an ntir function of logarithmic ordr, whr th function is valuatd ovr th natural numbrs and th prims, rspctivly, is normal in bas q. This is an xtnsion of rlatd rsults for polynomials ovr th ral numbrs stablishd by Naai and Shioawa. 1. Introduction Lt q 2 b a fixd intgr and θ = 0.a 1 a 2... b th q-ary xpansion of a ral numbr θ with 0 < θ < 1. W writ d 1...d l {0,1,...,q 1} l for a bloc of l digits in th q-ary xpansion. By Nθ;d 1...d l ;N w dnot th numbr of occurrncs of th bloc d 1...d l in th first N digits of th q-ary xpansion of θ. W call θ normal to th bas q if for vry fixd l 1 R N θ = R N,l θ = sup 1 N Nθ;d 1...d l ;N 1 = o1 d 1...d l as N, whr th suprmum is tan ovr all blocs d 1...d l {0,1,...,q 1} l. W want to loo at numbrs whos digits ar gnratd by th intgr part of ntir functions. Lt f b any function and [fn] q dnot th bas q xpansion of th intgr part of fn, thn dfin 1.1 θ q = θ q f = 0.[f1] q [f2] q [f3] q [f4] q [f5] q [f6] q..., τ q = τ q f = 0.[f2] q [f3] q [f5] q [f7] q [f11] q [f13] q..., whr th squncs of th argumnts run through th positiv intgrs and th prims, rspctivly. In this papr w considr th construction of normal numbrs in bas q as concatnation of q-ary intgr parts of crtain functions. Th first rsult on that topic was achivd by Champrnown [2], who was abl to show that is normal in bas 10. This construction can b asily gnralisd to any intgr bas q. Copland and Erdös [4] wr abl to show that is normal in bas 10. Ths xampls corrspond to th choic fx = x in 1.1. Davnport and Erdös [5] considrd th cas whr fx is a polynomial whos valus at x = 1,2,... ar always intgrs and showd that in this cas th numbrs θ q f and τ q f ar normal. For fx a polynomial with rational cofficints Schiffr [10] was abl to show that R N θ q f = O1/log N. Naai and Shioawa [8] xtndd his rsults and showd that R N τ q f = O1/log N. In th cas of ral cofficints Naai and Shioawa [7] provd th sam stimat for R N θ q f. In this papr w want to discuss th cas whr fx is a transcndntal ntir function i.., an ntir Supportd by th Austrian Rsarch Foundation FWF, Projct S9611-N13, that is part of th Austrian Rsarch Ntwor Analytic Combinatorics and Probabilistic Numbr Thory. 1

2 2 M. G. MADRITSCH, J. M. THUSWALDNER, AND R. F. TICHY function that is not a polynomial of small logarithmic ordr. Rcall that w say an incrasing function Sr has logarithmic ordr λ if 1.2 lim sup r log Sr log log r = λ. W dfin th maximum modulus of an ntir function f to b 1.3 Mr,f := max x r fx. If f is an ntir function and log Mr,f has logarithmic ordr λ, thn w call f an ntir function of logarithmic ordr λ. To achiv our rsults w combin th following ingrdints. Th first part of th proofs concrns th stimation for th numbr of solutions of th quation fx = a whr a C cf. [3], [11, Sction 8.21] for ntir functions of zro ordr. Following th mthods of Naai and Shioawa [7, 8] w rformulat th problm in an stimation of xponntial sums. Finally, th rsulting xponntial sums ar tratd by an xponntial sum stimat of Bar [1], which was originally usd to show that th squncs fn n 1 and fp p prim ar uniformly distributd modulo 1 for f an ntir function with logarithmic ordr 1 < α < 4 3. Th main rsults of our paprs ar as follows. Thorm 1. Lt fx b a transcndntal ntir function which tas ral valus on th ral lin. Suppos that th logarithmic ordr α = αf of f satisfis 1 < α < 4 3. Thn for any bloc d 1...d l {0,1,...,q 1} l, w hav Nθ q f;d 1...d l ;N = 1 N + on as N tnds to. Th implid constant dpnds only on f, q, and l. For prims w show that τ q f is normal in th following thorm. Thorm 2. Lt fx b a transcndntal ntir function which tas ral valus on th ral lin. Suppos that th logarithmic ordr α = αf of f satisfis 1 < α < 4 3. Thn for any bloc d 1...d l {0,1,...,q 1}, w hav Nτ q f;d 1...d l,n = 1 N + on as N tnds to. Th implid constant dpnds only on f, q, and l. 2. Notation Throughout th papr lt f b a transcndntal ntir function of logarithmic ordr α satisfying 1 < α < 4 3 and taing ral valus on th ral lin. Lt fx = a x =1 b th powr sris xpansion of f. By log x and log q x w dnot th natural logarithm and th logarithm with rspct to bas q, rspctivly. Morovr, w st β := xp2πiβ. Lt p always dnot a prim and b a sum ovr prims. By an intgr intrval I w man a st of th form I = {a,a + 1,...,b 1,b} for arbitrary intgrs a and b. Furthrmor, w dnot by nr,f th numbr of zros of fx for x r.

3 NORMALITY OF NUMBERS GENERATED BY THE VALUES OF ENTIRE FUNCTIONS 3 3. Lmmas First w stat th abov-mntiond rsult of Bar that will prmit us to stimat xponntial sums ovr ntir functions with small logarithmic ordr by choosing th occurring paramtrs appropriatly. Lmma 3.1 [1, Thorm 4]. Lt d and h b intgrs, with 8 h d. Lt a 1,...,a d b ral numbrs and suppos that N h log N 3.1 xp 20 log log N 2 < a h < xp 10 3 h 2, 3.2 Suppos furthr that 3.3 log N a xp 20 log log N 2 log N 10 5 d 3 log d 5. h < d. Thn, writing gx = a d x d + + a 1 x, w hav S = gn N xp 1 2 log N N a h 1/10h. n N Lmma 3.2 [1, Thorm 3]. Undr th hypothss of Lmma 3.1 w hav S = gp P xp clog log P 2 + Plog P 1 a h 1/10h, whr c is a constant dpnding on g. Th following lmma du to Vinogradov provids an stimat of th Fourir cofficints of crtain Urysohn functions. Lmma 3.3 [12, Lmma 12]. Lt α, β, b ral numbrs satisfying 0 < < 1 2, β α 1. Thn thr xists a priodic function ψx with priod 1, satisfying 1 ψx = 1 in th intrval α x β 1 2, 2 ψx = 0 in th intrval β x 1 + α 1 2, 3 0 ψx 1 in th rmaindr of th intrval α 1 2 x 1 + α 1 2, 4 ψx has a Fourir sris xpansion of th form ψx = β α + Aννx, whr 1 Aν min ν,β α, 1 ν 2. Finally, w giv an asy rsult on th limit of quotints of squncs that will b usd in our proof. Lmma 3.4. Lt a n n 1 and b n n 1 b two squncs with 0 < a n b n for all n and a n 3.5 lim = 0. n b n Thn n lim a i n n b = 0. i

4 4 M. G. MADRITSCH, J. M. THUSWALDNER, AND R. F. TICHY Proof. Lt ε > 0 b arbitrary. Thn by 3.5 thr xists an n 0 such that 3.6 a n b n < ε/2 for n > n 0. Lt AN := N n=1 a n and BN := N n=1 b n. W show that thr xists a n 1 such that An/Bn < ε for n > n 1. Thrfor w dfin CN := N n=n b 0+1 n. As 3.6 implis that a n < ε 2 b n for n > n 0 w gt An Bn = An n 0 + i=n a 0+1 i Bn 0 + n i=n 0+1 b i < An 0 + ε 2 Cn Bn 0 + Cn. As b n > 0 w hav that Cn for n. Thus An 0 + ε 2 lim Cn n Bn 0 + Cn = ε 2. Thrfor thr is a n 1 n 0 such that An/Bn ε for n > n 1 which provs th lmma. 4. Valu Distribution of Entir Functions Bfor w start with th proof of th thorms, w nd an stimation of th numbr of solutions for th quation fx = a with f a transcndntal ntir function and a C. In this sction w want to show th following rsult. Proposition 1. Lt f b a transcndntal ntir function of logarithmic ordr α. Thn for th numbr of solutions of th quation fx = a th following stimat holds. 4.1 nr,f a log r α 1. As usual in Nvanlinna Thory w do not dal with nr,f a dirctly but us a strongly rlatd function, which is dfind by 4.2 Nr,f = r 1 nt,f n0,f dt n0,flog r t in ordr to prov th proposition. Th connction btwn nr,f a and Nr,f a is illustratd in th following lmma. Lmma 4.1 [3, Thorm 4.1]. Lt fx b a non-constant mromorphic function in C. For ach a C, Nr,f a is of logarithmic ordr λ + 1, whr λ is th logarithmic ordr of nr,f a. Th nxt lmma provids us with a vry good stimation of th ordr of Nr,f a. Lmma 4.2 [9, Thorm]. If f is an ntir function of logarithmic ordr α whr 1 < α 2, thn for all valus a C log Mr,f Nr,f a log M rlog r 2 α N rlog r 2 α. Now it is asy to prov Proposition 1. Proof of Proposition 1. As f fulfills th assumptions of Lmma 4.2 w hav that 4.3 Nr,f a Mr,f log r α. Thus w hav that Nr,f a is of logarithmic ordr α and thrfor by Lmma 4.1 w gt that nr,f a is of logarithmic ordr α 1.

5 NORMALITY OF NUMBERS GENERATED BY THE VALUES OF ENTIRE FUNCTIONS 5 5. Proof of Thorm 1 W fix th bloc d 1...d l throughout th proof. Morovr, w adopt th following notation. Lt Nfn b th numbr of occurrncs of th bloc d 1...d l in th q-ary xpansion of th intgr part fn. Furthrmor, dnot by lm th lngth of th q-ary xpansion of th intgr m, i.., lm = log q m + 1. Dfin M by 5.1 M 1 n=1 lfn < N M lfn. n=1 Bcaus f is of logarithmic ordr α < 4 3 w asily s that lfn log M α 1 n M. Thus Nθ qf;d 1...d l ;N M Nfn lm W dnot by J and J th maximum lngth and th avrag lngth of fn for n {1,...,N}, rspctivly, i.., 5.2 n=1 J := max l fn log 1 Mα, J := 1 M l fn log M α, M n=1 whr stands for both and. Not that from ths dfinitions w immdiatly s that 5.3 N = M J + Olog M α. Thus in ordr to prov th thorm it suffics to show M 5.4 Nfn = 1 N + on. n=1 In ordr to count th occurrncs of th bloc d 1...d l in th q-ary xpansion of fn 1 n M w dfin th indicator function { 1 if l It = d iq i t t < l d iq q l, othrwis which is an 1-priodic function. Indd, writ fn in q-ary xpansion for vry n {1,...,M}, i.., fn = b r q r + b r 1 q r b 1 q + b 0 + b 1 q , thn th function It is dfind in a way that Iq j fn = 1 d 1...d l = b j 1...b j l. In ordr to writ Nfn proprly in trms of I w dfin th substs I l,...,i J of {1,...,M} by n I j fn q j l j J. Evry I j consists of thos n {1,...,M} for which w can shift th q-ary xpansion of fn at last j digits to th right to count th occurrncs of th bloc d 1...d l. Using ths sts w gt J fn 5.6 Nfn =. I j=l n I j In th nxt stp w fix j and show that I j = I j M consists of intgr intrvals which ar of asymptotically incrasing lngth for M incrasing. As I j consists of all n such that fn q j ths n hav to b btwn two zros of th quation fx = q j. By Proposition 1 th numbr of q j

6 6 M. G. MADRITSCH, J. M. THUSWALDNER, AND R. F. TICHY solutions for this quation is nm,f q j log M α 1. Thrfor w can split I j into intgr subintrvals I j = {n ji,...,n j m ji 1} whr m ji is th lngth of th intgr intrval and log M α 1. Thus th lngth of th intgr intrvals is incrasing, i.., Mlog M 1 α m ji M. Thus w gt that 5.7 Nfn = J j I j=l n ji n<n ji+m ji fn Following Naai and Shioawa [7, 8] w want to approximat I from abov and from blow by two 1-priodic functions having small Fourir cofficints. In particular, w st 5.8 α = α + = l d λ q λ + 2δ i 1, β = λ=1 l d λ q λ 2δ i 1, β + = λ=1 q j. l d λ q λ + q l 2δ i 1, λ=1 l d λ q λ + q l + 2δ i 1, λ=1 = i, + = i. W apply Lmma 3.3 with α,β, = α,β, and α,β, = α +,β +, +, rspctivly, in ordr to gt two functions I and I +. By th choics of α ±,β ±, ± it is immdiat that 5.9 I t It I + t t R. Lmma 3.3 also implis that ths two functions hav Fourir xpansions 5.10 I ± t = q l ± A ± ννt satisfying 5.11 A ± ν min ν 1,δ i ν 2. In a nxt stp w want to rplac I by I + in 5.6. To this mattr w obsrv, using 5.9, that It I + t I + t I t A ± ννt. Togthr with 5.6 this implis that Nfn = J j j=l n ji n<n ji+m ji fn I + q j + O Insrting th Fourir xpansion of I + this yilds J j 5.12 Nfn = 1 + O j=l n ji n<n ji+m ji A ± ν ν fn q j. A ± ν ν fn q j. Bcaus of th dfinition of M and J in 5.1 and 5.2, rspctivly, and th stimat in 5.3 w gt that 5.13 J j j=l n ji n<n ji+m ji 1 = JM + OlM = N + OlM.

7 NORMALITY OF NUMBERS GENERATED BY THE VALUES OF ENTIRE FUNCTIONS 7 Insrting this in 5.12 and subtracting th main part Nq l w obtain Nfn N J j 5.14 j=l n ji n<n ji+m ji Now w considr th cofficints A ± ν. Noting 5.11 on ss that { ν 1 for ν δ i, A ± ν δ i ν 2 for ν > δ i. Estimating trivially all summands with ν > δ w gt ν A ± ν q j fn δ i 5.15 ν 1 ν q j fn + i. Using this in 5.14 and changing th ordr of summation yilds Nfn N J j δ i 5.16 j=l n ji n<n ji+m ji ν A ± ν q j fn + lm. ν ν 1 q j fn + lm. Th crucial part is now to stimat th xponntial sum containing th ntir function f. Dfin SX := ν 5.17 q j fn. n X W now trat th sum SX by a similar rasoning as in th proof of Bar [1, Thorm 2]. W will show that th sum only dpnds on f and X. To this mattr w lt th paramtr d occurring in Lmma 3.1 b a function of X, in particular, w st 5.18 d = dx = 10 2 log X 1/3 log log X 2, which tnds to infinity with X s quation 11 of [1]. Morovr, w dfin th polynomial g j x = ν q j a 1x + + a d x d by th first d summands of th powr sris of ν q f. Th paramtr h of Lmma 3.1 will also b a j function of X. In particular, w st h = hx to b th largst positiv intgr such that h d and 5.19 X h+ 1 ν 2 < q j a h. As shown in [1], h also tnds to infinity with X. Up to now w hav not chosn a valu for δ i. For th momnt, w just assum that δ i h bcaus this choic implis that th summation indx ν varis only ovr positiv intgrs that ar lss than h. Thus th logarithmic ordr of ν q fn is lss than 4 j 3. Indd, ν 5.20 log q j fn < log h j log q + log fn < log log X + log X α < log Xᾱ whr ᾱ = α + ε < 4 3. Not that g j satisfis th conditions of Lmma 3.1. Th stimat for th ν logarithmic ordr of q fn will nabl us to rplac f by g j j in 5.17 causing only a small rror trm. This will thn prmit us to apply Lmma 3.1 in ordr to stimat SX. By 5.20, quation 15 of [1] implis that for d as in 5.18 ν q j a 5.21 t Xt < 2X 1 t>d

8 8 M. G. MADRITSCH, J. M. THUSWALDNER, AND R. F. TICHY and thrfor s [1] n X fn ν q j g j n + π. By this w can us Bar s stimations for xponntial sums ovr ntir functions containd in Lmma 3.1 and gt with d = dx and h = hx dfind in 5.18 and 5.19, rspctivly, 5.22 n X SX X xp 1 2 log X X xp h. Now it is tim to st δ i for vry i. As ν changs th cofficints of th function undr considration w calculat for vry ν = 1,...,dm ji th corrsponding h ν m ji. In ordr to fulfill th constraint on th logarithmic ordr w nd to chos δ i smallr than th smallst h ν m ji with ν δ i. Thus w st 5.23 δ i := max{r dm ji : r min{h ν m ji : ν r}}. This is always possibl sinc h ν m ji 1. For this choic w also hav δ i h ν m ji and δ i as m ji bcaus th minimum of th h ν m ji tnds to infinity for m ji. Doing this for vry i = 1,..., i.., for vry intgr intrval comprising th st I j w can apply 5.22 with X = m ji and us th fact that δ i is th smallst h ν m ji for i. This yilds j δ i ν ν 1 q j fn j δ i ν 1 Sm ji n ji n<n ji+m ji δ i ν 1 m ji xp 1 2 log m ji mji xp δ i j m ji xp 1 2 log m ji mji xp δ i log δ i. As w do not now th asymptotic bhavior of δ i w hav to distinguish th cass whthr xp δ i is gratr or smallr than xp 1 2 log m ji 1 3. In both cass w can assum that m ji is sufficintly larg. Suppos first that xp 1 2 log m ji 1 3 > xp δ i holds. As δ i dm ji log m ji 1/3 w gt xp 1 2 log m ji 1 3 log δi xp 1 2 log m ji 1 3 log log mji xp 1 3 log m ji 1 3 and thus xp 1 2 log m ji xp δi log δ i xp 1 3 log m ji xp δi /2. For th scond cas assum that xp 1 2 log m ji 1 3 xp δ i holds. This implis that log δ i log log m ji and w gt xp 1 2 log m ji 1 3 log δi xp 1 2 log m ji 1 3 log log mji xp 1 3 log m ji 1 3. Thrfor w also hav xp 1 2 log m ji xp δi log δ i xp 1 3 log m ji xp δi /2. By this w hav th stimation 5.24 δ i ν 1 n ji n<n ji+m ji By 5.16 w gt that Nfn N ν q j fn J j j=l n ji n<n ji+m ji m ji xp 1 3 log m ji xp δi /2. δ ν ν 1 q j fn + lm

9 NORMALITY OF NUMBERS GENERATED BY THE VALUES OF ENTIRE FUNCTIONS 9 Thus it rmains to show that 5.25 and 5.26 n ji n<n ji+m ji = δ i n ji n<n ji+m ji m ji δ i = o I j. ν ν 1 q j fn = o I j, whr I j = m ji th sum of th lngths of th intgr intrvals. First w considr Thrfor w st a i = mji δ i and b i = m ji. By noting that ai b i = i 0 w ar abl to apply Lmma 3.4 and gt 0 m ji δ i m ji 0. Finally w hav to show W again want to apply Lmma 3.4 by stting a i := m ji xp 1 3 log m ji mji xp δ i /2, b i := m ji. As Mlog M 1 α m ji M w gt that both xp 1 3 log m ji 1 3 and xp δ i /2 tnd to zro. Thus w hav that ai b i 0 for M. An application of Lmma 3.4 togthr with 5.24 givs δ ν 1 n ν ji n<n ji+m ji q fn 0 j m ji xp 1 3 log m ji xp δ i /2 I j m 0 ji for M and thus 5.26 holds. W put 5.25 and 5.26 in our stimat 5.16 and gt togthr with 5.13 that Nfn N J j δ ν ν 1 q j fn + lm j=l n ji n<n ji+m ji J o I j + lm = o JM = on. j=l Thus by 5.4 th thorm is provn. 6. Proof of Thorm 2 Throughout th proof p will always dnot a prim and πx will dnot th numbr of prims lss than or qual to x. As in th proof of Thorm 1 w fix th bloc d 1...d l and writ Nfp for th numbr of occurrncs of this bloc in th q-ary xpansion of fp. By lm w dnot th lngth of th q-ary xpansion of an intgr m. W dfin an intgr P by 6.1 As abov w gt that 1 l fp < N l fp. l fp log P α 2 p P. Again w st J th gratst and J th avrag lngth of th q-ary xpansions ovr th prims. Thus J := max l fp log Pα prim J := 1 l fp log P α. πp

10 10 M. G. MADRITSCH, J. M. THUSWALDNER, AND R. F. TICHY Not that by ths dfinitions w hav 6.4 N = JP + Olog P α. Thus by th sam rasoning as in th proof of Thorm 1 it sufficis to show that 6.5 Nfp = N + on. W dfin th indicator function as in 5.5 and also th substs I l,...,i J of {2,...,P } by n I j fn q j Following th proof of Thorm 1 w s that 6.6 Nfp = J j=l p I j I fp l j J. q j + O lπp. Now w fix j and split I j into intgr intrvals of lngth m ji for i = 1,...,. Thus I j = {n ji,n j 1,...,n j m ji 1} By Proposition 1 w again gt that log P α 1. Thus th lngth of th m ji is asymptotically incrasing for P, indd, w hav Plog P 1 α m ji P. Now w can rwrit 6.6 by 6.7 Nfp = J j I j=l n ji p<n ji+m ji fp q j + O lπp. Following Naai and Shioawa [7, 8] again w gt as in th proof of Thorm 1 that thr xist two functions I and I +. W rplac I by I + in 6.7 and togthr with th Fourir xpansion of I + in 5.10 w gt in th sam mannr as in 5.12 that 6.8 Nfp = J j j=l n ji p<n ji+m ji By 6.1 and 6.2 togthr with 6.4 w hav 6.9 J j j=l 1 q j + O n ji p<n ji+m ji 1 = JπP + OlπP = N + OlπP. W subtract th main part Nq l in 6.8 and gt by 6.9 Nfp N J j 6.10 j=l n ji p<n ji+m ji A ± ν ν fn q j. ν A ± ν q j fn + lπp. W stimat th cofficints A ± ν in th sam way as in Thn 6.10 simplifis to Nfp N J j δ i ν 6.11 ν 1 q j fp + lπp. j=l n ji p<n ji+m ji Again th crucial part is th stimation of an xponntial sum ovr th prims. W apply quit th sam rasoning as in th proof of Thorm 1. W st S X := ν 6.12 q j fp. p X

11 NORMALITY OF NUMBERS GENERATED BY THE VALUES OF ENTIRE FUNCTIONS 11 and us th functions dx and hx dfind in 5.18 and 5.19, rspctivly. If w assum that δ i hx thn w gt that th logarithmic ordr of ν q fx is lss than 4 j 3 as in W st By 5.21 w also gt that p X g j x = ν q j a dx d + + a 1 x. fp ν q j W can apply Lmma 3.2 to gt th stimat 6.13 p X S X X xp c ν log log X 2 + g j p + π. X log X xp h, whr c ν is a constant dpnding on ν and h = hx is th function dfind in Now w fix i and for vry ν = 1,...,dm ji w calculat th corrsponding h ν m ji and c ν. W st 6.14 δ i := max{r dm ji : r min{h ν m ji : ν r}}, c i := min{c ν : ν = 1,...,δ i }. By th abov rasoning w hav that δ i for m ji and thrfor for P. By this w gt a δ i for vry i = 1,..., and w can stimat th xponntial sum in 6.11 with hlp of 6.13 and th dfinitions of δ i and c i in 6.14 to gt 6.15 δ i n ji p<n ji+m ji ν ν 1 q j fp δ i ν 1 S m ji δ i ν 1 m ji xp c i log log m ji 2 + xp δ i log m ji m ji xp c i log log m ji 2 + xp δ i log m ji log δ i. As w do not now th asymptotic bhavior of δ i w want to mrg it with th xprssion in th parathsis and thrfor hav to distinguish two cass according whthr xp δ i log m ji 1 is gratr or smallr than xp c i log log m ji 2. If xp c i log log m ji 2 > xp δ i log m ji 1 thn as δ i log P 1/3 w hav that xp c i log log m ji 2 log δ i xp c i log log m ji 2 log log m ji < xp c i /2log log m ji 2. Thus xp ci log log m ji 2 + xp δ i log m ji 1 log δ i xp c i /2log log m ji 2 + xp δ i /2log m ji 1. On th contrary w hav xp c i log log m ji 2 xp δ i log m ji 1 and this implis δ i clog log m ji 2 for a positiv constant c. Thrfor w gt xp c i log log m ji 2 log δ i xp c i log log m ji 2 clog log m ji 2 < xp c i /2log log m ji 2. W again hav xp ci log log m ji 2 + xp δ i log m ji 1 log δ i xp c i /2log log m ji 2 + xp δ i /2log m ji 1.

12 12 M. G. MADRITSCH, J. M. THUSWALDNER, AND R. F. TICHY By this w hav 6.16 j δ i ν ν 1 q j fp n ji p<n ji+m ji m ji xp ci /2log log m ji 2 + xp δ i /2log m ji 1. Th considrations abov can b usd in 6.11 in ordr to obtain Nfp N J j δ i ν ν 1 q j fp + lπp. Thus it rmains to show that 6.17 and 6.18 δ i j=l n ji p<n ji+m ji j ν 1 j n i p<n i+m ji n ji p<n ji+m ji i = oπi j ν q j fp = oπi j, whr πi j stands for th numbr of prims in th intrval I j. First w hav to stimat th numbr of prims in I j for vry j. Thrfor w st m ji := π {n ji,...,n j m ji 1}. Thus th numbr of prims in I j is th sum of th m ji, i.. πi j = j m ji. As 6.19 Plog P 1 α m ji P i = 1,..., holds w considr an intgr intrval [x y,x] Z with xlog x 1 α y < x. W st y := xβ 1 and gt < β log x α 1. To stimat th numbr of prims w apply th Prim Numbr Thorm in th following form which is a war rsult than in Chaptr 11 of [6]. πx = x x 6.21 log x + O log x 2. Thus w gt with 6.20 and π [x y,x] Z = πx πx y = x log x x xβ 1 x logx xβ 1 + O = x log x x xβ 1 log x + O β 1 + O log x 2 x log x 2 = x log x x xβ 1 log x 1 + O β 1 log x 1 + O = y x log x + O log x 2. Now w rformulat 6.22 by stting x = P and y = m ji and gt with 6.19 m ji = π {n i,...,n m ji 1} = m ji P 6.23 log P + O log P 2. x log x 2

13 NORMALITY OF NUMBERS GENERATED BY THE VALUES OF ENTIRE FUNCTIONS 13 Now w us th stimation 6.23 in ordr to show By stting a i = m ji δ i and b i = m ji w not that as m ji w gt that m ji which implis ai b i 0. Thrfor w can apply Lmma 3.4 and gt 0 p I j πi j Finally w show that 6.18 holds. W st = m ji δ i m ji 0. a i = m ji xp c i /2log log m ji 2 + xp δ i /2log m ji 1, b i = m ji. By th stimation in 6.23 w gt that ai b i 0 for P and w ar abl to apply Lmma 3.4. Thus with 6.16 w gt 0 j δ ν 1 n ji p<n ji+m ji ν q fp j πi j j m jixp c i /2log log m ji 2 + xp δ i /2log m ji 1 j 0. m ji Thus by putting 6.11, 6.18, and 6.17 togthr w gt Nfp N J j δ i j=l n ji p<n ji+m ji J oπi j + lπp o JP on, j=l which, togthr with 6.5, provs Thorm 2. Rfrncs ν ν 1 q j fp + lπp [1] R. C. Bar, Entir functions and uniform distribution modulo 1, Proc. London Math. Soc , no. 1, [2] D. G. Champrnown, Th Construction of Dcimals Normal in th Scal of Tn., J. London Math. Soc [3] P. T.-Y. Chrn, On mromorphic functions with finit logarithmic ordr, Trans. Amr. Math. Soc , no. 2, lctronic. [4] A. H. Copland and P. Erdös, Not on normal numbrs, Bull. Amr. Math. Soc , [5] H. Davnport and P. Erdös, Not on normal dcimals, Canadian J. Math , [6] H. Davnport, Multiplicativ numbr thory, scond d., Graduat Txts in Mathmatics, vol. 74, Springr- Vrlag, Nw Yor, 1980, Rvisd by Hugh L. Montgomry. [7] Y. Naai and I. Shioawa, Discrpancy stimats for a class of normal numbrs, Acta Arith , no. 3, [8], Normality of numbrs gnratd by th valus of polynomials at prims, Acta Arith , no. 4, [9] Q. I. Rahman, On a class of intgral functions of zro ordr, J. London Math. Soc , [10] J.Schiffr, Discrpancy of normal numbrs, Acta Arith , no. 2, [11] E. C. Titchmarsh, Th thory of functions, 2nd d., Oxford Univrsity Prss, London, [12] I. M. Vinogradov, Th mthod of trigonomtrical sums in th thory of numbrs, Intrscinc Publishrs, London and Nw Yor., no yar givn, Translatd, rvisd and annotatd by K. F. Roth and Ann Davnport. M. G. Madritsch Dpartmnts of Mathmatics A, Graz Univrsity of Tchnology addrss: madritsch@finanz.math.tugraz.at J. M. Thuswaldnr Dpartmnt Mathmati und Informationstchnologi, MU Lobn addrss: Jorg.Thuswaldnr@mu-lobn.at R. F. Tichy Dpartmnts of Mathmatics A, Graz Univrsity of Tchnology addrss: tichy@tugraz.at