HermitePade Approximants to Exponential Functions and an Inequality of Mahler

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1 Journal of Nuber Theory 74, (999) Article ID jnth , available online at on HeritePade Approxiants to Exponential Functions and an Inequality of Mahler Franck Wielonsky INRIA, 2004, Route des Lucioles, B.P. 93, Sophia Antipolis Cedex, France Counicated by M. Waldschidt Received Noveber 3, 997; June 0, 998 We iprove Mahler's inequality e g &a >g &33g, a # N, where g is any sufficiently large positive integer by decreasing the constant 33 to This we do by coputing precise asyptotics for a set of approxiants to the exponential which is slightly different fro the classical HeritePade approxiants. These approxiants are related to the Legendre-type polynoials studied by Hata, which allows us to use his results about the arithetic of the coefficients. 999 Acadeic Press. INTRODUCTION In a series of papers [8, 9, 0], Mahler studied the HeritePade syste of approxiants (of so-called Latin type or type I) to exponential or logariths functions, fro which he deduced nuerous results such as a transcendence easure for e iproving those of Borel and Popken, an irrationality easure for?, lower bounds for the approxiation of logariths of algebraic nubers or powers of e. As an auxiliary result, Mahler obtained in [9] a lower bound for the distance of integral exponents of e to the nearest integer. This lower bound was iproved in [0] and the following assertion was established. For any sufficiently large positive integer g, e g &a >g &33g, a # N. (.) This eleentary inequality ay be seen as a particular case of ore general inequalities involving algebraic nubers instead of integers and fro which transcendence easures for nubers related to the exponential function can be derived [4, 5, 2, 4, 5]. The ai of this paper is to iprove (.). Naely, we shall prove X Copyright 999 by Acadeic Press All rights of reproduction in any for reserved. 230

2 AN INEQUALITY OF MAHLER 23 Theore.. Also, Let g be any sufficiently large positive integer. Then e g &a >g &9.83g, a # N. log g&a >g &9.83 log log g, a # N. As in the original proof, the ethod consists in coputing the asyptotics of HeritePade approxiants to the exponential function along with a study of the arithetic of the coefficients (Sections 4 and 5, respectively). The asyptotics will be obtained by eans of the saddle point ethod as applied to the integral expressions of the HeritePade approxiants. Actually, the approxiants that we choose differ fro the usual HeritePade approxiants and are akin to the Legendre-type polynoials studied by Hata [6]. His results concerning the arithetic of the coefficients together with our asyptotics yield Theore.. In [], Mignotte displays a proof of (.) with a constant 7.7 instead of 33. However, the proof contains errors which rule out the assertion. Nevertheless, one ay check that Mignotte's arguent leads to an inequality siilar to the one that can be obtained by our ethod when applied to ordinary HeritePade approxiants (Section 3). Finally, we ention that the convergence in the coplex plane of HeritePade approxiants of the exponential is studied in [6]. 2. THE INEQUALITY OF MAHLER: METHOD OF PROOF HeritePade approxiants to the exponential function and its powers are defined as a set of polynoials A 0,..., A, not all identically zero, of degree less than n and such that R(z) :=: p=0 A p (z) e pz (2.) vanishes at the origin to the order (+) n&. The polynoials A 0,..., A are obtained by solving a linear syste of (+) n& hoogeneous equations with (+) n unknown coefficients. Thus, nontrivial solutions to (2.) always exist. Moreover, such non-trivial solutions A 0,..., A can be given by explicit expressions, the coefficients of which are rational nubers. Multiplying (2.) by the least coon ultiple of the denoinators of the coefficients of the A p,0p, we get R (z)= : p=0 A p(z) e pz, (2.2)

3 232 FRANCK WIELONSKY where all polynoials A p, 0p, have integral coefficients. In fact, the sharp value of this least coon ultiple is difficult to copute. Usually, one uses instead a coon ultiple which coes out fro siple derivations (see [9]). We shall evaluate (2.2) at soe large positive integer g # N. Actually the paraeters and n will be functions of g, both tending to infinity as g does. Naely it appears appropriate to choose n=[g;], =[: log n], (2.3) where [x] denotes the integral part of x and : and ; are positive real nubers to be deterined in soe optial way. We write e g =a+$, a the closest integer to e g, $ <2. Taking into account the relations (2.3) between g, n, and, one can easily check that, for p, (a+$) p =a p + p$a p& (+o()) as g +, where the convergence of the o()-ter to zero is unifor with respect to p. Thus equation (2.2) becoes R (g)= : p=0 A p(g) a p + : p= A p(g) p$a p& (+o()). (2.4) If : and ; satisfy soe condition, R (g) tends to 0 as g becoes large. Assue this condition fulfilled and choose g so large that R ( g) 2. As the first su in the right-hand side of (2.4) is an integer, it iplies that R (g) } : A p(g) p$a p& (+o()) } p= (+o()) $ : p= pa p& A p(g). (2.5) In order to get the desired lower bound on $, we need estiates both fro below and above of R (g) (the second one to ensure the condition R (g) 2) and an estiate fro above of the A p(g), p, allowing us to obtain an upper bound for the second su in (2.5). 3. HERMITE-PADE APPROXIMANTS TO EXPONENTIALS The estiates just alluded to above can be derived fro precise asyptotics of the HeritePade approxiants (2.). To achieve this, we need explicit expressions that are, for exaple, given in [9]. Let us denote

4 AN INEQUALITY OF MAHLER 233 by C 0 and C two circles both centered at the origin and of radius less than and greater than respectively. The following expressions A p (z)= 2i? e`z d`, 0p, (3.) C0 > l=0 (`+p&l)n R(z)= e`z d` 2i?, (3.2) C > l=0 (`&l)n satisfy (2.) as can be seen by perforing a change of variable in the A p 's and using hootopy invariance of Cauchy integral. Clearly, they also satisfy the prescribed hypothesis about the degrees of the A p 's and the order of vanishing of R at the origin. Applying the saddle point ethod to the contour integrals in (3.2) and (3.) leads to exact asyptotics as n and =[: log n] of the reainder ter R(;n) and of the approxiants A p (;n), 0 p. This gives the first ingredient in order to derive Mahler's inequality. The second ingredient consists of an estiation of a coon ultiple of the denoinators of the coefficients of the A p 's, the rationality of which is clear fro (3.). Such an estiate can be found, for exaple, in [9, p. 203]. Using these two ingredients and following the ethod of proof described in Section 2, it ay be shown that for g, a large positive integer, e g &a >g &2.02g, a # N. (3.3) This yields a first iproveent of (.) which, actually, can also be obtained fro Mignotte's coputations in []. We take this opportunity to point out that inequality (i) on p. 27 of [] is false. It is a consequence of the inequality derived in the appendix II on p. 30 of this sae paper but the factor x n in it is issing in (i) p. 27. Keeping track of this correction in Mignotte's proof leads to the sae inequality as (3.3). 4. ASYMPTOTICS OF NON-DIAGONAL HERMITEPADE APPROXIMANTS In the following, we study a set of approxiants different fro (3.). Naely, we consider the following polynoials, B p (z)= 2i? e`z d`, 0p, (4.) C0 > l=0 (`+p&l)n > &q l=q (`+p&l)sn

5 234 FRANCK WIELONSKY where s and q=[+] are positive integers with 0+2, a fixed real nuber. Note that the case +=0 corresponds to the classical approxiants (3.) with n replaced by (s+) n. By analogy with the terinology used in the theory of Pade approxiants, we ay ter the set of polynoials B p non-diagonal HeritePade approxiants, eaning that their degree ay change with p. Indeed, the B p 's are of degree n& when 0p<q or &q<p and of degree (s+) n& when qp&q. The polynoials B p are also related to the Legendre type polynoials studied by Hata [6]. In Section 5, we shall use his results concerning arithetical properties of their coefficients. The B p 's satisfy an equality S(z)= : p=0 B p (z) e pz, (4.2) where S vanishes at the origin to the order n(+)(s+)&2nqs&. Clearly, S adits the following integral representation obtained after plugging (4.) into (4.2) and using hootopy invariance of Cauchy integrals, S(z)= 2i? e`z d`. (4.3) C > l=0 (`&l)n > &q l=q (`&l)sn The faily of polynoials (4.) will prove to be of soe interest since they allow us to iprove inequality (3.3), which is the goal of the ensuing sections. We study the asyptotics as n and =[: log n] of the reainder ter S(;n) and of the approxiants B p (;n), 0p by applying the saddle point ethod to the contour integrals given in (4.3) and (4.) respectively. We recall the ain result that we shall need in a for convenient for our purpose (cf. [3, 3]). Theore (Saddle Point Method). Let h and g be analytic functions in a siply connected open set 2 and assue g has no zeros in 2. Let be a sooth oriented path with a finite length and endpoints a and b, lying in 2. Moreover, let I n = h(`)g(`) n d`.

6 AN INEQUALITY OF MAHLER 235 (i) Assue that in` # g(`) is attained at the endpoint a only and g$(a){0. Then, if h(a){0, h(a) I n = ng$(a) g(a) \ n& +O \ n++ as n. (ii) Assue that a point `0 of, different fro an endpoint, is a nondegenerate critical point of g (i.e., g$(`0)=0, g"(`0){0) and let be the phase corresponding to the direction of the tangent to the oriented path at `0. Suppose further that in` # g(`) is attained at the point `0 only. Then, if h(`0){0, h(`0) I n =- 2?g(`0)g"(`0) - ng(`0) \ n +O \ n++, (4.4) as n, where the phase 0 of g"(`0)g(`0) is chosen to satisfy 0 +2?2. Since g(`)&g(`0)t( g"(`0))2)(`&`0) 2 as ` `0 along, and g(`)g(`0), it is always possible to choose 0 uniquely in this way. Before applying this theore, we introduce the usual Pochhaer notation; we set (z) =z(z+) } }} (z+&), > Asyptotics of the reainder ter S(;n) Theore 4.. As n, and q tend to infinity, with =[:log n] and q=[+], 0+2, one has the following estiate S(;n)=e &n(+;$2+s(+;"2)) e nqs;" _ \ e (a 0 &)+q+ \ ns(&2q+) e (a 0 &)+ n(+) e O(n), (4.5) where a 0 is the unique real root lying in (, ) of e ; (a&)(a++&) s =a(a&+) s (4.6) and ;$=log \ a 0 a 0 &+, ;"=log a \ 0 &q (a 0 &)+q+. (4.7) Reark. We ay replace S(;n) bys(g) in (4.5). Indeed, by the relation n=[g;], this is equivalent to odify the paraeter ; by a quantity which

7 236 FRANCK WIELONSKY is less than ;n in odulus. It is easily checked that such a perturbation does not change the given estiate. Proof. We use the saddle point ethod as applied to the integral expression (4.3) of S(;n). With the sae notations, we have here g(`)=e &;`(`&) + (`&+q) s &2q+, which depends on the paraeter n since does. We first copute the expression (4.4) corresponding to S(;n) and then, briefly explain why the dependence of with respect to n does not odify the obtained asyptotics. The critical points of g satisfy &2q ;= : `&+k + : s `&+q+k k=0 k=0 =(`+)&(`&)+s(`&q+)&s(`&+q) (4.8) where denotes the logarithic derivative of or digaa function (cf. [3] for a definition and soe properties). This equation has + real roots, of which lie in the segent (0, ) whereas the last one, say `0 is larger than. As the contour C has to encopass all points 0,...,, we choose `0 as the critical point that C should go through. The function adits the expansion (z)=log(z)& 2z +O \, z (&, 0]. (4.9) z 2+ Using the approxiation (z)tlog(z), we get fro (4.8) that, as, `0 is equivalent to a where a satisfies or equivalently ;=log \ a&+ a +s log \ a++&+ a&+, (4.0) e ; (a&)(a++&) s =a(a&+) s. This equation has a unique real root a 0, larger than and when s is odd, another real root exists, lying in (0, ). Hence, the unique critical point of g larger than satisfies `0 ta 0 as. The contour C should be such that the odulus of g there attains its iniu at `0 only. To achieve this, we consider the level curves of e &;` and (`&) + (`&+q) s &2q+ at `0. The first one is a line D of direction i while the second one is a leniscate L, of degree ++s(&2q+) tangent to D at `0, contained in the left half plane deliited by D. Then, a convenient contour C is any

8 AN INEQUALITY OF MAHLER 237 curve contained in this left half plane, surrounding all points 0,..., as well as the leniscate L. The saddle point ethod applies and we need an estiation of g(`0)=e &;`0 (`0&) + (`0&+q) s &2q+. Fro the recurrence forula for the gaa function, (z+)=z(z), we know that g(`0)=e &;`0 (`0+) (`0&) s (`0&q+) s (`0&+q). (4.) Using the well-known Stirling's forula for (see [3, p. 294]) and substituting `0 by its approxiation a 0, one can check after soe coputations that s(+(2&q);")++;$2\ g(`0)=e (a + 0&) e + (&2q+)s _ 0+q \a (+O()), (4.2) e + where ;$ and ;" are defined by (4.7). Next, we copute g"(`0)g(`0). Differentiating (4.), we get g"(`0)g(`0)=$(`0+)&$(`0&)+s$(`0&q+)&s$(`0&+q). Thus, with the equivalence `0 ta 0 and the expansion (4.9), we obtain g"(`0)g(`0)t \ (2+&)s (a 0 &+)(a 0 &++) & a 0 (a 0 &)+. (4.3) Making use of (4.2) and (4.3) to evaluate expression (4.4) leads to (4.5). It should be noted that this expression does not take into account the actual value of g"(`0)g(`0) since the logarith of its odulus is of an order less than O(n). As the point `0, the contour C and the function g depend on, this last paraeter increasing with n like :log n, a justification of (4.5) is in order. This entails screening the proof of the Laplace's ethod for contour integral (cf. [3]), checking that the arguents reain valid. This will not be detailed. We only ention two facts that are relevant here. The first one is that the increasing length of C, which has to surround L, as well as all of the points 0,..., is of order the degree of L (cf. [] for an estiate giving the correct rate of growth of the arc length of a leniscate). As this order is less than e =n, =>0, this eans that the ain contribution in integral (4.3) coes as usual fro its restriction to soe arc in the

9 238 FRANCK WIELONSKY neighborhood of `0. Perforing locally the change of variable v=log g(`)&log g(`0), the integral under study becoes e &n log g(`0 ) e&nv f(v) dv with f(v)= d` dv = g(`) g$(`). For sall v, we check that f(v) has an expansion with respect to `&`0 and such that f(v)=o(v &2 ). This is the second iportant fact here, allowing us to estiate the integral of S(;n) by the sae ethod as in [3, Chapter 4, Section 6], leading eventually to (4.5). K 4.2. Ray asyptotics of the nubers B p (;n) Theore 4.2. Let p be an integer depending on such that, as n, p * for soe fixed * #(0,) distinct fro + and &+. For exaple, choose p=[*]. Then, p=*+o(). As, let B * denote the following expansion, B * =(+s(&2+))(log &)+* log *+(&*) log(&*) +s(*&+) log *&+ +s(&+&*) log &+&* +O(log ). As n, =[: log n] and q=[+], we have B p (;n) =e &nb *. (4.4) Moreover, in (4.4), the O-ter of expansion B * is uniforly doinated by log with respect to *. Reark. It is possible to deterine the sign of B p (;n) asp varies. Since the knowledge of this sign is not needed in the paper, it will not be given here. Proof. To obtain the estiate of B p (;n), we again use the saddle point ethod, now with the integral representation (4.). Here, we have g(`)=e &;`(`+p&) + (`+p&+q) s &2q+. (4.5) We defor the circle of integration C 0 to a rectangle R included in the region [&<Re(z)<] with vertices (&a$, &r), (a, &r), (a, r), (&a$, r) where r is a positive real nuber increasing with like 2 - and a and a$ lie in (0, ). We shall choose &a$ anda as the two critical points of g in (&, ). Then, the iniu of (`+p&) + (`+p&+q) s &2q+ on the vertical segent joining (&a$, r) to (&a$, &r) (resp. (a, &r) to (a, r)) is attained at the point &a$ (resp. a) only. With the assuption ade on r, the value of (`+p&) + (`+p&+q) s &2q+ on the two

10 AN INEQUALITY OF MAHLER 239 reaining horizontal segents is larger than both values at a and &a$. Indeed, this will be true if for any 0p, (p+) 2 r 2 +(p&) 2.In particular, it is satisfied when the inequality holds for p= i.e. 4r 2. The exponential factor e &;z is of constant odulus on the two vertical segents and introduces only a constant independent of n on the horizontal segents. Observe also that the length of the contour R which is O(- ) will not affect the geoetric estiate in n of the integral near the critical points. These critical points are aong the solutions of ;= (`+p&)$ + (`+p&) + +s (`+p&+q)$ &2q+ (`+p&+q) &2q+ = : k=0 `+p&+k +s : &q k=q `+p&+k. We rewrite this equation as ;=(`+ p+)&(`+ p&)+s(`+p&q+)&s(`+ p&+q). (4.6) Using the reflection forula for, (z)=(&z)&?cot?z, we get and ;=(`+ p+)&(&`& p+)+? cot?`+s(&`& p+q) &s(&`& p+&q) if p<q, ;=(`+ p+)&(&`& p+)+s(`+ p&q+) &s(&`&p+&q)+2? cot?` if qp&q, ;=(`+ p+)&(&`& p+)+? cot?`+s(`+ p&q+) &s(`+p&+q) if &q<p. Using the approxiation (`)tlog ` for ` large, one can check that the three previous equations lead for any 0p to tan?`= ;+log \&* * } &+&* +&* }? +O(log n). (4.7) s+

11 240 FRANCK WIELONSKY Denote by f 0 the fraction in the right-hand side of (4.7). Since we choose * # (0, ) different fro + and &+, f 0 lies in [&, ]&[0]. Let arctan denotes the inverse of the tan function, whose iage is the segent [0,?) with the convention that arctan(\)=?2. Then, there are two critical points in (&, ), x and x 2, lying in (&, 0) and (0, ) respectively, x =x 2 &+O(log n), x 2 =? arctan f 0+O(log n). We choose &a$=x, a=x 2 and deterine which one of x and x 2 gives the sallest odulus for g by coputing the quotient g(x ) x + p& g(x 2 ) =e;+o(log n) x 2 + p \ x s + p&+q x 2 + p&q + _ p& ` k= p& x +k+ x 2 +k p&q& ` k= p&+q \ x s +k+. (4.8) x 2 +k + To analyze the two products over k in the right-hand side of (4.8), we write an expression of the difference x &x 2 at order O(log n): x &x 2 =&+b 0 log n+o(log 2 n), where the actual value of b 0 can be coputed fro (4.6). Then, the first product, say, equals p& ` k= p& \ + b 0 (x 2 +k) log n +O(log2 n) +, which tends to as n tends to infinity. Indeed, this follows fro the agnitude of the bounds of suation p& and p& which is O(log n) and the convergence to of the ore eleentary sequence whose ter of order j is > j k= (+a(kj)), where a is soe fixed real nuber. Since the sae conclusion applies to the second product over k in (4.8), we obtain, as n, g(x ) *& e; g(x 2 ) * \ *++& s. (4.9) *&+ + It ay be seen that the liit in (4.9), as a function of *, has odulus one, two or three ties in (0, ) according to the values of 0;, 0+2 ands # N. Note that a value of * such that the liit in (4.9)

12 AN INEQUALITY OF MAHLER 24 has odulus, akes the denoinator of f 0 vanish, i.e., we have x 2 =2+O(log n) and x =&2+O(log n) in this case. Now, let us copute an expansion of g(x 2 ), in ters of. Fro (4.5) and the recurrence forula for the gaa function, (z+)=z(z), we have g(x 2 )=e &;x 2 (x 2 + p+) (x 2 + p&) s (x 2 + p&q+) s (x 2 + p&+q). Using the reflection forula, (z) (&z)=?sin?z, z Z, together with the expansion log (z)=(z&2) log z&z+o(), z (&, 0], one can check that, as, g(x 2 ) =exp ((+s(&2+)) log +(* log *+(&*) log(&*) +s(*&+) log *&+ +s(&+&*) log &+&* +s(2+&)&)+o(log )), (4.20) where the O-ter is uniforly doinated by log with respect to the paraeter * # (0, ). Fro (4.9), we see that g(x ) adits the sae expansion. We now apply assertion (ii) of the saddle point ethod with x or x 2 as critical point `0 according to which one of the nubers g(x ) and g(x 2 ) has the largest odulus. To copute an expansion of order e O(n) or larger, the only relevant factor in (4.4) is the exponent g(`0) n. Hence, taking the nth power of (4.20), we eventually get (4.4). K 5. ARITHMETIC OF THE COEFFICIENTS As already entioned, the coputation of a coon ultiple to the denoinators of the rational coefficients of the classical polynoials A p, 0p, ay be found in [9, p. 203]. In this section, we perfor such a coputation for the polynoials B p. The precise results obtained by Hata concerning the asyptotics of the coefficients of Legendre type polynoials (cf. [6, 7]) will be an iportant ingredient in the analysis. Set T = lc p, l=0,,..., p{l ( p&l)=lc(,..., ),

13 242 FRANCK WIELONSKY where lc denotes the least coon ultiple and M l = ` j=0 j{l &q M l, q = ` j=q j{l & (l& j)=(&) &l l!(&l)!=(&) &l! l+ \, 0l, (l& j)=(&) &q&l (l&q)! (&q&l)! & =(&) &q&l (&2q)! \&2q, ql&q, l&q + so that lcm l =! and lcm l, q =(&2q)!. We rewrite the integral expression (4.3) of S(z) as S(z)= 2i? C \ : l=0 On the other hand, we can write n M l (`&l)+ &q \ : l=q sn M l, q (`&l)+ e`z d`. (5.) \ : l=0! M l (`&l)+\ : &q l=q (&2q)! M l, q (`&l)+ = : l=0 c, l &q `&l + : l=q d, l (`&l) 2 with rational coefficients c, l and d, l. Now, for any subset X/N, put J(X)= ` l. l # X Then, fro [7, Lea 5.], we know that [d, l, l=q,..., &q]/j(x &q, q )Z, [c, l, l=0,..., ]/ J(X &q, q) T Z, (5.2) where X &q, q is the set of prie nubers p>- satisfying { &q p = + { q p= > and { &q p = > { q p=, [x] denotes the fractional part of x. Moreover, the asyptotic behavior of J(X &q, q ) as has been deterined by Hata. For any subset E/(0, )_(0, ) put E(&)=[x>0; ([x], [&x])#e]. Put also Y=[(x, y) # (0, )_(0, ); x+ y> and y>x].

14 AN INEQUALITY OF MAHLER 243 Then, fro [6, Lea 2.5], we have where \=\(&) := li &q log J(X &q, q)= dy & Y(&) y, (5.3) 2 &= li &q q =&+ +. Assue & is a rational nuber &=_%, _, % # N. Then, one ay copute the integral in (5.3) by the sae way as in [7, p. 64]. It consists in the following. Let R j be the affine apping defined by R j :(x, y) ((x+ j)%. y) and put Y = %& R j=0 j(y). Then ([%x], y)#y if and only if ([x], y)#y. Hence, substituting y=%x in (5.3), we get \= _ Y (_) dx x 2= _ Z d(x), (5.4) where Z=Y (_) & (0, ). As explained in [7], the set Z is a finite disjoint union of open intervals, which coincides with the projected set onto x-axis of the intersection of Y and the line segents y=[_x]. Therefore \ can be expressed as a finite su of the values of (x) at rational points. Let us return to the study of (5.). Expanding the nth power of : l=0 and using the eleentary identities `&l show that the quantity! M l (`&l) `&j = l& j \ `&l & `&j+, l{ j, T n& \ : l=0 n! M l (`&l)+ adits a partial fraction decoposition with integral coefficients. Multiplying this decoposition by &q : l=q (&2q)! M l, q (`&l)

15 244 FRANCK WIELONSKY (sn) ties and applying Hata's result (5.2) for each of these products, we deduce that D n, :=! n (&2q)! sn T sn+n& J sn (X &q, q ) is a coon ultiple to the denoinators of the rational coefficients in the partial fraction decoposition of \ : l=0 n M l (`&l)+ &q \ : l=q sn. M l, q (`&l)+ Plugging this partial fraction decoposition in (5.) and evaluating the integral by eans of the identities 2i? e`z d` z*& e lz C (`&l) *=, *=,..., n, (*&)! we obtain that S :=(sn+n&)! D n, S is an exponential polynoial with integral coefficients. Equivalently, the B p := (sn+n&)! D n, B p, 0p, are polynoials with integral coefficients. Moreover, we have the wellknown upper bound T e (+o()), (5.5) as tends to infinity. Indeed, log T = : p prie p l l log p : p prie p log =(+o()), where the last equality coes fro the prie nuber theore. Fro Stirling's forula for the factorial, inequality (5.5) and the definition of \ in (5.3), one checks that an upper bound on the coon ultiple (sn+n&)! D n, of the denoinators of the coefficients of the B p, 0p, is given by n (s+) n n(+s(&2+)) (&2+) ns(&2+) e 2n+s e &s(&+) \n e o(n). (5.6)

16 AN INEQUALITY OF MAHLER APPLICATION TO THE INEQUALITY OF MAHLER Proof of Theore.. We follow the schee of proof described in Section 2. We first consider the condition S (g) 2. (6.) Fro Theore 4. and the upper bound (5.6), we have as g with n=[g;], =[:log n] andq=[+] that S (g) < \e(+s)(+:)&s(&+)\ a 0 & \ a 0&+ a 0 &+++ s+\ &2+ s(&2+) n(+o()). a 0 &+++ + Hence the condition (6.) is et for g large if e (+s)(+:)&s(&+)\ a 0 & \ a 0&+ a 0 &+++ s+\ &2+ s(&2+) <. (6.2) a 0 &+++ This is the first constraint. Let us now proceed to the analog of inequality (2.5), when replacing the usual HeritePade approxiants A p with our B p 's. As the noralization B p B p does not play any role here, it is sufficient to consider the inequality S( g) (+o()) $ : p= pa p& B p (g). (6.3) We establish an upper bound for the su in the right-hand side. Fro Theore 4.2, we have as n, : p= pa p& B p (g) =e &n(+s(&2+))(log &)+O(nlog ) : where * p = p, p and p= e nf + (* p ), (6.4) f + (*)=;*&* log *&(&*) log(&*)&s(*&+) log *&+ &s(&+&*)log &+&*, * #[0,]. About this equality, observe the following. First, the factor exp(o(nlog )) can be pulled out of the su because the O-ter is uniforly doinated with respect to the paraeter * p. Second, the factor pa & disappears as it is of order less than exp(o(nlog )) and third, replacing a with e g is possible as it does not affect the estiate. Differentiating f + with respect to *, we get f $ + (*)=;&log *+log(&*)&s log *&+ +s log &+&*.

17 246 FRANCK WIELONSKY Thus, critical points of f + are solutions of * *&+ s e ; = (&*) &+&* s. (6.5) One can check that this equation, already et when studying the odulus of the liit in (4.9), has, 2 or 3 solutions in (0, ) according to the values of ;>0, 0+2 ands # N. One of the lies in (+, &+) while the two other possible roots lie in (&+, ). When s is odd and +<*<&+, (6.5) is identical to (4.6). In all other cases, (6.5) is rewritten as e ; (&*)(*++&) s =*(*&+) s. The function f + increases fro the value s+log +&s(&+) log(&+) taken at the origin to a axiu et when * equals the first solution of (6.5), lying in (+, &+). Then, if (6.5) has or 2 ore solutions in (&+, ), the function f + still adits a iniu, then a axiu before decreasing to the value ;+s+ log +&s(&+) log(&+) et as * equals. Assue first that (6.5) adits only one solution in (0, ) and let 4 + be the value of * which corresponds to the axiu of f +. Then, an equivalent for the su in the right-hand side of (6.4), : p= e nf + (* p ) (6.6) is exp(nf + (4 + )). Indeed, denote by p 0 the largest rational nuber aong the p, p, less than or equal to 4 +. Then, divide the su by exp(nf + (4 + )) and split it into two parts p 0 : p= e n( f + (* p )&f + (4 + )) and : p= p 0 + e n( f + (* p )&f + (4 + )). The quotient (resp. inverse of the quotient) of a ter by the previous one in the second (resp. first) su is exp(&n f + (* p )&f + (* p+ ) ). This quotient is less than exp(&n in( f + (4 + )&f + (4 + &), f + (4 + )&f + (4 + +))). (6.7) The two differences f + (4 + )&f + (4 + &) and f + (4 + )&f + (4 + +) have the sae expansion C 2 \ 2+O, (6.8) 3+

18 AN INEQUALITY OF MAHLER 247 where C is soe constant independent fro. This shows that (6.7) tends to zero as n tends to infinity and =[:log n]. Therefore, the two previous sus are ajorized by any geoetric series, hence bounded. Actually, they even tend to zero (except the first one if p 0 =4 + ) and decrease at ost like (6.7). In regards of (6.8), this does not odify the asyptotic estiate exp(nf + (4 + )) since we copute an expansion up to order exp(o(nlog )) only (see (6.4)). Now, if f + adits ore than one critical point, it suffices to split the su (6.6) on each interval between two critical points and to repeat the previous arguent on each partial su. Coparing their contribution, we obtain that an equivalent for (6.6) is exp(nf + (4 + )), where 4 + is a value of * which corresponds to the axiu of f + in (0, ). Fro (6.3) along with Theore 4. and the previous estiate of the su (6.6) appearing in (6.4), we derive that (a 0 &) &n (a 0 &++) &ns(&+) (a 0 &+) ns+ $ e O(nlog ) e nf + (4 + ). (6.9) Since n=[g;] and =[: log n], (6.9) is rewritten as ((a 0 &)(a 0 &++) s(&+) (a 0 &+) &s+ e f + (4 + ) ) &(:;) g log g(+o()) $. (6.0) Considering the liit case in which both sides of (6.2) are equal, we obtain that :=& +s _ +s&s(&+)\&log(a 0&) +s log((&2+) (&2+) (a 0 &+) + (a 0 &++) &(&+) )&. (6.) In view of (6.0) and (6.), we want to iniize _ &(+s)( f +(4 + )+log(a 0 &)&s+ log(a 0 &+) +s(&+) log(a 0 &++)) & _ ;(+s&s(&+)\&log(a 0&) +s log((&2+) (&2+) (a 0 &+) + (a 0 &++) ))& &(&+) (6.2) over the set of paraeters ;, s and +. Perforing soe nuerical experients, one can see that a large value for s should be chosen. We shall take This gives s=50, ;=8.45, and +=0.07. := , a 0 = , 4 + = , f + (4 + )=

19 248 FRANCK WIELONSKY We estiate \=\(937) by coputing the second integral in (5.4). Setting and I=[(i, j)#n 2 ;0i92, 0 j6, &7<7i&93j43] J=[(i, j)#n 2 ;0i92, 0 j6, 43<7i&93j<86], one can easily check that \ is given by \= 93 \ : 92 i= Nuerically, this yields \ 93+ i & : j \+i & : j \i& (i, j)#i (i, j)#j \= Finally, we obtain for (6.2) the value of , which copletes the proof of the first inequality in Theore.. By the sae way as in [9], this inequality is easily seen to iply for any =>0, log g&a >g &(9.83+=) log log g, a # N, for any sufficiently large positive integer g. Since the first inequality actually holds with the constant 9.824, one has log g&a >g &9.83 log log g, a # N, for any sufficiently large positive integer g, which gives the second inequality in Theore.. K REFERENCES. P. B. Borwein, The arc length of the leniscate p(z) =, Proc. Aer. Math. Soc. 23 (995), G. V. Chudnovsky, HeritePade approxiations to exponential functions and eleentary estiates of the easure of irrationality of?, in ``Lecture Notes in Math.,'' Vol. 925, pp , Springer-Verlag, New York, N. G. De Bruijn, ``Asyptotic Methods in Analysis,'' 2nd ed., Wiley, New York, G. Diaz, Une nouvelle inoration de log :&;, :&exp ;, : et ; alge briques, Acta Arith. 64 (993), A. I. Galochkin, On the diophantine approxiation of values of the exponential function and of solutions of soe transcendental equations, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 3 (972), M. Hata, Legendre type polynoials and irrationality easures, J. Reine Angew. Math. 407 (990), 9925.

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