Nuis, J.

Ouwerkerk-Dijkers, M.P. van

English

Ouwerkerk-Dijkers, M.P.

NARCIS 

On the asymptotic behaviour of the solution of a differential-difference equation arising in number theory

CWI's Institutional Repository

STICHTING MATHEMATISCH CENTRUM 2e BOERHAAVESTRAAT 49 AMSTERDAM AFDELING TOEGEPASTE WISKUNDE TN 50 On the asY1J\Ptotic behaviour of the solution of a differential-difference equation arising in number theory by M.P. van Ouwerkerk-Dijkers and J. Nuis February 1968 The Mathematical Centre at Amsterdam, founded the 11th of February, 1946, is a non-profit institution aiming at the promotion of pure mathematics and its applications, and is sponsored by the Netherlands Government through the Netherlands Organization for the Advancement of Pure Research (Z. W. O.) and the Central Organization for Applied Scientific Research in the Netherlands (T. N. O.), by the Municipality of Amsterdam and by several industries. 1. In this report we consider the·asymptotic behaviour of the solution of a differential.;.difference equation which arose in a problem of number theory considered by·the·pure mathematics department. Let g(m) be the largest·prime factor of a natural number m and let G (n) denote the number of integers m 5,_ n, for which g(m) 5,_ma., a. where a. is a real number. Then a:ccording to VAN DE LUNE and WATTEL [8] the frequency G(a.) is defined as G(a.) = lim G (n)/n. a. ·. n-+<x> In their report the following differential-difference equation for H(x) = G(1/x) is derived ( 1 • 1 ) xH'(x) = -H(x-1), _co< X < 00 0 The function H(x) has to satisfy the condition ( 1. 2) H(x) = 1, 0 < X < 1. It follows from these equations that (1.3) H(x) = O, X < 0. In this report we shall investigate the asymptotic behaviour of H(x) for large values of x. 2. We apply a Laplace transformation to (1.1) with respect to x and define ( 2. 1 ) h(s) e-sx H(x)d.x = Jooo for Res> s 0• The existence of s 0 follows from the boundedness of H(x). Integration by parts of the integral in fx {~H'(~) + H(~ ·~ 1)}d~ = 0 x-1 gives the recurrence relation 2 Ix Jx-1 H(~ )d~ - xH(x) = H(~ )d~ - (x-1) H(x-1), x-1 x-2 from which follows, since H(x) < 0 if x < O, that Ix H(~)d~ = x H(x). x-1 Therefore, the assumption that there is a point x0 > 0 for which H(x0 ) = 0 leads to a contradiction, so that H(x) > 0 for x > o. From (1.1) follows that for x > 1 H(x) is· a monotonously decreasing function of x, so that H(x) < 1. Hence (2.2) 0 < H(x) < 1 and a s 0 exists. We remark that from its definition follows that H(x) in the original number theory problem can be interpreted as a probability and therefore trivially satisfies (2.2). With the aid of the transformation the differential-difference equation is transformed into the following simple ordinary differential equation (2.3) sh' (s) = (e-s - 1) h(s ), Since H(x) + 1 as x +Owe obtain the boundary condition sh(s) + 1 as s + oo. (2.4) where The general solution of (2.3) is h(s) =~exp - E1(s), E (s) d~f Joo e-cr dcr. 1 s rJ The boundary condition gives at once c = 1. From (2.4) follows that h(s) is an entire function. Hence (2.1) is valid in the wholes-plane, We note in passing that by substitution of s = 0 into (2.1) and 3 (2.4) the following curious relation is obtained (2.5) JooO H(x)dx = exp y, where y is Euler's constant. Applying the inverse Laplace transformation to (2.1), we get (2.6) H(x) = 2;i JL exp(sx - E1(s) - ln s)ds, where L denotes a vertical path in the s-plane. The asymptotic behaviour of H(x) as x + 00 is determined by apply-ing the saddle point method. Since (2.7) 1 J(s, x) = s - x(E1(s) + ln s) is an entire function of s, there are no further singularities. The saddle points are found from a~/as = 0 which gives (2.8) (2.9) 1 -s - e ---- = x. s In order to facilitate the computations, we write (2.8) as t e - 1 t = x, wheres= -t. This equation can be brought in the following more sym-metrical form by making the substitution This gives (2.10) 1 t = w - -. X - --W 1 X we = - e X which is of the form of an equation considered by WRIGHT [9]. There are two real roots and an infinity of complex roots. The trivial root -1 . . . w = x is of no importance to us but there exists a non-trivial one for which, however, no explicit analytical expression is available. 4 Hence there exists only one real saddle point. It will be shown later on that only the real saddle point contri-butes to the asymptotic behaviour of H(x). The value t 0 oft at the real saddle point ximated by means of an it~rative method such as With t = t we can write (2.8) in the form can be easily appro-gi ven in DE BRUIJN [3] • 0 (2.11) -t t 0 = ln x + ln t 0 -·ln(1 - e 0 ) and proceed in the following way. From the rough estimate ln X < t 0 < 2 ln x obtained from (2.9) fort= t 0 , we derive ln t 0 = O(ln ln x), X -+ oo, which is substituted into the right-hand side of (2.11). This will give a better estimate for ln t 0 , viz. O(ln ln x) ln to= ln ln X +. ln X , X -+ co• By making a further step in the iterative procedure we obtain the more precise result ln t = l l + ln ln x _ 1(ln ln x)2 + O(ln ln x) O n n x ln x 2 ln x l 2 ' n X X -+ 00• Hence the position of the saddle point is determined by = ln x + ln l + ln ln x ,(ln ln x)2 + O(ln ln X) x-+ oo. to n x ln x - 2 ln x 2 ' ln X According to ~ 1 the path of steepest descent is given by S =SQ+ iT, -oo < T < oo, where s0 = -t0• If we are satisfied with the first few terms of the asymptotic expansion it is of course permissible to use the slightly displaced line -00 < T < 00, passing through the point s1 = -ln x - ln ln x - lnl;nxx + !(lnl~nxx)2, 5 as the path of integration L, see [5]. In the neighbourhood of s = s 1 the approximation 2 2 (2.12) ~(s, x) ~ ~(s 1, x) _.!....a ~(s, x)I 2 2 as s=s, can still be used. From [6] we get for large negative values of s the following asymptotic behaviour of the exponential integral -s E1 ( s) + ln s = 7 ( 1 + O( s-1 ) ) • Further we need the following asymptotic relations ( ) ln ln x ,(ln ln x)2 ~ s 1 , x ,-.., -ln x - ln ln x - 1 + l! 1 + 1 n x n x and a2~(s, x) I 2 ~ 1 - I'ri'x as s=s, Thus in the vicinity of s = s 1, ~(s, x) is asymptotically equal to the following expression ~,(T, x) = -ln x - ln ln x - ln ln x + l(ln ln x)2 + 1 - T2(1 - _!_) ln x G ln x 2 ln x • This expression is substituted in~-~ which gives H(x),.,.., 2: J:co exp{x~1(T, x)}d, __ ex1.?{xf(x)} Jco 1 x,2( 1 )} -- - -co expl - 2 1 - y--- d T' 2rr(x ln x)x n x where (2.13) f(x) = 1 _ ln ln x + ~(ln ln x)2. . ln X ln X Finally, (2.14) H(x)~ ex;e{xf(x)} ( 1 + 1 ) \&;:: 2 ln X' y21rx(x ln x)x which is in very good agreement with the numerical calculations of VAN DE LUNE and WATTEL. 6 If we calculate the situation of the saddle point with greater accuracy this will result in the addition of a few small terms to the sum between the brackets and in the addition to f(x) of the further terms from the expansion of ~(s 0 , x) + ln x + ln ln x = = s - l{E (s ) + ln s 0l + ln x + ln ln x. 0 X 1 0 J 3. We consider next the possible contributions from the complex saddle points. The geometry of these points is given in a paper by LAUWERnm [r]. He considers the conformal map z = we -w and shows that the z-plane is mapped upon an infinite number of regions in thew-plane, which are designated 1n fig. 1 by I, II, III, IV, •••• If w = u + iv, 2 2· 2 2u then the curves u + v = r e in the complex w-plane correspond to circles I z I 1 = r 1n the z-plane. For large x we haver= - e X 1 X -1 < e 1 X so that the curve corresponding to the circle with radius r = - e X consists of two parts, a small closed curve around the origin and a parabolical branch. The roots of eq_uat ion ( 2. 1 0) -W X we = - e X are marked by crosses 1n fig. 1. 7 V .----- - - - -. 41T ;;;=--:.-::._ -_ --cp. = 21T ,· -;;;;;;::_-=..::., -. 1T Fig. 1. Conformal mapping icp -w z = re = we ; w = u + iv. w = u + iv III 2 2u =re -II The s-plane with the path of integration Land the line corres-ponding to the parabolical line containing the saddle-points in the w-plane are given in fig. 2. u 8 -----" 0 I rL I I fig, 2o 2 2 2 2u It is easy to deduce from u + v = r e that the complex saddle points in the s-plane are all situated on the left-hand side of the vertical through the real saddle point. Therefore the asympto-tic beha-viour of H(x) is determined only by the real saddle point. Since,however, the complex saddle points are rather near to the verti-cal line through s 0 their contributions to the remainder term can still be consiclerableo These contributions are studied in great detail by BEENAKKEH [2] who also gives explicit estimates. 9 Literature. 1. ABRAMOWITZ, M. and .I~A. STEGUN, Handbook of mathematical functions, N.B.S., 1965, Washington D.C., (5.1.11). 2" BEENAKKER, J .J .A •. The differentia•l-d;lffe,;i:-ence equation ax f'(x) + f(x-1) = O~ Thesis, Technical University, Eindhoven, 1966, Ch. VII, 3. DE BRUIJN, N.G., Asymptotic'methods· in analysis, Bibliotheca Mathematica IV, 1958, Amsterdam.;..Groningen, p, 25, 4. DE BRUIJN, N.G., Asymptotic methods in analysis, Bibliotheca Mathematica IV, 1958, Amsterdam-Groningen, p. 83. 5. DE BRUIJN, N.G., Asymptotic methods in analysis, Bibliotheca Mathematica IV, 1958, Amsterdam-Groningen, p. 91. 6. ERDELYI, A. et al., Higher transcendental functions, Vol. II, 1953, McGraw-Hill Book Cy. , Inc. , New York N. Y. , Ch, IX. 7. LAUWERIER, H.A., The Asymptotic Expansion of the Statistical Distribution of N.V. Smirnov, z. Wahrscheinlichkeitstheorie 2, 61-68 (1963). 8. VAN DE LUNE, J. and E. WATTEL, On the frequency of natural numbers m whose prime factors are smaller than ma, Mathematisch Centrum, 1968, Amsterdam (to appear). 9. WRIGHT, E.M., Solution of the equation zez = a, Bull. Am. Math. Soc. 65, 89-93 ( 1959). 

On the asymptotic behaviour of the solution of a differential-difference equation arising in number theory

Abstract

Similar works

Full text

Available Versions

NARCIS

CWI's Institutional Repository