Precise error estimate of the Brent-McMillan algorithm for the computation of Euler s constant

Transcription

1 Precise error estimate of the Brent-McMillan algorithm for the computation of Euler s constant Jean-Pierre D ly Université de Grenoble Alpes, Institut Fourier Abstract Brent and McMillan introduced in 98 a new algorithm for the computation of Euler s constant γ, based on the use of the Bessel functions I x and K x It is the fastest nown algorithm for the computation of γ The time complexity can still be improved by evaluating a certain divergent asymptotic expansion up to its minimal term Brent-McMillan conjectured in 98 that the error is of the same magnitude as the last computed term, and Brent-Johansson partially proved it in 5 They also gave some numerical evidence for a more precise estimate of the error term We find here an explicit expression of that optimal estimate, along with a complete self-contained formal proof and an even more precise error bound Key-words Euler s constant, Bessel functions, elliptic integral, integration by parts, asymptotic expansion, Euler-Maclaurin formula MSC Y6, 33B5, 33C, 33E5 Introduction and main results Let H n = n denote as usual the partial sums of the harmonic series The algorithm introduced by Brent-McMillan [BM8] for the computation of Euler s constant γ = lim n + H n log n is based on certain identities satisfied by the Bessel functions I α x and K x : I α x = + n= x α+n n! Γα + n +, K x = I αx α α= Experts will observe that x has been substituted to x in the conventional notation of Watson s treatise [Wat] As we will chec in, these functions satisfy the relations 3 K x = log x + γi x + S x I x = + n= x n n!, S x = As a consequence, Euler s constant can be written as + n= where H n x n n! γ = S x I x log x K x I x, and one can show easily that 5 < K x I x < π e for x

2 Precise error estimate of the Brent-McMillan algorithm for Euler s constant In the simpler version BM of the algorithm proposed by Brent-McMillan, the remainder term K x I x is neglected ; a precision d is then achieved for x d log + log π, and the power series I x, S x must be summed up to n = a x approximately, where a p is the unique positive root of the equation 6 a p log a p = p The calculation of I x = + x + x x n + x n requires arithmetic operations for each term, and that of S x H,N x H I x,n + x x H n n,n + x n H n,n + requires operations The time complexity of the algorithm BM is thus 7 BMd = a d log 6 d d However, as in Sweeney s more elementary method [Swe63], Brent and Mcmillan observed that the remainder term K x/i x can be evaluated by means of a divergent asymptotic expansion 8 I xk x N! 3! 6x Their idea is to truncate the asymptotic expansion precisely at the minimal term, which turns out to be obtained for = x if x is a positive integer We will chec, as was conjectured by Brent-McMillan [BM8] and partly proven by Brent and Johansson [BJ5], that the corresponding truncation error is then of an order of magnitude comparable to the minimal term = x, namely Theorem The truncation error e π x 9 x := I xk x admits when x + an equivalent 3/ by Stirling s formula x = x 5 e π x 3/, and more specifically x = e The approximate value 5 π x K x I x I x! 3! 6x + εx, εx < 863 3/ x x =! 3! 6x is thus affected by an error of magnitude x 3 I x 5 π x / e

3 Jean-Pierre D ly, Institut Fourier Grenoble 3 The refined version BM of the Brent-McMillan algorithm consists in evaluating the remainder term K x I x up to the accuracy e permitted by the approximation 3 This implies to tae x = 8 d log and leads to a time complexity 3 BM d = a 3 + log d 97 d, substantially better than 7 The proof of the above theorem requires many calculations The techniques developed here would probably even yield an asymptotic development for x, at least for the first few terms, but the required calculations seem very extensive Hopefully, further asymptotic expansions of the error might be useful to investigate the arithmetic properties of γ, especially its rationality or irrationality The present paper is an extended version of an original text [Dem85] written in June 98 and published in Gazette des Mathématiciens in 985 However, because of length constraints for such a mainstream publication, the main idea for obtaining the error estimate of the Brent-McMillan algorithm had only been hinted, and most of the details had been omitted After more than 3 years passed, we tae the opportunity to mae these details available and to improve the recent results of Brent-Johansson [BJ5] Proof of the basic identities Relations and 3 are obtained by using a derivation term by term of the series defining I α x in, along with the standard formula Γ n+ γn+ = H n γ, itself a consequence of the equalities Γ x + Γx + = x + Γ x and Γ = γ Γx Explicitly, we get I α x α hence and 3 function /Γ as = + n= log x x α+n n! Γα + n + Γ α + n + x α+n n! Γα + n +, Now, the Hanel integral formula see [Art3] expresses the Γz = ζ z e ζ dζ πi C where C is the open contour formed by a small circle ζ = εe iu, u [ π, π], concatenated with two half-lines ], ε] with respective arguments π and +π and opposite orientation This formula gives 3 + x α+n I α x = n! πi n= C = ζ α expx/ζ + ζx dζ πi = π C π e x cos u cosαu du ζ α n e ζ dζ = x α ζ α expx /ζ + ζ dζ πi C sin απ π e x cosh v e αv dv The integral expressing I α x in the second line above is obtained by means of a change of variable ζ ζx recall that x > ; the first integral of the third line comes from

4 Precise error estimate of the Brent-McMillan algorithm for Euler s constant the modified contour consisting of the circle {ζ = e iu } of center and radius, and the last integral comes from the corresponding two half-lines t ], ] written as t = e v, v ], + [ In particular, the following integral expressions and equivalents of I x, K x hold when x + : 5 I x = π K x = π e x cos u du hence I x x + e x cosh v dv hence K x x + πx e x, π e x Furthermore, one has I x > πx e x if x and K x < π e x if x > These estimates can be checed by means of changes of variables I x = ex π x K x = e x x e t t t/ dt, t = x cos u, e t t + t/ dt, t = xcosh v, along with the observation that t e t dt = Γ = π ; the lower bound for I x is obtained by the convexity inequality + t/ and an integration by parts t/ of the term t e t, which give e t dt Γ t t/ + t t Γ 3 + e t dt π π + 6x e 3 x + 3x > π x for x Inequality 5 is then obtained by combining these bounds Our starting point to evaluate K x more accurately is to use the integral formulas, 5 to express I xk x as a double integral 6 I xk x = exp xcos u cosh v du dv π { π<u<π, v>} A change of variables r e iθ = sin u + iv = cosu + iv = cos u cosh v + i sin u sinh v gives r = cosh v cos u, r u + iv eiθ = cos, u + iv u + iv r dr dθ = sin cos du dv = r r e iθ du dv, therefore 7 I xk x = π dθ exp r dr π r e iθ Let us denote by α αα α + =, α C! the generalized binomial coefficients For z = r e iθ and z = r < the binomial identity z / = + = z combined with the Parseval-Bessel formula yields the

5 Jean-Pierre D ly, Institut Fourier Grenoble 5 expansion 8 ϕr := π π dθ + r e iθ = w r for r <, where the coefficient / w := = =! 6! is closely related to the Wallis integral W p = π/ sin p x dx Indeed, the easily established induction relation W p = p p W p implies W = = π, W + = , whence w = π W The relations W W = π, W W + = with the monotonicity of W p imply π + < W < π, therefore π + < w < π π + together The main new ingredient of our analysis for estimating I xk x is the following integral formula derived from 7, 8 : I xk x = where 3 ϕr = + = ϕr = r ϕ = r e r ϕr dr w r for r <, + = w r for r > The last identity can be seen immediately by applying the change of variable θ θ in 8 It is also easily checed using that one has an equivalent ϕr + = r π = π log r when r, in particular the integral converges near r = later, we will need a more precise approximation, but more sophisticated arguments are required for this By an integration term by term on [, + [ of the series defining ϕr, and by ignoring the fact that the series diverges for r, one formally obtains a divergent asymptotic expansion I xk x N w! + N! 3! 6x If x is an integer, the general term of this expansion achieves its minimum exactly for = x, since the ratio of the -th and -st terms is 3 6x = 3 < iff x x

6 6 Precise error estimate of the Brent-McMillan algorithm for Euler s constant As already explained in the introduction, the idea is to truncate the asymptotic expansion precisely at = x, and to estimate the truncation error This can be done by means of our explicit integral formula Expression of the error in terms of elliptic integrals By 7 and the definition of x we have x = where δr := ϕr x = e r δr dr w r, so that δr = + =x+ w r for r < For r <, let us also observe that ϕr coincides with the elliptic integral of the first ind π/ π r sin θ / dθ, as follows again from the binomial formula and the expression of W We need to calculate the precise asymptotic behavior of ϕr when r This can be obtained by means of a well nown identity which we recall below By putting t = r, the change of variable u = tan θ gives 3 ϕr = π = π π/ r cos θ / dθ = π dv + v + t v + π t du + u t + u du dv + v t + v where the last line is obtained by splitting the integral du on the 3 intervals [, t], [t, ], [, + [, and by performing the respective changes of variable u = vt, u = v, u = /v the first and third pieces being then equal Thans to the binomial formula, the first integral of line 3 admits a development as a convergent series π dv + v + t v = + c π t, c = = / v dv + v The second integral can be expressed as the sum of a double series when we simultaneously expand both square roots : dv π v + v + t /v = / / v l t /v dv π l v t t,l The diagonal part = l yields a logarithmic term + / t log π t = π ϕt log t, = and the other terms can be collected in the form of an absolutely convergent double series [ ] / / t v l = / / t t l π l l π l l l t l After grouping the various powers t, the summation reduces to a power series π c t

7 Jean-Pierre D ly, Institut Fourier Grenoble 7 of radius of convergence, where due to the symmetry in, l c / / = l l l<+, l In fact, we see a priori from that and c π + = O 3/, c π + <l log l = O πl In total, if we put t = r, the above relation implies ϕr = ϕt log + π t + c t, c = c + c, = and this identity will produce an arbitrarily precise expansion of ϕr when r In order to compute the coefficients, we observe that / c = c + c = α with α = v dv + + +v A direct calculation gives Next, if we write c = α = v + v = v v +v l= dv + + +v / v l dv + l l= dv = log +v v v + v, + v = v + v l / and integrate by parts after factoring v, we get / [ α = + v ] + v l l v + v dv l= [ ] + / v + v l + v l l l= + / v v l + v dv l l This suggests to calculate α + α and to use the simplification v + v l= v + v = v + v l

8 8 Precise error estimate of the Brent-McMillan algorithm for Euler s constant We then infer / α + α = α + l l l= + / v + v l l l / v l v l dv l l= l= / / + + l= l l l= l A change of indices l = l in the sums corresponding to then eliminates almost all terms There only remains the term l = in the first summation, whence the induction relation / α + α = We get in this way c / / = α and the explicit expression, ie α = α 5 c = w l= log / l α = / ll = log l= l l l= l The remainder of the alternating series expressing log is bounded by half of last calculated term, namely /, thus according to we have < c < π if, and the radius of convergence of the series is From and we infer as r the well nown expansion of the elliptic integral 6 ϕr = + w t log + π t + c t, t = r, = with w =, w =, w = 9 6, c = log, c = log, c = 9 log 7 6 Let us compute explicitly the first terms of the asymptotic expansion at r = by putting r = +h, h For r = +h < h < we have t = r = h h = h +h/, where log t = log h + h/ = log h log h + 8 h + Oh, + = + = = w t = + h h h + Oh 3, c t = log + log h h log 7 h + Oh 3, l

9 Jean-Pierre D ly, Institut Fourier Grenoble 9 and ϕ + h = π h h + Oh 3 log h log h + 8 h + Oh 3 + log log 5 3 h + log h + Oh 3 6 If terms are written by decreasing order of magnitude, we get ϕ + h = log π h + 3 log h log 3 h log h h log 5 h + 6 log 7 7 h + O 6 h 3 log h For r = + h >, the identity ϕr = r ϕ r gives in a similar way ϕr = + w t log + h π t + + c t, t = r = h 3h + Oh 3 = = After a few simplifications, one can see that the expansion 7 is still valid for h > Passing to the limit r, t in 6 implies the relation c = π The following Lemma will be useful Lemma A For h >, the difference 8 9 ρh = ϕ + h log π h + 3 log h log 3 h log = ϕ + h h log h8 π + h admits the upper bound ρh h + log + h h Proof A use of the Taylor-Lagrange formula gives +h = h+θ h, t = r = h 3θ h, with θ i ], [, and we also find t h and log t = log r r r + = log + log + h log h = log h log + 3 h 7 8 θ 3h, θ ], [, + h log while the remainder terms w t and c t are bounded respectively by w t t w r h 5 56 h and c t t c r h < h if h, r = + h 5

10 Precise error estimate of the Brent-McMillan algorithm for Euler s constant For h we thus get an equality ϕ + h = π h + θ h + h 3θ h θ h + log + log log h 3θ h + θ 5h h log + 3 h 7 8 θ 3h with θ i ], [ In order to estimate ρh, we fully expand this expression and replace each term by an upper bound of its absolute value For h, this shows that ρh h 885 log h +, so that is satisfied For h, we write ρ h = ϕ + h log h π 8 +, ϕ r = h and by we get therefore + = π π This implies 7 < π π + = + w r, π r < ϕ r < + r π r < r = r, = hh + < ϕ + h < log 8 h + h π hh + log + 3π 9 log h 8 + < ρ h < π = hh +, < ρ h < π log 8 h + h + log [ ] < 5 on 9,, < ρ h < π log 3 < on [, + [, therefore ρ h π h log 8+ π h for h [, + [ Since ρ 9 < π, we see that ρh π h, and this shows that still holds on [, + [ A numerical calculation of ρh at sufficiently close points in the interval [, ] finally yields on that interval Now we split the integral on the intervals [, ] and [, + [, starting with the integral of ϕ on the interval [, + [ The change of variable r = + t/ provides e r ϕr dr = e e t ϕ + t dt,

11 Jean-Pierre D ly, Institut Fourier Grenoble and Lemma A 9 yields for this integral an approximation e t e t 8πx log t 3x + t dt = e log3x + γ + 8πx = e πx with an error bounded by Writing e log x + γ + 5 log log x e t t < log t + t we further see that t e t log t + dt t We infer with + log + t γ + 5 log dt e t t log t + t dt, = e + 6x t e t log t + dt t = log t + log + t log t + t, t e t log t + t dt = log γ 3 x e r ϕr dr = e log x + γ + 5 log log x + e πx R x, 3 R x < γ + 5 log + 3 log + x + + 8πx 6x < 83 x thans to a numerical evaluation of the sequence in a suitable range 3 Estimate of the truncated asymptotic expansion We now estimate the two integrals e r x+ By means of iterated integrations by parts, we get + e r r dr = e l l, 3 e r r dr = e l= + l= w r dr, e r if N x, x l + l w r dr Combining the identities,,, 3, 3 we find 33 x = e log x + γ + 5 log log x x π 8πx w + Sx + R x + R x with 3 Sx = + =x+ x l= w l x + + l = x l= = w l + l,

12 Precise error estimate of the Brent-McMillan algorithm for Euler s constant and 35 R x = + + =x+ l=x w l x + + l + = l=x w l + l In the final summation, terms of index l > are zero Formula 33 leads us to study the asymptotic expansion of x = w This development is easy to establish from 6 one could even calculate it at an arbitrarily large order Lemma B One has w = 36 π + ε where x w = 37 log x + 5 log + γ + R 3 x, π = < ε < πx < R 3x < 9 8πx 5,, 6 Proof The lower bound 36 is a consequence of the Euler-Maclaurin s formula [Eul5] applied to the function fx = log x x This yields log w = fi = C + fx dx + p f + b j j! f j + R p i= where C is a constant, and where the remainder term R p is the product of the next term by a factor [, ], namely b p+ p +! f p+ = p+ b p+ p + p + p+ p+ j= We have here fx dx = log log log = log log + log and the constant C can be computed by the Wallis formula Therefore, with b = 6, we have log w = log + log + + θ b π log π + l=3 The inequality e x x then gives w > π 6 > log l l π 3 = π + and the lower bound 36 follows for all In the other direction, we get log w < log π = log π + 3 3

13 Jean-Pierre D ly, Institut Fourier Grenoble 3 and the inequality e x x + x implies w < π whence by a difference of polynomials and a reduction to the same denominator w < π + 5 if 3 6 One can chec that the final inequality still holds for =,, and this implies the estimate 36 On the other hand, formula 6 yields + w + w r = ϕr π π log r = = π ϕt log r + π log + c t with t = r and ϕt = + O r By passing to the limit when r and t, we thus get + w + w = log π π We infer x+ w + x = = and the upper and lower bounds in 36 imply + + < π w The Euler-Maclaurin estimate 38 x = then finally yields 37 w + π π log = π w x+ x+ + π < x+ π = logx + γ + + b x b + = 8πx It remains to evaluate the sum Sx This is considerably more difficult, as a consequence of a partial cancellation of positive and negative terms The approximation 36 obtained in Lemma B implies 39 Sx = T x π Ux R, and if we agree as usual that the empty product l+ = for l = is equal to, we get 3 3 T x = Ux = x l= x l= + =x+ + =x+ l x + + l l x + l l= x l= = x = l + l, l + l,

14 Precise error estimate of the Brent-McMillan algorithm for Euler s constant where the new error term R x admits the upper bound 3 R x x l= + =x+ l x / + l + l= x = Application of discrete integration by parts l + l To evaluate the sums T x, Ux and R x, our method consists in performing first a summation over the index, and for this, we use discrete integrations by parts Set u a,b := + a + a + + b, a b agreeing that the denominator is if a = b Then u a,b u a,b + b + b + + a + a + + = + a + a + + b + b a + a + b + = + a + a + + b + The inequalities + a + a + b + + b + imply ua,b u a,b + + a + + b + b a + a + a + + b with an upward error and a downward error both equal to b a + + a + a + + b + In particular, through a summation + =x+ + =x+ + a + b with an upward error equal to b a + + =x+ u a,b u a,b + b a ua,b x+ b a = b a + a + b and an error on the error again upwards equal to b a+b a+ + =x+ In other words, we find + =x+ + a + b + a + b = b a b a+ 8, these inequalities imply + a + + b, + a + b + a + + b + a + b + a + b + θ b a + a,b 3, θ [, ] 8 + a + b If necessary, one could of course push further this development to an arbitrary number of terms p rather than 3 We will denote the corresponding expansion a,b p, and will use it here in the cases p =, 3 For the summations x =, we similarly define 3 v a,b = a a b +, a b,

15 Jean-Pierre D ly, Institut Fourier Grenoble 5 and obtain v a,b v a,b = a b + a a b b = a b + b a a b For a < b, the inequalities b a b a imply a b va,b v a,b b a a b + with an upward error and a downward error both equal to b a a b + By considering the sum x = with a downward error x = b a v a,b va,b b a, we obtain a b + va,b b a x = and an upward error on the error equal to b a b a x = ie there exists θ [, ] such that x a b + = a,b 3 with = = x va,b a b + va,b x a b + b a 8 v a,b x va,b + v a,b x v a,b v a,b x v a,b θ b a v a,b x 8 b a b a va,b x + va,b x θ b a v a,b x + C a,b 3 8, 5 a,b 3 Ca,b 3 b a va,b + va,b + b a 8 v a,b, v a,b, v a,b, especially C a,b 3 = if a = The simpler order case with an initial upward error gives x a b = v a,b x b a va,b θ v a,b x v a,b = 6a,b = b a a b + θ a b + + Ca,b In the order 3 case, it will be convenient to use a further change v a,b v a+,b+ = a b + a b = b av a+,b If we apply this equality to the values a, b, a, b and = x, we see that the a,b 3

16 6 Precise error estimate of the Brent-McMillan algorithm for Euler s constant development can be written in the equivalent form x a b + C a,b = = = 7 a,b 3 b a va+,b+ x + 3 va+,b 3 x + b a 8 v a+,b x θ v a,b x, b a a b + 3 a b + + b a a b + θ a b According to 3,,l,l 3 and 73, we get 8 T x = T x T x + R 5 x with 9 T x = T x = R 5 x 8 x l= x l= l x l= l l + + l l l + + l + 3, l + l, l + l l l l l The last term in the last line comes from formula 7,l 3, by observing that the inequalities l + l x imply l + 3 l + Similarly, thans to 3,,l and 6,l, we obtain the decomposition Ux = U x U x + R 6 x with 3 U x = x U x = 5 R 6 x l= x l + = x l= l x + l l= l l + l, negative term l = appearing in Ux, l + l + x l + 3 l l= The remainder terms R x [ resp R x ] can be bounded in the same way by means of,l and 6,l [ resp,l and 6,l ] and, 35, 3 lead to R x + + l + π + l + x l + l 6 π l=x =x+ + l=x l l=x = l + l l l,

17 7 8 R x + x l= x l= x = + =x+ l + Jean-Pierre D ly, Institut Fourier Grenoble 7 l x + l + l x + l + x + = l=3 l= x = l + l l + 3 l l [terms l =, in the summation] Finally, by 33, 37, 39 and 8, we get the decomposition x = e T x T x U x + U x log x πx 9 Lemma C The following inequalities hold : + π R x + R x R 3 x 5 R + R 5 x R 6 x log x < < log +, x = = < 3 log + log x + γ +, U x = log x + R 7x, < R 7 x < 37 x Proof To chec, we observe that the sum of the series is log and that the remainder of index x admits the upper bound + + = / + / < =x+ + =x+ < =x+ = According to the Euler-Maclaurin expansion 38, we get on the one hand x = = l= l l + x l= x l = < log + + = 3 log + log x + γ + whence, and on the other hand x = > log + > 3 log + = logx + γ + + logx + γ + x = + x x, x x log x + γ + 96x 9x

18 8 Precise error estimate of the Brent-McMillan algorithm for Euler s constant A straightforward numerical computation gives 3 log + γ + implies < 37, which then We will now chec that all remainder terms R i x are of a lower order of magnitude than the main terms, and in particular that they admit a bound O/x The easier term to estimate is R 6 x One can indeed use a very rough inequality 3 R 6 x x l= + x l= x + x < 6x Consider now R x We use Lemma C to bound both summations appearing in 8, and get in this way [[8]] log log + log x + γ + < 3 x this is clear for x large since log < 3 the precise chec uses a direct numerical calculation for smaller values of x By even more brutal estimates, we find x l= x l=3 l+ l x + l l= l+ l + l l+ This gives the final estimate x 3 l= R x 99 x 5 Further integral estimates log x+γ+ + l x 3x < 5 x, log x + γ 3x 3x < x In order to get an optimal bound of the other terms, and especially their differences, we are going to replace some summations by suitable integrals Before, we must estimate more precisely the partial products ± j, and for this, we use the power series expansion of their logarithms For t >, we have t t < log + t < t By taing t = j, we find jl j Since jl therefore 5 exp < log l + + l = jl log + j < j = ll+ and jl j = ll+l+ 6, we get ll+ < log 3x l+/ < For l x we have l + / l + /3 96x = jl l + +l < ll+ + ll+l+, l + +l < exp l + / j + jl j 3x l+/ + l+/3 96x l + / 5 x 6 l + /,

19 Jean-Pierre D ly, Institut Fourier Grenoble 9 hence after performing a suitable numerical calculation l + + l < exp 3x 5 l + / 5 6 l + + l < exp 3x 5 x / < 5 6 x x For l x, each new factor is at most 53 l=x + l=x +l 3, thus l + + l < 5 x + p= p < 3 3 x for l x, for l x On the other hand, the analogous inequality t 6 t /6 < log t < t applied with t = j / implies ll + 6 ll + l + l ll < log l < As exp/ > + /, we infer + l + 55 l + l l ll 3 exp < exp, and the ratio of two consecutive upper bounds associated with indices l, l + is less than exp l / e / if l = x and less than e / if l x +, thus + + l + 3 l exp x + + e / e p/ < 65 x As l, we deduce from 6 that 56 R x 769 π x < x but actually, one can see that R x even decays exponentially By means of a standard integral-series comparison, the inequalities, 5 and 5 also provide R 5 x x l + 8 exp 3x 5 l + / + l 3l 6 exp l l= + e 3 t + 3 exp 5 t dt + 3 e 3 + t exp t dt 83x 6 = 96x e 3 5 x π + 6 e 3 x < 9 for x 5 x It then follows from 39 and 5 that x T l l x = l + + l l l= = x l= x l= l l + + l l + / exp 3x + l j= p= l + /3 96 x j + j l + 96x ;

20 Precise error estimate of the Brent-McMillan algorithm for Euler s constant to get this, we have used here the inequality a j a j with a j = j <, and the identity jl j = ll+l+ 6 In the other direction, we have a lower bound aj a j, thus 53 implies T x = x l= l l l j l j= u = t x l= x l= x l= ll + exp l + / exp exp t3 96x = t l + /3 l + l + 78 x l + / l + /3 l + l + / 78 x 96x l + /3 l + l + / 78 x 96x We now evaluate these sums by comparing them to integrals This gives x T x e 3x exp t + t3 t + 3/ 96x 96x dt when we estimate the term of index l by the corresponding integral on the interval [l /, l + /] The change of variable t, du = t 5 t dt x u Moreover, a trivial convexity argument yields implies u 5 t, hence t v p + t p v if v ; if we tae v = x and p = 6 resp p = 3, we find t = u t / u + t 3 u + u, x x 5x dt = t du + t du + u du, t t 6x u 5x therefore 3 e u + 3 u + T x e 3x u 96x + x du u 5x 5x u This integral can be evaluated evaluated explicitly, its dominant term being equal to e 3x 96x e u u x du u x e u u du = π x / Moreover, the factor e 3x factor admits the very rough! upper bound + 35 x, whence an error bounded by π x / 35 x < x All other terms appearing in the integral involve terms O x with coefficients which are products of factors Γa, a, by coefficients whose sum is bounded by [ e ] 8 < 5 5

21 Jean-Pierre D ly, Institut Fourier Grenoble As Γa π, we obtain T x < π 77 + x/ x Similarly, one can obtain the following lower bound for T x : T x = x l= x+/ 3/ x x/ /x x/ /x x/ /x exp l + / l + /3 l + l + / 78 x 96x t3 t t / 78 x 96x dt t 3t/ 78 x 96x dt u 3 8 x / u / / 8 x / du exp t exp t t3 e u 8 8 u 3/ 78 x / e u 8 8 u 3/ 78 x / u e u / 8 u x/ 96x 8 u 3 x / u / / du x / u / 3 78 x 6x u e u / u x/ 7 x 6x du du u / u e u / du x/ The integral on the missing intervals is bounded on [, /x] by /x u / du = x/ 36 x, whilst the integral on [A, + [ = [x/, + [ satisfies A u α e u du = A α e A + This provides an estimate x A u e u / du exp x/ α u α e u du e A A α + αa α, α ], ] x Therefore, we obtain the explicit lower bound π T x > x / e / + x x > 6 e / x π 6 x/ x In the same manner, but now without any compensation of terms and with much simpler calculations, the estimates, 5, 53 provide an upper bound T x x 3 exp x l+/ l= + l+/3 96x + 3 3x exp l /

22 Precise error estimate of the Brent-McMillan algorithm for Euler s constant By using integral estimates very similar to those already used, this gives T x e 3 x x exp t + t3 96x dt + 3 x exp t dt e 3 x e 3 x e u + 5x u x du u + 3 x du e u + e 3 x u x < π 55 + / x, + 3 6x x du e u + 3 u 6x and we get liewise a lower bound T x x exp l + / + 3 exp l / l /3 7 l= x+/ exp t dt + 3 x / exp t t3 3/ / 7 dt x exp t dt + 3 x exp t t3 7 dt 9 8 x exp t dt 3 exp t t 3 7 dt 9 8 = x/ x du e u + u u du e 9 u 8 + x du e u 35 π u e x/ > / x All this finally yields the estimate 58 T x T x = 5 π x + R, 55 < R / 8 x < 33 x x There only remains to evaluate U x According to 3, a change of variable l = l + followed by a decomposition = l+l allows us to transform the second summation appearing in U x as U x = = x l= x l= l + l + l x + l l= l l x + + l Writing l+ = l ll+, one obtains l= x l= l + l+ l U x = T x R 9 x l + l l+ l + l+

23 Jean-Pierre D ly, Institut Fourier Grenoble 3 with R 9 x = x l= ll + and for x, we find an upper bound < R 9 x < l x + + l + + l= l l= l l+ l + l+, l=x ll + + x < 3 6x Thans to an explicit calculation of U x for x =,,3, we get the estimate 59 U x < 6 x Combining 3, 37, 9,, 3, and 56 59, we now obtain 5 x = e πx 5 π + Rx x/ with Rx = U x+π R x+r x R 3 x 5 R + R 5 x R 6 x+r 7 x+ R 8 x, whence 5 Rx < 835 x These estimates imply 3 The proof of the Theorem is complete References [Art3] [BJ5] [BM8] E Artin The Gamma Function ; Holt, Rinehart and Winston, 96, traduit de : Einführung in die Theorie der Gammafuntion, Teubner, 93 RP Brent & F Johansson A bound for the error term in the Brent- McMillan algorithm ; Math of Comp, v 8, 5, p RP Brent & EM McMillan Some new algorithms for high-precision computation of Euler s constant ; Math of Comp, v 3, January 98, p 35 3 [Dem85] J-P D ly Sur le calcul numérique de la constante d Euler ; Gazette des Mathématiciens, vol 7, 985, p 3 6 [Eul] [Eul5] [Swe63] L Euler De progressionibus harmonicis observationes ; Euleri Opera Omnia, Ser, v, Teubner, Leipzig and Berlin, 95, p 93 L Euler De summis serierum numeros Bernoullianos involventium ; Euleri Opera Omnia, Ser, v5, Teubner, Leipzig and Berlin, 97, p 9 3 See in particular p 5 The calculations are detailed on p DW Sweeney On the computation of Euler s constant ; Math, of Comp, v 7 963, p 7 78

24 Precise error estimate of the Brent-McMillan algorithm for Euler s constant [Wat] GN Watson A Treatise on the Theory of Bessel Functions ; nd edition, Cambridge Univ Press, London, 9 Jean-Pierre D ly Université de Grenoble Alpes, Institut Fourier UMR 558 du CNRS, rue des Maths 386 Gières, France October 6, revised in November 7