Lune, J. (Jan) van de

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stichting mathematisch centrum AFDELING ZUIVERE WISKUNDE J. VAN DE LUNE A NOTE ON EULER'S CONSTANT ✓ ~ MC ZN 62/75 AUGUST 2e boerhaavestraat 49 amsterdam BlBLIOTHEEK MATHEMATISCH CENTRUM · -AMSTERDAM-.- - ,,.. (')-PJun:te.d at :the. Mathema:ti.cai, Ce.ntll.e., 49, 2e. Boe!tha.a.vu.tna.a.t, Am.6,teJui.a.m. The Mathema:ticai, Ce.ntll.e., 6ounde.d -the. 11-:th 06 Fe.bll.u.aJty 1946, .l6 a. non-pll.o6U -ln6.tltu,tlon cumi.ng at -the. pll.omoUon 06 pUll.e. mathema:ti.C-6 a.nd -lt6 a.ppUc.a:ti.on6. 1,t .l6 .6pon6oll.e.d by :the. Nethe/Lta.nd.6 Gove.Jr.nme.n:t fuough :the Nethe/Lta.nd.6 011.ga.n-lza-tlon 6oll. -the. Adva.nc.eme.n:t 06 PUite. Rue.a.Jr.eh (Z.W.O), by :the. Mun-lc.-lpaU:ty 06 Am.6-te.Jr.da.m, by -the. Un-lve.ll.-6.lty 06 Am.6-te.Jr.da.m, by :the. Fll.e.e. Un-lve.ll.-6.lty at Am.6-te.Jr.da.m, a.nd by -lndU.6:tJr.iU. AMS(MOS) subject classification scheme (1970): 40A05, lOA40 A note on Euler's constant by J. van de Lune 0. INTRODUCTION Writing (1) H(n) = n I I k , k=l Euler's constant y is usually defined as the limit of the increasing sequence (2) 00 {H(n) - log(n+I)}n=I' or, equivalently, as the limit of the decreasing sequence (3) Since (4) H(n) - log(n+l) = n I I I {k - log(I + k)} k=I it follows that (5) (6) (7) and (8) 00 y = , I I l {k- log(l+k)}. k=l From (4) and (5) one may derive that Since -1- - _I_< y - {H(n) - log(n+l)} 2n+l 6nz I < -2n' (VnEJN). H(n) - log n I = H(n) - log(n+l) + log(l + -) n l I - - -- < n 2n2 I log(l + -) n I < -n 2 it follows from (6) that (9) I l 2 2n - 2n2 < H(n) - log n - y < 2n + 3n2. From (6) and (9) it is clear that the sequences (2) and (3) converge rather slowly and that, from the numerical point of view, it would be better to consider y as the connnon limit of, for example, the following two (monotonic) sequences (IO) {H(n) - log(n+l) and I }ro +-2n n=l - _I }ro (11) {H(n) - log n 2n n=l' where (IO) is decreasing and (II) is increasing • ... CESARO considered (cf. [2], p. 460) the sequence (12) I oo {H(n) - 2 log n(n+l)}n=l and showed that (13) 1 I O < H(n) - 2 log n(n+l) - y < 6n(n+l) • It was shown by LODGE (cf. [2], p. 460) that a very good approximation of then-th term of (12) is given by (14) y +------6{n(n+l) + ½} the error being of the order (15) -6 n . In this note we will consider a number of variations on Cesaro's e sequence (12). Some examples are: n+I (16) {H(n) - f 00 log x dx}n=I n I which approximates y from above, the error being less than --2 12n ( 17) I oo {H(n) - log(n + -) } 2 n=:I I which tends.decreasingly toy, the error being less than --2 ,. 24n (18) n+l 00 {H(n) + log(e - l)}n=I 3 which tends increasingly toy, the rapidity of convergence being about the same as that of (17). We will also determine all constants c > -1 for which .,, (19) {H(n) - log(n+c)}n=I is monotonic. For more refined methods to compute y numerically we refer to [I], [3], f4], [5] and [6]. We conclude this note by proving the remarkable identity 00 (20) I - y = l n=2 n (-1) [log n] n+I log 2 where[•] denotes the greatest integer function. l. (21) The general term of Cesaro's sequence may be written as H(n) _ log n + log(n+I) 2 . h. h h log n + log(n+I) b 'd d .d 1 in w ic t e term 2 may e consi ere as a trapezoi a ap-. . f Jn+I 1 d proximation o og x x. n Because of the concavity of log x we have (22) log n + log(n+I) 2 log x dx. n 4 Next we observe that (23 ) H(n) _ log n +2log(n+l) > y. In order to see this it suffices to prove that (12) is decreasing inn. Hence, we want to show that (24) or (25) or (26) 1 I H(n) - 2 log n(n+l) > H(n+l) - 2 log(n+l)(n+2) 2 log(n+2) - log n > n+l 2 > --n+l which is true by the wellknown inequality (27) 00 2n+ I log(l+x) - log(l-x) = 2 L ~n+ 1 > 2x, n=O (O<x<l). After these observations it seems natural to investigate the behaviour of the sequence (28) Since (29) we define n+l {H(n) - f n n H(n) = I k=l 1 = f 0 00 log x dx}n=l. I 1 n f k-1 - = I X dx = k k=l 0 00 n f I -nt - X - e dx = dt, - X t 0 e - I co (30) H(s) I l _ -st = __ e __ dt, et - I 0 (s>-1), and instead of (28) we will consider, more generally, the function (31) s+I y(s) def H(s) - f log x dx, s We first prove the following + PROPOSITION I. I. y(s) is decreasing on lR • PROOF. Since 00 (s>O). (32) log a. = f e - t -t e -a. t dt, (a>O), 0 the derivative of y(s) may be written as (33) y' (s) = H' (s) - log(s+I) + logs = 00 00 -(s+l)t -t f t f -st dt - e e dt + = e t t 0 e - I 0 co = f 0 -st { t e ----t e - I - e-t} t dt, (s>O). Now observe that fort> 0 we have 00 t2n (34) t2 2 2 I < t + (2n)! = n=2 t2 t3 t4 t2 t3 t4 = t +zr+JT+--z;r+ ... ) + c-t +zr-JT+--z;r- + ••. ) = t (e -t -I) ·-t t -I)• = (e -I) + = ( 1-e ) (e so that 00 f -t e - e t 0 5 -st dt = 6 -t (35) t - e ---< ----et - I t (t>O) From (33) and. (35) it follows that (36) y'(s) < 0, (s>0). proving the proposition. D Next we have PROPOSITION I • 2. (36) lim y(s) = y. s-+<x> PROOF. In view of proposition I.I it suffices to show that (38) lim y(n) = y, (nElN). n-+<x> Since we clearly have that (39) H(n) - log(n+I) < y(n) < H(n) - log n the proposition follows. D As to the rapidity of convergence we have PROPOSITION I . 3. (40) I y < y(s) < Y + --2 ' 12s (s>0). PROOF. From propositions I.I and 1.2 it is clear that y < y(s) for all s > 0. From (33) we infer that for a,b > 0 we have b (41) y(b) - y(a) = I y' (s) ds = a b (X) I I t -t -st - e } = e { dt ds = t t a 0 e -I 00 b -t I { t t - e } I -st ds dt = e t e -0 a (X) = I e-at ~ e-,bt 0 t { t e - I Replacing a bys and letting b ➔ co we obtain co (42) y(s) = y + f _e -_ts_t {-1 _-_t_e_-_t 0 Now observe that (43) -t (O <) _I_-_e_ t t dt, (t>O). In order to see this we may argue as follows: If n ~ 3 then (44) so that 24 < 1 4 27 3 • 3 5 3 6 n n + -<-3 = 3 3 (45) n-2 24. 3 < n(n+l)(n+2) ••• (2n) = or (46) Hence, if O < t ~ 3 and n ~ then n + 2 3 (2n)! (n-1)! = -t - e } dt. t (s>O). n + n 3 7 81 (48) t2n 24 . (2n) ! < n+2 t (n-1) ! Consequently_ we have (49) 24 . 00 t2n 00 n+2 r (2n)! < r c!-1)! ' n=3 n=3 (O<t~}). t2 ._2 2+2 Since 24 . -,-,--;..,..-,,- = (2t_l).' , it follows that (2.2)! (50) (O<t~}), from which it is easily seen that (51) t -t 2 t3 t e + e - 2 - t < IT (e -1), (O<t~3), so that (52) -t t 2 t2 t (1-e )(e -1) - t < 12 . t(e -1), or, equivalently, -t (53) - e t (O<t~3). If t > 3 then certainly (54) Since we obviously have that (55) -t - .e t t ---<-et - I t (t>O), it follows that also (56) -t - e t t t2 ---<-t 12 ' e - I (t>3). Combining (53) and (56) it follows that (43) holds. From (42) and (43) it is clear now that 00 (57) f -st -t ( ) -et {l - e ys =y+ - --t 0 t2 t - -} dt + et_ I 12 t2 . 12 dt < 00 l I -st < y + 12 e t dt = 0 I y +--12/ (s>O), completing the proof. D REMARK. From (42) one may derive the following asymptotic expansion where the y(s) ~ y + ; ..!_ {(-l)n - B} l n n+l n -n s = = y + I n=2 I + - -2s oo B I , n -n (s+I) log(l +-) - l - s s n=2 n B are Bernoulli's numbers defined by n 00 B t I nf n <ltl<2n). = t , t e - I n=O 2. In this section we consider the sequence (58) 00 {H(n) - log(n+c)}n=l where c is some constant in the open interval (-1, 00). ( s-+<x>) I PROPOSITION 2.1. If -1 < c ~ 2 then the sequence (58) tends decrea.singly to y. Before proving this proposition we establish the following LEMMA 2. I. The function (59) f(x) = ---ex - I X , (x>O) 9 10 is increa.sing~ Moreover, (60) I lim f(x) = - 2 . x+O PROOF. Since for !xi < 2~ we have (6 I) f(x) X = - {--- - I} = X = - { (I 2 X X --+--+ 2 12 X it is clear that lim f(x) x+o I = - 2· - I} + In order to see that f(x) is increasing on lR we may argue as follows. Since (62) it suffices to show that (63) (x>O) or, equivalently, that (64) 2x X I 2 X (x>O). e - 2e + > X e ' Since 00 2n -(65) 2x X I l 2 n e - 2e + = n! X ' n=2 and 00 n (66) 2 X I X X e = (n-2)! ' n=2 ,. and (67) 2n - 2 I for n 2 and n 3, n! = (n-2)! = = we are done if we can show that (68) Zn - z l for n ~ 4, n! > (n-Z)! or, equivalently, that (69) Zn - Z > n(n-1), (n~4). It is easily seen by induction that (70) n 2 Z ~ n , (n~4), so that 2n - Z ~ nz - Z > n2 - n = n(n-1) 1 (n~4) completing the proof of the letmna. D REMARK. Letmna 2.1 may also be proved by means of the identity which may be deduced from X ( I - e zn) n TT k=I X (x>O), -x I - e by logarithmic differentiation and taking the limit for n + 00 • I l PROOF OF PROPOSITION 2.1. Fix any c such that -1 < c ~½- In order to show that (58) is decreasing we have to prove that for all n E JN (71) H(n) - log(n+c) > H(n+l) - log (n+l+c), or, equivalently, that (72) I log(l + n+c) > n+ I ' or 12 (73) I c<-----+l, a I (a=-). n+l I In view of lemma 2.1 and our assumption that c < -- it follows that (73) is = 2 true indeed. Since it is obvious that (58) has the limit y this completes the proof of the proposition. PROPOSITION 2.2. If c > -- - I (= 0.54149 ... ) then the sequence (58) e 2 - I tends increasingly toy. PROOF. Similarly as in the proof of proposition 2.1 it suffices to show that for all n E ]N we have (74) C > + I ' a I (a=n+I). Since a= n!I ~ ½, (74) follows from lemma 2.1, completing the proof. 0 PROPOSITION 2.3. If½< c < ~-- - I then the sequence e2 -00 (75) {H(n) - log(n+c)}n=I is eventually increasing. I PROOF. Fix any c > 2. Similarly as before we have (76) H(n) - log(n+c) < H(n+I) - log(n+l+c) if and only if (77) C > 1 ---- - (n+l) + 1 = --- - - + I, ea - 1 a n+I e - 1 I (a=-). n+l If follows from lemma 2.1 and our assumption that c >½that (77) holds if n is large enough. 0 A somewhat closer examination of the above argument reveals that for all n E ]N (78) H(n) - log(I + 11 ) < y. en - I More precisely we have PROPOSITION 2.4. The sequence (79) {H(n)-log(I+ / )}:=I en - 1 converges increasingly toy. PROOF. It is easy to see that (80) y = lim {H(n) n-+<» log(l +-)}. In order to see that (79) is increasing we may argue as follows. In order to prove that (81) H(n) - )< H(n+I) - log(I+ 11 ) we may just as well show that (82) I + {en+ I - 1} - I log I n -1 I + {e - I} or n+l {e n+I I }-I (83) e -log 1 I n n I }-I e {e -or n - I (84) log e < -n ,, n+I e <--n + 1 <--n + I n+l e - 1 13 14 or (85) or or n e - n ---< e n+l e (87) {en+l - l}-l - (n+l) < {en - l}-I - n or (88) I I -----<---ea_ I a eS _ S I where a= --- < - = s. Hence, the proposition follows from lennna 2.1. n + n PROPOSITION 2.5. The sequence (89) n+ I 00 {H(n) + log (e - l)}n=l tends increa.singZy to y. PROOF. Observe that (90) n+l H(n) + log(e - I)= H(n+I) ----n + n+l + log(e - I)= n+l e = H(n+I) - log --1-- = en+! - I = H(n+l) - log (1 + ---) n+l e - I so that our assertion follows from proposition 2.4. 11 D I Concerning the case c = 2 in proposition 2.1 we have PROPOSITION 2.6. (91) I 1 y < H(n) - log(n +-2) < y + --24n2 PROOF. First observe that for s > 0 (92) H(s) - I log(s+ 2) = I 00 00 -(s +-)t f I -st f -t 2 - e e - e = dt - dt = t - I t 0 e 0 t 00 00 --f ( t I _ e:t) dt + f -st {C- l dt = e 1f - I t . t 0 e e -0 t 00 -2 f -st Je l dt. = y + e l-t- - 1f t e -0 From proposition 2.1 or from the fact that t -2 (93) e > 0, (t>O) ---t t e - I (the proof of which is left to the reader) it is clear that I y < H(s) - log(s +2), (s>O). Now observe that t -2 (94) e t (t>O). --- < t t 24 • e - I In order to see this we may argue as follows: (95) First let t ~ 4. Then t 2 t -2 -2 I e e e t ------<--<--<-< t =~ 4 28 24 15 = 16 so that (94) holds fort~ 4. Now let O < t < 4. If in addition n .?_ 3, then O < t < n+I, so that n+l )n+I • • · · 1 t < (n+I , from which it is easi y seen that (96) n-1 2n 24 . t < 2 (n+I) (n+2) •.. (2n+I), or (97) 2n+I n+2 t t --=----- < --=-:----a-22n ( 2n+ I)! 24.n! · Since (97) also holds for n = 2 and O < t < 4 we have 00 (!_)2n+l 00 n+2 (98) 2 I 2 I I t (2n+l)! < 24 ~ n=2 n=2 from which it follows that (99) or ( 100) or ( I 01) t t 2 2 t 2 t e - e - t < 24 (e -1) t 2 t 2 t (e t-I) e ( I) - t < 24 e -t 2 e t -t- - _e_t ___ 1 < 24 ' (O<t<4). Combining (95) and (IOI) we find that (94) is true. Consequently we have t ( l 02) completing I H(s) - log(s + 2) 00 -st t 00 Y + j e-st 0 00 -st + J e 24 dt < y + _l J 24 e 0 0 the proof of the proposition. {C - I t t e -t 1 - 24J dt + t dt = y + 24.s 2 D REMARK. Numerically it turns out that (for rather small n) the rapidity of convergence of (79) (or (89)) is about the same as that in (91). 3. In this section we prove PROPOSITION 3.1. The sequence (103) ) 1 }oo {H(n + log log(l + n) n=l tends decreasing 7y to y. PROOF. Since e = lim(l + _!_)n and n n-+= ( 104) H(n) + log log(l + .!..) = H(n) - log n + log log(l + _!_)n n n 17 it is clear that (103) tends toy. In order to show that (103) is decreasing we have to prove that ( 105) or ( 106) or (107) or ( I 08) H(n) + log log(l + !) > H(n+l) + log log (I + n1l) 1 log ( I + n) I log-----> --log( I + _l_l) n + 1 n+ log(l +¼) n+l -----> e log ( I + - 1-1) n+ 1 -log(l -n+I°) I log(l + n+l) n+l > e Hence it certainly suffices to show that (109) -log( 1-x) log ( 1-x) X > e • (O<x<l), 18 or (I I 0) X X -log(I-x) > e {-log(l - l+x)}. Since for O < t < I we have (I 11) t2 t 3 t 2 I t 3 -log(I-t) = t + -2 + -3 + ... < t + -- + - --2 3 I - t ' it suffices to show that 2 3 2 I 3 I l X X X X (I I 2) x{ x I x+-+-+ ... >e T+x+z +- . - 2-I 2 3 (l+x) 2 3 3 (l+x) l+x or (I 13) or (114) 2 (l+x) 00 xJ x I 2 X = e lT+x + z (I+x/ 00 n-1 \ x x 3 I 2 l -n- > e .{I + 2 x + j x }, n=l 00 I 3 X } +-3 2 , (l+x) I n=3 1 2 I (--+-+--) n+I n n-1 n X > t c-1 + 3 + 3.c~-2)!) n! 2.(n-1)! n=3 It is easily seen that ( 115) I 2 I I 3 I --+-+-->-+---,----,--..,...+-~--,--..,... n+l n n-1 n! 2.(n-1)! 3.(n-2)!' for all n ~ 3, so that (114) follows, completing the proof. D REMARK. Since (116) we have (117) ,, I n+-(l +_!_) 2 n > e, (nEJN) I H(n) + log log(!+-) = n 1 l I n+- I = H(n) - log(n + 2) + log log ( I + n) 2 > H(n) - log(n + 2). n X • = According to proposition 2.5 we have I H(n) - log(n+ 2) > Y, so that the only interesting thing in proposition 3.1 is the monotonicity of (IOI). 4. We conclude this note by proving the following remarkable identity 00 ( I 18) I - y = l n=2 2 (-I )n [ log n] n+l 2 where log n denotes the logarithm of n in the base 2 whereas[] denotes the greatest integer function. Since the general term of the series in (118) tends to zero the con-vergence of this series follows from the convergence of (119) 2N+I 2 S(N) def li' (-l)n [ log n] l n+I (N~). n=2 19 It is easily seen that S(N) is increasing so that it suffices to prove the convergence of S(2N), (N~). Wr.iting (120) n K(n) = l k=l (-1 l+ I I k • we have (121) K(2n) = H(2n) - H(n). Now observe thaL 2N+I 2 +I ( 122) S(2N) = I (-1 )n [ log n] n=2 n+I I 1 I I I I = ---+ 2(---+---) 3 4 5 6 7 8 = + ... + 20 + N( I 2N + I + - • • . + ----- - N+ I) = 2N + 2 2N + 2N - 2 N+I N = {K(4) - K(2)} + 2{K(8) - K(4)} + •.• + N{K(2 ) - K(2 )} = = -{K(2) + (K(4) + ••• + K(2N)} + N.K(2N+l) = = -{(H(2) - H(I)) + (H(4) - H(2)) + .•• + (H(2N) - H(2N))} + ] =I+ N. {(N+I) log 2 + y + 0(2N+I)} + - (N+I) {N log 2 + y + 0(-1 ) = 2N = 1 - y + 0( ~), 2 from which (118) is immediate. D REFERENCES [1] ADAMS, Note on the value of Euler's constant, Proc. Royal Soc. of London, 28 (1878) pp. 88-94. [2] BROMWICH, An introduction to the theory of infinites eries, MacMillan and Co., London, 1949. [3] GLAISHER, On the calculation of Euler's constant, Proc. Royal Soc. of London, .!.2_ (1871) pp. 514-524, 21 [4] KNOPP, Theory a:nd application of infinite series, Blackie and Son Ltd., London, 1928. [5] KNUTH, Euler's constant to 1271 places, Math. Comp., Vol. 16, no. 77 (1962) pp.275-281. [6] SHANKS, On the calculation of the numerical value of Euler's constant, Proc. Royal Soc. of London, 20 (1872) pp.27-34. ADDENDUM. Just before the printing of this note we found that (118) is equivalent to a similar relation given by SANDHAM in the Amer. Math. Monthly, Vol. 56 (1949) p. 414. A proof of SANDHAM's formula (by BARROW) may be found in the Amer. Math. Monthly, Vol. 58 (1951) p. 117. 

A note on euler's constant

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