Lune, J. (Jan) van de

Riele, H.J.J. (Herman) te

English

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stichting mathematisch centrum AFDELING ZUIVERE WiSKUNDE J. VAN DE LUNE & H.J.J. TE RIELE QN A CONJECTURE OF ERDOS (ll) zw 56/75 ~ MC OCTOBER 2e boerhaavestraat 49 amsterdam Ptu.n:te.d a.:t .the. Ma.:the.ma,ti,c.a..l Ce.n:tJt.e., 49, Ze. BoeJ1.ha.a.ve1.>:tJr.,a.c/..t, Am.6.teJl.dam. The. Mathe.ma.tic.al Ce.n,tJie., 6ou.nde.d .the. 11-.th of; Fe.bnuoJty 1946, ..U a. n.on.-pM6U inll.t-Ltu;ti.on. cu.ming at .the. pnomo;t,lon. 06 pwr..e. ma.the.ma,ti,C/2 a.n.d U.6 a.pp.Uc.a.tioM. It ..U .oponl,otr.e..d by .the. Ne..theJr1.a.n.d6 GoveJl.n.me.n.t :t.hJ/..ough :the. Ne.:t.heJri.a.ndo Otr.ga.n.iza,ti,on. &on the. Adva.nc.e.me.n.t a,& Pwr..e. Re..oe.Mc.h (Z .W.O), by the. MwuupaLLty 06 Am.ti.teJl.dam, by .the. UniveMliy a,& Am.oteJl.dam, by .the. F ne.e. Un,lveN.>Uy a.:t Am.ti.teJl.dam, a.n.d by in.dU.6.ttueJ.> . AMS(MOS) subject classification scheme 0970): 26A86 On a conjecture of Erdos (II) . van de Lune & H.J.J. te Riele ABSTRACT For any integer n ~ 2 let m = m(n) be determined by ( l _ .!_) n l ( I _ _L_) n • m > 2 > m-l In this note it will be shown that n n • • • + m > (m+ I) and .. for almost all n. Compare the conjecture of ERDOS stated in the Amer. Math. Monthly, Vol. 56 (1949), p.343 (Advanced Problem 431+7). KEY WORDS & PHRASES: Inequalities, sums of powers of integers.; uniform dis-tribu t·ion. 0. INTRODUCTION .. In [I] ERDOS proposed the following problem: Prove that if m and n are positive integers such that (0.0) then (0. l) and (0.2) (0.3) I > - > 2 (I __ l_)n m-l n n n n l + 2 + • • • + ( m- 2) < ( m- 1 ) In+ 2n + ••• + mn > (m+l)n. Show also that 1n infinitely many instances and that (0.4) n n n n I + 2 + ••• + (m-1) > m 1n infinitely many instances. The first partial solution of this problem was recently given by the first named author [4]. He showed by elementary means that (0.1) is true indeed and, in a similar fashion, he also proved the related inequality ( 0. 5) n n n n I + 2 + • • • + ( m+ 1) > ( m+ 2 ) • In the meanwhile TIJDEMAN has simplified the proof of (O.l) consider-ably (see [4; addendum]). In this paper we will investigate the remaining inequalities (0.2), (0.3) and (0.4). It will be shown that the natural density of all n for which (0.2), resp. (0.3), 1s true is equal to I, so that (0~3) certainly holds true 2 in infinitely many instances. However, we have not succeeded in finding any n for which either (0.2) or (0.3) is false. Also, we have no example in which (0.4) is true. 1. PRELIMINARIES AND THE MAIN THEOREM In [4] it was already shown that we may assume n ~ 2 and that from (O.O) it follows that for any given n the number m = m(n) is uniquely de-termined by ( 1. 1) A(n) < m(n) < A(n) + I or, equivalently, by (1. 2) m(n) = [A(n)] + !, where (1.3) A(n) = l I -1/n - I+ / 1 - 2 2 1 n -From (I.I), (1.3) and [4; lemma 3.3] it follows that (1.4) m(n) > A(n) = l + l > I + { n _ .!._} > __ n_ exp(log 2) _ 1 log 2 2 log 2 n so that ( l • 5) n m(n) < log 2. Also, by (l.1), (l.3) and [4; lemma 3.3] we have ( I. 6) m(n) < l + A(n) = 2 + 1 < 2 + { n exp(~) _ 1 log 2 n < n +l ~ = log 2 2 + 24 1 - - + 2 log 2} < 12 n Since m(2) = 4 and (L 7) n 3 log 2 2n, + -· + 24 < log 2 2 (n~)) it follows that ( l • 8) n l --> -m(n) = 2' (n~2). Moreover, from (1,4) and (1.6) it is clear that (I. 9) lim _n_ = log 2. m(n) n-+<xi Similarly as in [4] we define m (1.10) CJ (n) = m I k=I In [5] it was shown that for all m,n E: N (m, nE N). (I.I l) mn+l(m+l)n mn(m+l)n+I ___ ....;_. _____ <CJ (n) < ---'---''----( I ) n+ l n+ I m ( 1 ) n+ l n+ I • m+ - m m+ - m We now define 8 = 8(m,n) by ( I • 12) o (n) m n n m (m+l) (m+8) = -------( l) n+I n+I m+ -m or, more explicitly, by (1.13) G(m,n) = -m + a (n) (m+l) m {l - (~)n+I} n m+l m so that by (l.1 I) we have (1.14) 0 < 0(m,n) < I. Since (for a proof, see [SJ) 3 4 ( L 15) l m0 + 1(m+J) 0 + mn(m+l)n+l a (n) > - ----------------- = m = 2 ( 1 ) n+ I n+ l m+ - m n n 1 m (m+l) (m+:,) ( l) n+l n+l • m+ - m (m, nE N) we even have ( l. 16) l 2 ~ e(m,n) < I. Concerning the function 8(m,n) we have the following (MAIN) THEOREM 1. If for n ~ 2 the number m = m(n) is determined by (0.0) then (1.17) lim 8(m,n) = 2(1-log 2), n-+<x> Before proving this theorem we first examine the sums o (n) somewhat m closer, By means of the Euler-Maclaurin sunnnation formula we readily ob-tain (see [2; p.527]) ( 1. 18) a (n) m n+ I I m n =--+-m + n+l 2 [E-] 2 B , ...2E_( n )mn-2r+ I l 2r 2r-l r=l or, equivalently (1.19) a (n) 1 m m --=--+-+ n n+l 2 m [~] 2 I r=l B2r 2 I ( n ) - r+ 2r 2r-1 m ' where the Bernoulli numbers Bare defined by ( l , 20) z --= oo B l r r -z r! r=O <lzl<21r). It is well known that for any real a# 0 (see [2; p.528]) (l.21) where (1.22) l I I k B2r 2r-l rL (2r) ! a. + ¾: (a) ---=---+ a. I a. 2 e -11< (a.) = a. ax 2k+ l ( 1 J O P Zk+ 1 (x) e dx ea-1 so that (1. 23) m 1 -----+-= en/m_ 1 n 2 t B2r (E.)2r-l n l ( 2r) ! m + \: (m) • r=l Taking k =[%]in (1.23) it follows from (1.19) and (1.23) that ( I • 24) = I B2r (-mn)2r-1{ 1 _ n(n-l) ••• (n-2r+2)} + R._ (-mn) = r=2 (2r)! n2r-l -1< k B2r = l --,-...,-;- (E_)2r-l 8 (2r-2) + R (-mn) r=2 (2r)! m n k where o (·) is defined by n ( I , 25) o (a) = l - ( I - .!..) ( l - I) ... (l - ~), n n n m From (1.25) it is easily seen that for any fixed a EN (l. 26) lim n 8 (a) n l = I + 2 + ••• +a= 2 a(a+l), n-+oo Also, by mathematical induction, it is easily shown that (I. 27) (0 < ) o (a) n n As a consequence we have I B2 cE./r-1 on(2r-2)1 ( l • 28) (2r~! m IB2rl (2n/r-1 =- (2r)! 2n m , so that, in view of (l.5), ( l~a<n;n~2). IB2rl (E.) 2r- l 2r-2 2 < (2r)! - m n = 5 a (n) m ---= n m 6 (1.29) I I B2r I 2 l ( 2 log 2) r- . < 2 (2r) ! ' the right hand side of (1.29) being the general term of a convergent series with positive terms (see (l.20) and note that log 2 < TT). Hence~ by a uni-form convergence argument (or by Lebesgue's dominated convergence theorem) we obtain (l. 30) ~ B2r (E.)2r-ln lim l ( 2r)! m 8n(2r-2) = n+oo r=2 00 B2r 2 l I 1 -,--...,........ (log 2) r- -(2r-2) (2r-l) == l (2r)! 2 r=2 l 2 00 B2r 2 3 = 2(log 2) r12 (2r)!(2r-1)(2r-2)(log 2) r- = 1 2 d2 { 00 B2r 2r-l} = 2(log2) - 2 I (2r)!x • dx r=2 x = log 2 Now observe that (see [2; p.204]) (l.31) B2 2 B4 4 x cot x = I - 2T (2x) + 4!" (2x) - + ••• from which it 1s easily seen that ( I • 32) oo B 1 2r 2r-l l --x r=2 (2r)! i ix l x = - cot - - - - -2 2 X 12 so that (1.33) j_~ { I B2r x2r-l} = £ {i.. cot ix _ _!_ _ ~} = 2 (2r) ! dx2 2 2 x 12 dx r=2 2 d { l } 2 e -x - ex = - 3 + dx 4( . ix)2 = - 3 - ( -x/2 x/2)4 x s1.n 2 x e -e which, for x = log 2, takes the value ( l • 34) £ {i.. cot ix - ..!_ - ..!..} d 2 2 2 X 12 l 2 x x= og -2 = ----3 + 6. (log 2) Hence, defining (l.35) P (n) k B2r 2 = ~ __ (-n-) r-1 ( ) n l (2r)! m(n) 8n 2r-2 r=2 it follows from (1.24) that form= m(n) (I. 36) a (n) _m __ = --- + 1 _ m _ p (n) _ R_ (-mn) mn en/m_ 1 n(n+l) n k where, in view of (1.30), (1.33) and (1.34) (1.37) lim p(n) = .!_(log 2) 2 { - 2 + 6} = n-+oo 2 (log 2) 3 As to I\(;) we have the following estimate ( I • 38) (~)2k+l 11\C;) I ~ :/m e -l Joi Ip 2k+ I (x) I em dx. nx Since 2 log 2 + 3(1og 2) • ( I. 39) max 0.2._x~l Ip 2k+ I (x) I 4 <----= (2TT)2k+I ' (see [2; p.527]) and ( I. 40) it follows from (1.5), (l.8) and (l .38) that (1.41) IR_ (-mn) I __ < (log 2)n 8 k 2TI le- l so that ~ (;) tends exponentially fast to zero as n • 00 , As a simple consequence of (1.36), (1.37) and (1.41) we have ~i.42) a (n) lim _m __ = ---- + I n-+oo mn e log 2 - I = 2 (a relation which may also be proved by much simpler means). 7 8 PROOF OF THEOREM I. From (1.13) it follows that (I. 43) {o (n) } o (n) m ( ( 1 )n+l) m (l (1-m+ll)n+l).· 8 (m, n) = m n I - 1 - - - 1 + -- -m+l n m m Since (1.44) lim (I+ o.(n))n = ea if lim o.(n) n-+<x> n n-+<» = a it follows from (1.9) and (1.42) that ( 1. 45) o (n) lim _m __ (1 - (1--1-)n+l) = 2 • (1 - 21) = n m+l n-+<x> m so that, in order to determine lim 8(m,n), we only need to study the asymp-totic behaviour of n-+oo ( I .46) {om(n) l I ( l (1--)n+) m--;- - m+l m _J( I m _ p(n) _ (n))( _ (l --1-)n+l) _ 1} = = wl en/m_ 1 + 1 - n(n+l) n ~ m 1 m+l _ -m{ m + p(n)} {i _ (l __ l_)n+t1.J + - n(n+l) n m+l + m{( / + 1)(1 - (1--1-)n+l) - 1} - m R (~) {1 - (1 --1-)n+t}. n m 1 m+l -1< m m+l e -Since ~(i) tends exponentially fast to zero as n + 00 and m(n) = O(n) it follows easily that (1.47) { I n+I} I - ( I - m+l) = O. Next we observe that (1.48) { m p (n)} . { m2 m } !!: m n(n+l) + -n- = !: n(n+l) + 'ii' n(n) = = __ 12_)..,,..2 + lo~ 2 {- lo~ 2 + 3 (log 2)2} = 3 log 2, (log 9 so that (i .49) Finally we have ( I • 50) mf (-,-.-- + l ) ( I - ( 1 - _l_) n+ l ) l en/m _ 1 m+l 1 _ en/m(l __ l_) n+ 1 1 m+l lfl = m -----,----- = n/m 1 e --e-n""'t-m-_-1 { l - exp(~ + (n+ I) log(] - m!l))} = m = - ---,---en/m - I n 1 exp(-+ (n+l)log(l --1)) - l { l m m+ • ; + (n+l)log(l - m!l)f (O#) .!!. + (n+l)log(l --1- 1) m m+ so that, in view of (1.51) lim {; + (n+l)log(I - m!l)} = n-+<io I log 2 + log 2 = 0 it follows that ( 1.52) lim ( L 50) n-+<io = - lim m{: + (n+l) log( l - m!I)} = n-+<io = - lim m{; - (n+l)(m!l + l 2 + O(~))} = n-+<io 2 (m+ I) m I I = - (log 2 - l - 2 log 2) = - 2 log 2. Combining (1.45) through (1.52) with (1.43) it follows that (I. 53) lim 8(m,n) = l + 0 - flog 2 + (1 - ½ log 2) = 2(1-log 2) n-+<io completing the proof of the theorem. 0 10 2. APPLICATIONS TO ERDOS' CONJECTURE THEOREM 2.1. The set of all n EN for which inequality (0.2) is false has natural density equal to zero. THEOREM 2.2. The set of all n EN for which inequality (0.3) is false has natural density equal to zero. Before proving these theorems we study the numbers m(n) - \(n) some-what closer. LEMMA 2.1. If the real sequence {a(n)}:=I is unifoY'mly distributed modulo (u.d. mod 1) and if {S(n)}:=l is any convergent real sequence then also 00 {a(n) + S(n)}n=l is u.d. mod I. PROOF. Exercise. 00 LEMMA 2.2. The (real) sequence {a(n)}n=l is u.d. mod 1 if and only if the sequence {-a(n)}:=l is u.d. mod I. PROOF, Exercise. 00 LEMMA 2.3. The sequence {m(n) - \(n)}n=2 is uniformly distributed on the intewal (0, J). PROOF. Since m(n) EN and \(n) < m(n) < A(n) + l it suffices to show that 00 {-\(n)}n=2 is u.d. mod l. In view of lemma 2.2 it therefore suffices to 00 show that {\(n)}n=2 is u.d. mod I. Observing that ( 2. I) \(n) ] l I + 1 = + 2 l /n - I = = log 2 exp( ) -n l n I oc.!.)) n 1 o(.!.) = + (log 2 --+ = + - + 2 n log 2 2 n ' ( n-+oo) 00 it follows from lemma 2.1 and the irrationality of log 2 that {\(n)}n=2 is u.d. mod I (compare [3; p.92, Satz 9]), proving the lemma. D 1 1 LEMMA 2.4. If {a(n)}:=l is uniformly distributed on the interval (0,1) and {a(I\_)}~=l is any convergent subsequence then the natural density of {nk}~=l is equal to zero. PROOF. Exercise. PROOF OF THEOREM 2.1. If (0.2) is false for only finitely many n E :N then we are done. Therefore, we assume (0.2) to be false for infinitely many n. For these n we have (2.2) n n n n 1 + 2 + ••• + m < (m+l) or, equivalently, (2.3) cr (n) < (m+l)n. m Hence, writing 6 instead of 6(m,n), (2.4) _m_n_(m_+_l __ )_0_(_m_+_e...,..) < ( 1 ) n+l n+l m+ -m so that (2.5) or, equivalently , (2.6) which may be rewritten as (2. 7) 1 m < ----,--,--....---= (2 +~/1/n+l) _ I m From this it follows that 12 (2.8) 0 < m(n) - }..(n) = - 1 + m(n) - 1 < 21/n_ 1 = < -1 1 I + (Z+!)(I/n+l) _ 1 2 l /n - 1 = = m 1 1 = - + = 1 e I exp(-1 log(2 +-)) - 1 exp(- log 2) - 1 n+ m n = -1 + { n + 1 - .!. + o(l)} - { n - .!. + o(.!.)} = log(Z +~) 2 n log 2 2 n m = -1 + ---- + n{ 1 - ----,-2} + e e log log(2+-) log(2+-) m m 1 = -1 + ----,,--log ( 2 + !) m e n log( 1 +2m) e log 2 log(2 +-) m In view of theorem we have (2.9) e lim n log( 1 + 2m) n-+<><> en = lim log( I + Zm)n = n 1 . en log exp 1m 2m = n-+oo 1 . en = 1m- = 2m n-+<» (I-log 2)•log 2 so that, if n runs through those positive integers for which (0.2) is false, we have (2. I 0) 1 0 ~ lim sup{m(n) - A(n)} ~ -I + --log 2 from which it is clear that (2. l I) lim{m(n) - A(n)} = O, n-+oo where n is such that (0.2) is false. (I-log 2)log 2 = O (log 2/ From this and lemmas (2.3) and (2.4) it follows that the set of all n for which (0.2) is false, has natural density equal to zero, completing the proof of theorem 2.1. 0 13 PROOF OF THEOREM 2.2. Suppose that (0.3) is false for infinitely many n E :N. For these n we have ( 2. 12) ln + 2n + ··• + or, equivalently (2. 13) n am-I (n) ~ m • n n (m-1) ~ m Writing 8 instead of 8(m-l,n) we have in view of (1.12) that (2.14) n ~l ~l (m-1) (m-1+8) > m - (m-1) which may be rewritten as (2.15) m~ l + I (Z+-8-)(l/n+l) _ I m-1 It follows that (2.16) l > m(n) - >,.(n) > I + 1 - (l + I ) = = ( 2 +_8_/1/n+l) _ l 21/n_ l m-1 I =---------(2+-8-/1/n+I) _ I m-1 2 1 /n _ and similarly as in the proof of theorem 2.1 it follows that (2.17) lim{m(n) - >,.(n)} = n-+«> where n is such that (0.3) is false. Again, utilizing lennnas (2.3) and (2.4) this completes the proof of theorem 2.2. D FINAL REMARK. In a forthcoming paper the first named author will demonstrate how the technique of this paper may be applied to the diophantine equation n n n n I + 2 + ••• + M = (M+I) 14 or, more generally, to n n n n J + 2 + • • • + M = G • (M+ l ) where G is any given positive rational number. REFERENCES [I] ERDOS, P., Ad-l)anced Problem 434?, Amer. Math. Monthly Vol. 56 (1949), p.343. [2] KNOPP, K., Theory and application of infinite series, Blackie & Son Ltd., 1928. [3] KOKSMA, J.F., Diophantische approximationen, Springer (!936). [4] van de LUNE, J., On a conjecture of Erdos (I), Mathematical Centre~ Amsterdam, Report ZW 54/75. [SJ van de LUNE, J., Monotonic approximation of integrals in relation to some inequalities for sums of powers of integers, Mathematical Centre, Amsterdam, Report ZW 39/75. 

On a conjecture of Erdös (II)

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