AN ASYMPTOTIC ROBIN INEQUALITY. Patrick Solé CNRS/LAGA, Université Paris 8, Saint-Denis, France.

Transcription

1 #A8 INTEGERS 6 (206) AN ASYMPTOTIC ROBIN INEQUALITY Patrick Solé CNRS/LAGA, Uiversité Paris 8, Sait-Deis, Frace. sole@est.fr Yuyag Zhu Departmet of Math ad Physics, Hefei Uiversity, Hefei, Chia zhuyy@hfuu.edu.c Received: /4/6, Accepted: /6/6, Published: /23/6 Abstract The cojectured Robi iequality for a iteger > 7! is () < e log log, where deotes the Euler costat, ad () = P d d. Robi proved that this cojecture is equivalet to the Riema hypothesis (RH). Writig D() = e log log (), ad d() = D(), we prove ucoditioally that lim if! d() = 0. The mai igrediets of the proof are a estimate for the Chebyshev summatory fuctio, ad a e ective versio of Mertes third theorem due to Rosser ad Schoefeld. A ew criterio for RH depedig solely o lim if! D() is derived.. Itroductio.. History The cojectured Robi s iequality for a iteger > 7! = 5040 is () < e log log, where deotes the Euler costat, ad is the sum-of-divisors fuctio () = P d d. This iequality has bee show to hold ucoditioally for 7! < apple N with N e e26 [], ad for ifiite families of itegers that are odd ad greater tha 9 [4]; square-free ad greater tha 30 [4]; a sum of two squares ad greater tha 720 [2]; ot divisible by the fifth power of a prime [4]; ot divisible by the seveth power of a prime []; ot divisible by the eleveth power of a prime [3].

2 INTEGERS: 6 (206) 2 Ramauja showed that the Riema Hypothesis implies Robi s iequality for large eough [8]. Robi proved the coverse statemet [9], thus makig that cojecture a criterio for RH. This criterio was made popular by [6] which derives a alterate criterio ivolvig harmoic umbers..2. Cotributio Deote the di erece betwee the right-had side ad the left-had side of Robi s iequality by D() = e log log (). Let d() = D(). The mai result of this ote is Theorem. We have lim if! d() = 0. The proof of Theorem will deped o the followig itermediate result. Theorem 2. For large, the quatity lim if! d() is fiite ad oegative. The mai igrediets of the proof of Theorem 2 are a combiatorial iequality betwee arithmetic fuctios (Lemma ), a e ective versio of Mertes third theorem due to Rosser ad Schoefeld (Lemma 2), ad a asymptotic estimate of Chebyshev s first summatory fuctio (Lemma 4). Also eeded is a result of Ramauja of 95, first published i 997 [8]. We also study the asymptotic behavior of D(). Recall that a umber is colossally abudat (CA for short) if it is a left-to-right maximum for the fuctio with domai the set of itegers x 7! (x) x, where is a real parameter. Thus is CA if m < + (m) etails m < () +. + Theorem 3. We have the followig limits whe rages over the set of CA umbers: If RH is false the lim if! D() = If RH is true the lim! D() =. This result costitutes a ew criterio for RH. Its proof will deped, for the RH false part, o a oscillatio theorem of Robi [9], modelled after ad depedig upo a oscillatio theorem of Nicolas [7] for the Euler totiet fuctio. For the RH true case, we use a result of Ramauja from 95, first published i 997 [8]..3. Orgaizatio The material is arraged as follows. The ext sectio cotais the proof of Theorem, Sectio 3 that of Theorem 2, ad Sectio 4 that of Theorem 3. Sectio 5 cocludes

3 INTEGERS: 6 (206) 3 ad gives some ope problems. 2. Proof of Theorem The result will follow from Theorem 2 if we exhibit a sequece of itegers m with lim m! D( m ) = 0. We follow the approach of [4, 4, proof of Lemma 4., ), p. 366]. Cosider of the shape = Q papplex pt, with t > iteger ad x real, both goig to ifiity, ad to be specified later. By this referece, we have d() = e log log (t) + o t(), with deotig the Riema zeta fuctio. The error term ca be made e ective as follows. By [0, (3.28),(3.30)] we have e log x( 2 log 2 x ) apple Y papplex ( p ) apple e log x( + log 2 x ). From the Euler product of ad [4, Lemma 6.4] we derive (t) apple Y papplex ( t tx exp( ) apple pt (t) t ). Combiig these four bouds together we ca take o t () = O exp( tx t ) = O(x t ). Now it is elemetary to show that for a iteger t > we have (t) = + h(t), with h(t) = O(/2 t ). Ideed Thus, summarizig, we get 2 t apple (t) apple X m= 2 mt = 2 t 2 t. d() = e log log O(/2 t ) + O(x t ). To achieve d()! 0, we eed both log log 2 t, ad log log x t, where stads for o() ( little-oh ) otatio. This is esured if we take x = p m, ad t = m +. I that case we have log log = log m + log (p m ). By Lemma 4 below, log (p m ). O the other had, p m m log m as is well-kow (see e.g. [5]). Combiig the last two estimates we see that log log 2 log m << 2 m. Similarly, log log << p m m. t

4 INTEGERS: 6 (206) 4 3. Proof of Theorem 2 If lim if! d() = the lim! D() =, ad, by Robi s criterio, RH holds. We kow the by [8, p.25] that the sequece d() p log admits fiite upper ad lower limits whe rages over the set of CA umbers (see 4), which is a cotradictio. Assume therefore that lim if! d() is fiite, ad let us show that it is oegative. For ay iteger write its decompositio ito prime powers as = i= where the q i s are prime umbers, idexed by icreasig order, ad a i s are positive itegers. Deote by p i the i th prime umber, ad for ay iteger, let = i= q ai i, p ai i. Note that, by defiitio, for each i =, 2,, m we have q i. With this otatio observe that I particular () ( ) = i= d( ) apple p ai+ i p i i= = i= p i p i p i apple 2m, p ai i p i. ad, likewise, We ca thus assume whe cosiderig lim if! d() that m!. We prepare for the proof of Theorem 2 by a series of Lemmas. p i, ad that, therefore, apple 2. Thus, if m is bouded ad!, we see that d()!. Lemma. For ay iteger, we have d() d( ). Proof. Let d() = f () f 2 (), with f () = e log log, ad f 2 () = (). The mootoicity of the log ad yields f () f ( ). Write f 2 () = Q m i= g(a i, q i ), where g(a, x) = x x a x. Writig g(a, x) = + x + + xa x a = we see that, for fixed a, the fuctio x 7! g(a, x) is oicreasig i x. This implies that g(a i, q i ) apple g(a i, p i ) for each i =, 2,, m ad, therefore, multiplyig m iequalities betwee oegative umbers, that f 2 () apple f 2 ( ). The result follows the by d() = f () f 2 (). 2 ax i=0 x i,

5 INTEGERS: 6 (206) 5 Lemma 2. For ay large eough we have i= ( ) < e ( + log 2 p m ). Proof. Note that, with the otatio of the proof of Lemma, we have g(a, x) apple for x 2 ad a, ad, therefore p i f 2 () = g(a i, q i ) apple p i. i= x x, The result follows the by [0, Th. 8, (39)]. 2 Recall the Chebyshev summatory fuctio #(x) = P papplex log(p). Lemma 3. For all, we have log #(p m ). Proof. By defiitio mx log = a i log p i i= m X i= log p i = #(p m ). 2 A classical result, related to the Prime Number Theorem, is Lemma 4. For large x, we have #(x) = x + O( x log x ). Proof. A e ective versio is i [0, Th. 4]. See for istace [5, Th. 4.7] for a p sharper error term i O(x exp( log x 5 )). 2 We are ow ready for the proof of Theorem. Proof. By Lemma d() d( ). By Lemma 2 we have ( ) > e log p m( + log 2 ). () p m By Lemma 3 ad 4 we have p m e log log e log #(p m ) = e + O( ) = (2) e + log( + O( )) = e log(p m ) + O( ), (3) where the last equality results from log(+u) u for u! 0. Addig up iequalities ad 3, after cacellatio of the terms i, we obtai the iequality d( ) = e log log ( ) O( ) e, the right had side of which goes to zero for large. 2

6 INTEGERS: 6 (206) 6 4. Proof of Theorem 3 Recall the stadard otatio for oscillatio theorems [5, p. 94]. If f, g are two real valued fuctios of a real variable x, with g > 0, the we write f(x) = + (g(x)), if lim sup x! f(x)/g(x) > 0 f(x) = (g(x)), if lim if x! f(x)/g(x) < 0 f(x) = ± (g(x)), if both f(x) = + (g(x)), ad f(x) = (g(x)) hold By [9, Propositio, 4] if RH is false the, for CA umbers we have log log D() = ± ( (log ) b ), for some b 2 (0, ). This would imply, usig the ifiitude of CA umbers [9], that lim if! D() =. If RH holds the by [8, p.25] the sequece D()p log admits upper ad lower limits for colossally abudat that are fiite ad greater tha 0. Thus there are reals greater tha 0 say A, ad B such that A apple D() apple B log log, whe is CA. Therefore lim! D() =. 5. Coclusio ad Ope Problems I this ote we have studied the quatity D() which is the di erece betwee the two sides of Robi s iequality, ad its ormalizatio d() = D(). While the asymptotic behavior of d() ca be determied ucoditioally (Theorem ), that of D() depeds crucially o the truth of RH (Theorem 3). It would be desirable to exted Theorem 3 to itegers that are ot CA. It seems impossible to use Theorem ad Theorem 3 together to prove that RH holds. For istace, oe caot rule p out the case that D() behaves like whe!, which would ot cotradict the fact that lim if! d() = 0. Ackowledgmets. Both authors are grateful to Mijia Shi for puttig them i touch. They thak Gourab Batthacharya, Floria Luca, Jea-Louis Nicolas, Pieter Moree for helpful discussios, ad the aoymous referee for careful readig.

7 INTEGERS: 6 (206) 7 Refereces [] K. Briggs, Abudat umbers ad the Riema hypothesis, Experimet. Math. 5, (2006), o. 2, [2] W. D. Baks, D. Hart, P. Moree, C. W. Nevas, The Nicolas ad Robi iequalities with sums of two squares, Moatsh. Math. 57, (2009), o. 4, [3] K. Brougha, T. Trudgia, Robi s iequality for -free itegers. Itegers 5 (205), Paper No. A2. [4] Y-J. Choie, N. Lichiardopol, P. Moree, P. Solé, O Robi s criterio for the Riema hypothesis, J. Théor. Nombres Bordeaux 9 (2007), o. 2, [5] W. J. Elliso, M. Medès-Frace, Les Nombres Premiers, Herma, Paris (975). [6] J. C. Lagarias, A elemetary problem equivalet to the Riema hypothesis, Amer. Math. Mothly 09 (2002), [7] J.-L. Nicolas, Petites valeurs de la foctio d Euler, J. Number Theory 7 (983), o. 3, [8] S. Ramauja, Highly composite umbers, The Ramauja Joural (997), [9] G. Robi, Grades valeurs de la foctio somme des diviseurs et hypothèse de Riema, J. Math. Pures Appl. (9), 63 (984), o. 2, [0] J. B. Rosser, L. Schoefeld, Approximate formulas for some fuctios of prime umbers, Illiois J. Math. 6 (962), [] P. Solé, M. Plaat, The Robi iequality for 7-free itegers, Itegers 2, article A65.