Math 680 Fall 07 Chebyshev s Estimates Here we will prove Chebyshev s estimates for the prime coutig fuctio. These estimates are superseded by the Prime Number Theorem, of course, but are iterestig from both a historical perspective i the methods ivolved. Before we get to the actual estimates themselves, it s useful to recall two relevat defiitios from advaced calculus. Defiitio: Suppose { } is a real-valued sequece. The limit superior or limsup of the sequece is L we write = L if for every ɛ > 0 there is a N such that < L + ɛ for all > N > L ɛ for ifiitely may > N. I other words, L is a accumulatio poit for the sequece, is actually the supremum of all accumulatio poits of the sequece. There is a aalogous otio for limit iferior. Moreover, both ca be eteded to iclude L = ± i the obvious maer. Lemma: Both the limit iferior limit superior of a sequece eist we have lim if. Morever lim eists if oly if the limif limsup are equal, i which case the limit is equal to the limif. Recall that the vo Magoldt Lambda fuctio { if = p e for some prime p e 0, Λ() = 0 otherwise. We will ivestigate the followig two fuctios that are closely related to the prime coutig fuctio : Θ() := Propositio: We have, := Λ() = lim if p m m Θ() = lim if = lim if / log. Θ() = = / log. Proof: We first make some obvious observatios. Clearly Θ(), so that () Θ().
Also, if p is a prime p m < p m+, the occurs i the sum for eactly m times. Thus = p m m = log = log. [ ] log (Here [ ] deotes the greatest iteger fuctio, as usual.) This immediately implies that () Now fi a α (0, ). Assumig that > we have Θ() = / log. α <. Notice that all primes p occurrig i the secod sum here ecessarily satisfy > α log. Hece so that Θ() > α log α < = α log ( π( α ) ) > α log ( α), (3) But for all α (0, ) we have Θ() > α / log α log α. lim α log = 0. α This observatio together with (3) yields Sice this is true for all α (0, ), we get (4) Θ() Θ() α / log. / log.
We ow tur to the limifs. Similar to () (), we have ( ) lim if ( ) lim if Usig (3) oce more, we get (4 ) lim if Θ() / log. Θ() / log. All together, (), (), (4), ( ), ( ) (4 ) combie to prove the propositio. Theorem(Chebyshev): We have 4 log / log I particular, there eist positive real umbers m M such that for all sufficietly large. M > Proof: We will cosider the biomial coefficiet ( ) for positive itegers. We ote that / log > m log. / log = ()! ()( ) ( + ) = (!)! = ( + ) = ( ) > i Moreover, ( ) is the greatest summ here, so that ( ) < ( + ). Therefore (5) > ( ) > +. ( ). Now suppose p is a prime satisfyig < p. Such a prime is clearly ot a factor of!, but is a factor of ()!. Thus every such prime is a divisor of ( ). I particular, we have ( ) p 3 <p
log (( )) <p = Θ() Θ(). (Here, as usual, empty products are iterpreted to be empty sums are iterpreted to be zero.) Combiig this with (5) gives (6) Θ() Θ() < log. Net for all positive itegers m we see via (6) that Θ( m ) = Θ( m ) Θ() = < m m Θ( i ) Θ( i ) i log m = log i = log ( m ) < m+ log. We ote that Θ() is a o-decreasig fuctio. Thus for ay > we may write m < m for some positive iteger m get Now by the Propositio We et cosider the sum Θ() Θ( m ) < m+ log 4 log. S() := Recallig that log = d, we have S() = Θ() = 4 log. / log d log / log. / d. Writig = qd + r with 0 r < d, we see that {d, d,..., qd} is the set of satisfyig d. This implies that S() = d = d / d / =. [ ] d d / [ d] ([ ] [ + d d]) + (/)<d (/)<d [ ] d 4
Hece (7) S() = log / log. Now sice log t is a icreasig fuctio, + log t dt log I particular, S() log t dt (/)+ log. log t dt log t dt = ( ( ) + log + log ( ( + )/ ) + = log + + log( + ) + + log > log ( /( + ) ) log( + ) + log. ) + Combiig this with (7) the Propositio gives lim if = lim if / log S() > lim log ( /( + ) ) log( + ) + log = log. 5