A Simple Proof Of The Prime Number Theorem N. A. Carella

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1 N. A. Carella Abstract: It is show that the Mea Value Theorem for arithmetic fuctios, a simple properties of the eta fuctio are sufficiet to assemble proofs of the Prime Number Theorem, a Dirichlet Theorem. These are amog the simplest proofs of the asymptotic formulas for the correspoig prime coutig fuctios. Mathematics Subject Classificatios: A4, N05, N3. Keywors: Prime Numbers, Distributio of Primes, Prime Number Theorem.. Itrouctio The Prime Number Theorem is a asymptotic formula ( ) log o( log ) for the umber of primes p up to a umber. The uatity E() o( log) is calle the error term. Oe of the objectives of prime umber theory is to reuce the error term to the optimal E() O( log ). Geerally, the various proofs of the Prime Number Theorem are erive from the erofree regios of the eta fuctio (s) of a comple umber s C. The first proofs, iepeetly achieve by Haamar, a elavallee Poussi, are base o the erofree regio s C : e(s). Sice the, may other relate proofs have bee fou. Furthermore, the erofree regios of the eta fuctio, a the error term E() o( log) have bee slightly improve, see [FD]. The eceptio to this rule are the more recetly iscovere elemetary proofs of the Prime Number Theorem, which are erive from the average orers of various arithmetic fuctios. The elemetary methos o ot reuire ay iformatio o the eta fuctio, refer to [SB], [ES], a [DD]. I this ote, a simple proof of the Prime Number Theorem is costructe from Mea Value Theorem for arithmetic fuctios, a basic properties of the eta fuctio. This proof oes ot reuire ay kowlege of the prime umbers, a oes ot reuire ay ifficult to prove erofree regio beyo the well kow, a simplest erofree regio s C : e(s) >. I aitio, a

2 similar proof of the Dirichlet Theorem for primes i arithmetic progressios is iclue. Theorem. (Prime Number Theorem) For all sufficietly large umber, the prime coutig fuctio satisfies the asymptotic formula () # prime p log o( ). () log Proof: For N, let f ( ) ( ) be the weighte characteristic fuctio of the prime umbers, see (0). The correspoig geeratig fuctio is () s (s) (s) (s) (s) () (s) for s C, e(s) >, see Lemma 3. By Lemma 4, the fuctio ( s) ( s) g( ) coverges at s =. Therefore, by Theorem 7 or 9, the mea value of f ( ) ( ) is give by M( f ) lim () (), (3) () see (5) a (8). This a Theorem 9 immeiately imply that () o(). (4) Net, by partial summatio, it follows that s () () log () log log o( log ) (5) p k, k as claime. This is probably oe of the simplest proof of the Prime Number Theorem, there is a claim for the simplest proof i [DF, p. 80]. Etesio to Dirichlet Theorem As the mea value theorem for arithmetic fuctios has a atural etesio to arithmetic progressio, it is atural to cosier the et result.

3 Theorem. (Dirichlet Theorem) For all sufficietly large umber, a a pair of itegers a,, gc(a, ) =, the prime coutig fuctio satisfies the asymptotic formula (, a, ) # prime p a mo O( ) o( ), (6) log log where c 0 is a costat. Proof: Assume is prime, a cosier the Mobius pair f ( ) g( ), a g( ) ( ) log. (7) By Theorem 0, the mea value of f ( ) ( ) over arithmetic progressio + a : N is give by M ( f ) lim ( ) () () ( ) ( ), a mo g( ) ( )log ( ) ( ) ( )log ( )log. (8) This a Theorem 9 immeiately imply that ) O( ) o( ), (9) (, a mo where M ( f ) O( ). This proves the claim. Note. The proof of Theorem 0 has o limitatios o the rage of values of. Thus, this result is probably a improvemet o the Siegel-Walfis Theorem, which states that (, a, ) ()log O(e log ) with = O(log B ), B > 0, a 0 < < costats. 3

4 . Powers Series Epasios of the Zeta Fuctio Let N = 0,,, 3, be a oegative iteger. The Mobius fuctio is efie by () ()v if p p p v, 0 if suarefree. (0) A the vomagolt fuctio is efie by () log p if pk, k, 0 if p k, k. () Lemma 3. Let : N C be the vomagolt fuctio. The, e(s) >. (s) (s) (), () Proof: Sice the eta fuctio is coverget o the comple half plae e(s) >, it has a Euler prouct Takig the logarithm erivative yiels s p s () (s) p s. (3) s ( s) log ( s) ( s) s p p p s s log p p s log p s p ( ) s. (4) The seco lie follows from the absolute coverges o the comple half plae e(s) >, rearragig the ouble sums, a the efiitio of the vomagolt fuctio. s The eta fuctio ( s) is a etire fuctio of a comple variable s C with a simple pole at s =. At each fie comple umber s0 C, it has a Taylor series epasio of the form 4

5 ( s ) ( ) s 0 ( ) ( s s ( s )! ) 0, (5) 0 where () (s) is the th erivative. The power series epasios at the eros, a some other special values of the eta fuctio (s) are simpler to etermie, for eample, s0 Z = : 0. The well kow power series epasio at s = has the series (s) (s ) (s ) 0 (s ).0 (s ) (s ), (6) see where k is the kth Stieltjes costat, see [KR], a [DL, 5..4]. A at s = 0, it has the epasio (s) (s ) () s, (7) 0 see [DL]. The power series epasios at the eros s0 are some sort of eta moular forms sice the first coefficiet a0 = (s0) = 0. Lemma 4. The comple value fuctio (s) (s) is aalytic a erofree o the comple half plae e(s), a at s =. Proof: Sice the eta fuctio is erofree o the comple half plae e(s) >, it is sufficiet to cosier it at the simple pole at s =. Towar this e, rewrite the fuctio as a ratio (s) (s) (s ) (s ) (s ) 0 (s ) 0 (s ), (8) where the power series of the eta fuctio, a its erivative have the power series epasios (s) s (s ) 0 (s ), (9) (s) (s ) (s ) (s ) 3, Respectively, see (5). Takig the limit yiels () () lim (s ) (s). (0) s (s ) (s) 5

6 This shows that (s) (s) is well efie at each poit o the comple half plae e(s) >, a at s =. Lemma 5. Let, : N C be the Mobius, a the vomagolt arithmetic fuctios. The () ( ) o(). () Proof: By Lemmas 3 a 4, the series (s) (s) () s () s ()( ) () coverges o the comple half plae e(s) >, a at s =. Therefore, by Lemma 3, the summatory fuctio f ( ) s ( ) ( ) o( ) (3) with f ( ) ( ) ( ), for large. 3. Mea Values of Arithmetic Fuctios Let f : N C be a comple value arithmetic fuctio o the set of oegative itegers. The mea value of a fuctio is efie by M ( f ) lim f ( ). (4) The mea value of a arithmetic fuctio is sort of a weighte esity of the subset of itegers Supp ( f ) : f ( ) 0 N, which is the support of the fuctio f, see [SR, p. 46] for a iscussio of the mea value. The esity of a subset of itegers A N is efie by (A) lim # : A. 3.. Some Results o Mea Values of Arithmetic Fuctios This sectio ivestigates the two cases of coverget series, a iverget series 6

7 f () a f (), (5) respectively, a the correspoig mea values M ( f ) lim f ( ) 0 a M( f ) lim f () 0. (6) First, a result o the case of coverget series is cosiere here. Lemma 6. Let f : N C be a arithmetic fuctio. If the series f () coverges, the its mea value M ( f ) lim f ( ) 0 vaishes. Proof Cosier the pair of fiite sums f (), a f (). By hypothesis f () c o(), c 0 costat, for large. Therefore, f () f () tr(t) R() R() o(), R(t) t (7) where R() f () c o(). By the efiitio of the mea value of a fuctio M ( f ) lim f ( ) 0, this cofirms that f() has mea value ero. This is staar material i the literature, see [PS, p. 4]. The seco case is covere by a few results o iverget series, which are cosiere et. Let f ( ) g( ), a let the series c g() be absolutely coverget. Uer these coitios, the mea value of the fuctio f() ca be etermie iirectly from the properties of the fuctio g(). Theorem 7. (Witer) Cosier the arithmetic fuctios f, g : N C, a assume that the associate geeratig series are eta multiple f () s g() (s). (8) s 7

8 The, the followigs hol. (i) If the series f () s is efie for e(s) >, a the series c g() is absolutely coverget, the the mea value, a the partial sum are give by (ii) If the series g() M ( f ) a f () c o(). (9) f () s is efie for e(s) >, a the series c g() is absolutely coverget, the for ay > 0, the mea value, a the partial sum are give by g() M ( f ) a f () c O( ). (30) Proof: The partial sum of the series f () s (s) g() s is rearrage as f () g() g() g(), g() g() g() O g(), (3) where (( )) = [ ] is the fractioal part fuctio. The first lie arises from the covolutio of the power series (s) s, a g(). This is followe by reversig the orer of summatio. Moreover, the first fiite sum is because g() g() o() (3) g() is absolutely coverget, a the seco fiite sum is 8

9 g() g() g() g() g() g() (33) O( ) o() o(). Agai, this follows from the absolute covergece of the series g(). Similar proofs appear i [PV, p. 38], [DF, p. 83], a [HD, p. 7]. Aother erivatio of the Witer Theorem from the Wieer-Ikehara Theorem is also give i [PV, p. 39]. Theorem 8. (Aer) Let f, g : N C be arithmetic fuctios a assume that the associate geeratig series are eta multiple f () s g() (s). (34) If the series f () s coverges for e(s) >, a g() O(), the the mea value, a the partial sum are give by s g() M ( f ) a f () c o(). (35) The goal of the et result is to stregthe Witer Theorem by removig the absolutely covergece coitio. Theorem 9. eta multiple Suppose that the arithmetic fuctios f, g : N C have geeratig series that are f () s g() (s). (36) s The, the followigs hol. If the series f () s is efie for e(s) >, a the series c g( is coverget, the the mea value, a the partial sum are give by 0 ) g() M ( f ) a f ( ) c ( ) 0 o. (37) 9

10 Proof: The partial sum of the series f () s (s) g() s is rearrage as f () g() g() g() o( ) g() o, g(). (38) The first lie is a rearragemet of the orer of summatio. The seco lie follows from Lemma. Sice the series for c 0 g() g, the partial sum satisfies g() c 0 o(), c0 > 0 costat, for large, see the proof of Lemma 3. I light of this observatio, it follows that f () c o() 0 o( ) c o() 0 c 0 o( ). (39) Now, the origial partial sum f ( ) c o( ) is recovere by partial summatio. Etesio To Arithmetic Progressios The mea value of theorem arithmetic fuctios over arithmetic progressios + a : facilitates aother simple proof of Dirichlet Theorem. Theorem 0. Let f ( ) g( ), a let the series c g( ) 0 be absolutely coverget. The M( f ) lim, amo f () c (a) g(), (40) where c ( ) gc(, k k ) e k. The proof is give i [PV, p. 43], seems to have o limitatios o the rage of values of. Thus, this result is probably a improvemet o the Siegel-Walfis Theorem, which states that (, a, ) ()log O(e log ) with = O(log B ), B > 0, a 0 < < costats. 0

11 For the parameter a <, a gc(a, ) =, the mea value reuces to mo, ) ( ) ( ) ( lim ) ( a g f f M. (4) 4. Powers Sums Over Arithmetic Progressios Let N = 0,,, 3, be the set of oegative itegers, a let N + a = + a : 0 be the arithmetic progressio efie by a pair of itegers a 0, a. The sums of powers over arithmetic progressios is oe of the possible geeraliatios of the sums of powers k, k 0, over the itegers. A few estimates of the powers sums over arithmetic progressios are compute here. Lemma. Let a 0, a be fie itegers. Let be a sufficietly large real umber. The (i), amo o( ). (ii) ) ( mo, O a. (4) Proof: The itegers i a liear arithmetic progressio are of the form = m + a, with 0 m ( a). Isertig this ito the fiite sum prouces, amo m m (a) a m ( a) a a a a. (43) where [ ] = be the largest iteger fuctio. Put = a, a epa the epressio to obtai: ). ( o a a a (44) Puttig = a back ito the (4) yiels the result. The above estimates are sufficiet for the itee applicatios. A sharper estimate of the form

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