ON THE PRIME NUMBER LEMMA OF SELBERG
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1 MATH. SCAND. 03 (2008, 5 0 ON THE PRIME NUMBER LEMMA OF SELBERG FILIP SAIDAK (Dedicaed o he memory of Ale Selberg Absrac The key resul eeded i almos all elemeary proofs of he Prime Number Theorem is a prime umber lemma proved by Ale Selberg i 948. Wihou resricig ourselves o purely elemeary echiques we show how he error erm i Selberg s fudameal lemma relaes o he error erm i he Prime Number Theorem. I spie of all he ieres i his opic over he las siy years his paricular quesio seems o have bee overlooked i he pas.. Prime Number Theorem Le π( := p. I is simples form he Prime Number Theorem (PNT assers ha, as, { ɛ } d ( π( lim + ɛ 0 0 +ɛ log ψ(, where ψ( = ( = p m log p. The impora error fucio E(, E( := ψ( is closely coeced o zero-free regios of ζ(s i he criical srip. I fac, sice ψ( ψ( 0 = ρ 2 ρ ζ ζ (0 ( 2 log 2 ρ ad 0<Im(ρ<T = O(log T 2, boh sums beig eeded over all orivial zeros of ζ(s, i is easy o see ha (e.g. [5], 8: E( = O( (log 2 ρ if ζ(s = 0 for Re(s>. However as Selberg [9] ad Erdős [7] discovered Received Jauary 29, 2007.
2 6 filip saidak i 948, aalyic mehods are o eclusively eeded for esimaio of E( or he proof of PNT. 2. Selberg s lemma The followig remarkable lemma of Ale Selberg is used i almos all kow elemeary proofs of (: Lemma 2. (Selberg [9], p For all >0we have ( (2 ψ( log + ψ dψ( = 2 log + O(. I his origial proof of Lemma 2. (see [9], p. 307 Selberg used oly he rough elemeary Chebyshev-ype boud ϑ( := p log p = O(, which is equivale o he weak esimaes π( = O(/log ad ψ( = O(. I seems aural o ask wheher a somewha beer esimae for ψ( could give us a sharper resul ha (2. Moreover, if we are willig o abado he idea of ecessiy of purely elemeary meas, ad jus look a Selberg s lemma as a fudameal resul cocerig prime umbers worh sudyig i is ow righ, he i seems of ieres o ivesigae he quesio of wha he sharpes possible esimaes cocerig ψ( may yield i his direcio. This coverse problem is eacly wha we ivesigae below. The purpose of his paper is o give a simple proof he followig: Theorem 2.2. If we assume ha ψ( = + E(, he we have ( ψ(log + ψ dψ( = 2 log (2γ + + O(E((log 2. Corollary 2.3. The sharpes kow versio of he Prime Number Theorem (sill due o Korobov [8] ad Viogradov [22], from 958 implies ha here eiss a cosa A>0, such ha for all ɛ>0we have ψ(log + ( ψ dψ( = 2 log (2γ + ( + O ep ( A(log 3/5 (log log /5. Corollary 2.4. Uder he assumpio of he Riema Hypohesis [7] ( ψ(log + ψ dψ( = 2 log (2γ + + O ( (log 3.
3 o he prime umber lemma of selberg 7 Remark 2.5. Sarig wih Kuh [9] ad va der Corpu [4] may resuls cocerig he error (remaider erm i he Prime Number Theorem by elemeary mehods have bee published over he las si decades, icludig he works of Bombieri [2], Wirsig [23], Diamod & Seiig [6], Lavrik & Sobirov [0] ad Lu [3], o meio jus a few. Moreover, various eesios ad geeralizaios of Lemma 2. were obaied by Shapiro [20], Tauzawa & Iseki [2], Popke [5], Breusch [3] ad ohers. However hese coribuios did o ivolve sharpeig of Selberg s lemma per se, ad hey did o eplicily sudy he relaioship of is error erm o ha of he error i he Prime Number Theorem eiher. Remark 2.6. The sharpes versio of (2 is due o V. Nevalia [4]: ( (3 ψ( log + ψ dψ( = 2 log (2γ + + o(. 3. Lemmas The proof of our mai heorem is shor ad sraigh-forward, bu i relies o some basic lemmas which, for coveiece, we lis below wih appropriae eplaaios. (For all R, we deoe by [] he ieger par of. Lemma 3. (Euler. Le γ := ζ( =2 ( = = Ɣ ( be Euler s cosa. The, for all >0, we have = log + γ + O ( ad log = log + O(log. Proof. Boh resuls follow immediaely from simple applicaios of he Euler-MacLauri Summaio, see [] Theorem 2.4 ad Theorem 2.3, respecively. Lemma 3.2. For posiive iegers, defie he Möbius fucio μ( as if = μ( := ( k if = p p 2 p k 0 if p 2 The ψ( E( M( := μ( E(.
4 8 filip saidak Proof. I fac, uder he give assumpio, eve a lile bi sroger boud o M( could be proved see Ladau [2], Vol. II, Chapers 0 2. For he Chebyshev fucio ψ( we will eed he followig wo lemmas: Lemma 3.3. Le ψ( = + E(, he for >we have ( dψ( E( (4 = log γ + O. Proof. For all real >, iegraig by pars gives us dψ( by Lemma 3.. = ψ( = + ψ( 2 d[] + O d[] = + E( + E( + d[] 2 ( ( E( E( = log γ + O, Lemma 3.4 (Tauzawa & Iseki [2]. Cosider wo real valued fucios f(ad g( relaed via he ideiy g( = log ( f. The, for all >, we have ( (5 f ( log + f dψ( = ( g dm(. Corollary 3.5 (Nevalia [4]. The fucio ψ( ca be wrie i he form [ ] (6 ψ( = log dm( log. Proof. Take f(= i (5. The g( = [] log ad he resul is clear.
5 o he prime umber lemma of selberg 9 4. Proof of he Theorem The key idea is o compare he wo sides of Lemma 3.4 whe f(= ψ(. Wih his paricular choice of f(, he lef had side (LHS of (5 becomes ( ( LHS := (ψ( log + ψ dψ( ( dψ( = ψ(log + ψ dψ( log ( = ψ(log + ψ dψ( 2 log + γ+ O(E(, by Lemma 3.3. O he oher had, he wo esimaes i our Lemma 3. imply g( = log ( ( ψ = log log log = ( log 2 log + O(log 2 ( log 2 + γlog + O(log = (γ + [] log + h( log, where h( is a fucio ha saisfies he boud h( = O(log. Therefore, he righ had side (RHS of (5 ca be ow rewrie as: ( RHS := g dm( [ ] = (γ + log ( dm( + h log dm( = (γ + (log + ψ( + O ((log 2 dm( = (γ + + O(E( + O(E((log 2, by Lemma 3.2 ad Corollary 3.5. Equaig he wo sides proves he heorem. Ackowledgemes. I would like o hak he referee for several helpful commes. REFERENCES. Ayoub, R., A Iroducio o he Aalyic Theory of Numbers, Chaper 2, Mah. Surveys Moogr. 0 ( Bombieri, E., Maggiorazioe del reso el Primzahlsaz col meodo di Erdős-Selberg, Is. Lombardo Accad. Sci. Le. Red. A, 96 (962,
6 0 filip saidak 3. Breusch, R., Elemeary proof of he prime umber heorem wih remaider erm, Pacific J. Mah. 0 (960, va der Corpu, G. J., Sur le rese das la démosraio élémeaire du héorème des ombres premiers, Colloque sur la Théorie des Nombres, Bruelles, 955, Davepor, H., Muliplicaive umber heory, 3rd Ed., Grad. Tes i Mah. 74 ( Diamod, H. G. & Seiig, J., A elemeary proof of he prime umber heorem wih a remaider erm, Ive. Mah., 970, Erdős, P., O a ew mehod i elemeary umber heory which leads o a elemeary proof of he prime umber heorem, Proc. Na. Acad. Sci. U.S.A. 35 (949, Korobov, N. M., Esimaes for rigoomeric sums ad heir applicaios, Uspehi Ma. Nauk 3 (958, Kuh, P., Eie Verbesserug des Resgliedes beim elemeare Beweis des Primzahlsazes, Mah. Scad. 3 (955, Lavrik, A. F. & Sobirov, A. S., O he remaider erm i he elemeary proof of he prime umber heorem, Dokl. Akad. Nauk SSSR 2 (973, Ladau, E., Hadbuch der Lehre vo der Vereilug der Primzahle, Teuber, Leipzig, 909. (Repried 974 by Chelsea, New York. 2. Ladau, E., Vorlesuge über Zahleheorie, Vol. II, Chelsea, New York, Lu, W. C., O he elemeary proof of he prime umber heorem wih a remaider erm, Rocky Mouai J. Mah. 29 (999, o. 3, Nevalia, V., A refieme of Selberg s asympoic equaio, Pacific J. Mah. 2, o. 3 ( Popke, J., O covoluios i umber heory, Idag. Mah. 7 (955, Posikov, A. G. & Romaov, N. P., A simplificaio of A. Selberg s elemeary proof of he asympoic law of disribuio of prime umbers, Uspehi Ma. Nauk (N.S. 0 (955, o. 4, (66, Riema, B., Über die Azahl der Primzahle uer eier gegebee Grösse (859, Colleced Works, p. 45, Teuber, Leipzig, Saidak, F., A elemeary proof of a heorem of Delage, C. R. Mah. Acad. Sci. Soc. R. Ca. 24, 4 (2002, Selberg, A., A elemeary proof of he prime-umber heorem, A. of Mah. (2 50 (949, Shapiro, H., O a heorem of Selberg ad geeralizaios, A. of Mah. (2 5 (950, Tauzawa, T. & Iseki, K., O Selberg s elemeary proof of he prime-umber heorem, Proc. Japa Acad. 27 (95, Viogradov, I. M., A ew esimae for ζ( + i, Izv. Akad. Nauk SSSR, Ser. Ma. 22 (958, Wirsig, E., Elemeare Beweise des Primzahlsazes mi Resglied. I. & II., J. Reie Agew. Mah. 2 (962, , ad 24/25 (964, 8. DEPARTMENT OF MATHEMATICS UNIVERSITY OF NORTH CAROLINA GREENSBORO, NC 2740 USA f saidak@ucg.edu
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