Aales Uiv. Sci. Budapest., Sect. Comp. 39 2013) 459 469 NORMALIZATION OF BEURLING GENERALIZED PRIMES WITH RIEMANN HYPOTHESIS We-Bi Zhag Chug Ma Pig) Guagzhou, People s Republic of Chia) Dedicated to Professor K.-H. Idlekofer o his sevetieth birthday Commuicated by Imre Kátai Received November 25, 2012; accepted Jauary 14, 2013) Abstract. Two Beurlig geeralized umber systems are give such that the primes of both systems up to ay assiged large umber are exactly all the ratioal primes ot exceedig the umber ad both cosist of oly ratioal umbers of the form +l/2. The coutig fuctios of g-itegers of both satisfy Nx) = x + x 1/2 Oexp{clog x) 2/3 }). The first realizes the RH ad the secod realizes exactly the classical zero-free regio of the Riema zeta fuctio ad the de la Vallée Poussi error term. 1. Itroductio Let P = {p i } be a ubouded sequece of real umbers satisfyig 1 < p 1 p 2. We call P a sequece of Beurlig geeralized primes heceforth, g-primes) ad the free multiplicative semigroup N geerated by P a system of Beurlig geeralized itegers g-itegers). Note that it is ot assumed that either P or N lies i the positive itegers whole umbers) N or that the uique factorizatio is i force i N. Key words ad phrases: Beurlig geeralized primes, prime umber theorem, Riema Hypothesis, de la Vallée Poussi error term. 2010 Mathematics Subject Classificatio: 11M41, 11N80, 11N05.
460 W.-B. Zhag Let πx) = π P x) := #{P [1, x]} ad Nx) = N P x) := #{N [1, x]} be the coutig fuctios of g-primes i P ad g-itegers i N, respectively. The geeral questio is how hypotheses o oe of Nx) ad πx) imply coclusios o the aother. Also, let ζs) = ζ P s) := x s dnx) = exp log 1 x s ) 1 dπx), 1 σ := Rs > 1 be the zeta fuctio associated with the g-iteger system N, which is the aalytic term combiig Nx) ad πx). A simple example of g-primes is the sequece P of all odd primes i N. Here N is just the sequece of odd whole umbers. Hece N P = 1/2)x + O1) ad π P x) is the coutig fuctio of odd primes. The omissio of the oly eve prime 2 cuts the desity of whole umbers i half. I case N = N, we have the classical coutig fuctios of ratioal primes ad whole umbers ad the Riema zeta fuctio. I 1899 de la Vallée Poussi [5] proved that 1.1) ζs) 0 for σ > 1 c/ log t + 4) with some costat c > 0, where s = σ + it. From this so-called classical zero-free regio, he could deduce the prime umber theorem PNT) with a error term as 1.2) πx) = lix) + Ox exp{ clog x) 1/2 }), where c is also a positive costat but eed ot be the same c i 1.1)). Later it was proved that if the ζs) 0, for σ > 1/2 the Riema Hypothesis) πx) = lix) + Ox 1/2 log x). The determiatio of the truth of RH is oe of most importat problems i today s mathematics. I geeral case, i 1937, Beurlig [2] showed that if 1.3) Nx) = Ax + Ox log γ x) with costats A > 0 ad γ > 3/2 the 1.4) πx) x/ log x, i.e., the aalogue of the PNT holds for this g-umber system. 1
Normalizatio of Beurlig primes 461 Better PNT error terms tha 1.2) for whole umbers are kow these days, with ay expoet less tha 3/5 i place of 1/2. All these improvemets are made i usig the well-spacig of whole umbers, i.e., the additive structure of positive itegers. Beurlig s result ca be viewed as a abstractio of a earlier proof by E. Ladau [4] of the prime ideal theorem for algebraic umber fields; but with the error term as 1.2). Ladau s argumet aalyzes the aalytic coectio betwee the coutig fuctios of itegral ideals ad prime ideals accordig to the orms of ideals. The orm fuctio o algebraic umber fields is multiplicative but ot additive. From this view-poit, it has log bee cojectured [1] that 1.2) may be optimal for g-umbers. I previous works [3] ad [6], examples are give i which the classical zero-free regio 1.1) ad the PNT error term 1.2) are exactly realized for Beurlig g-umber systems. Also, i the latter, examples are give i which the Riema Hypothesis are realized for Beurlig g-umber systems. It is the of iterest to view how close these examples could be to the atural umbers N. We shall show that, o the basis of g-primes P R ad P B give i [6], g-primes with the same properties ad more features imitatig the primes i N see i) ad ii) below) ca be costructed further. 1.5) Theorem 1. Give ay whole umber R, there is g-primes P R such that i) P R cosists of umbers i the sequece { v k = + l } 2 for k = 2 + l, = 0, 1, 2,, 0 l < 2 ad P R [1, R ] cosists exactly of primes i N ot exceedig R ; ii) the coutig fuctio N R x) of the resultig g-itegers satisfies N R x) = x + Ox 1/2 exp{clog x) 2/3 }); iii) the associated zeta fuctio ζ R s) is aalytic for σ > 1/2 except a simple pole at s = 1 with residue 1; iv) the fuctio ζ R s) has o zeros o the half plae σ > 1/2 the Riema Hypothesis); v) the g-prime coutig fuctio π R x) satisfies π R x) = lix) + Ox 1/2 ).
462 W.-B. Zhag Theorem 2. Give ay whole umber B, there is g-primes P B such that the fuctios N B x), π B x) ad ζ B s) have the properties i) iii) of Theorem 1 ad iv) the fuctio ζ B s) has ifiitely may zeros o the curve σ = 1 1/ log t, t e 2 ad o zeros to its right; v) the Chebyshev fuctio ψ B x) satisfies lim sup x lim if x ψ B x) x x exp{ 2 log x} = 2 ψ B x) x x exp{ 2 log x} = 2. Remark. The g-primes P R ad P B cosist of ratioal umbers of the form +l/2. We are uable to costruct P R or P B cosistig of oly positive itegers. 2. Normalizatio of g-primes We shall give oly the proof of Theorem 1 because Theorem 2 ca be proved i the same way. Without loss of geerality, we may assume that R > 4. The proof is a costructio procedure cosistig of a series of successively addig chose g-primes to a system ad deletig other chose g-primes from a system. Hece, durig the procedure, P, ζ, π ad N with subscripts 1, 2, 3,... will deote a series of particular g-prime systems, the associated zeta fuctios, the coutig fuctios of g-primes, ad the coutig fuctios of the g-itegers, respectively. We first isert all ratioal primes up to R that are ot already i P R give i [6] ad delete all g-primes up to R that are ot ratioal primes. I this way the ew system P 1 satisfies properties i), iv), v) of Theorem 1 ad ii) 1 the coutig fuctio N 1 x) of the resultig g-itegers satisfies with κ 1 > 0; N 1 x) = κ 1 x + Ox 1/2 exp{clog x) 2/3 }) iii) 1 the associated zeta fuctio ζ 1 s) is aalytic for σ > 1/2 except a simple pole at s = 1 with residue κ 1.
Normalizatio of Beurlig primes 463 The desired estimates of ii) 1 ad iii) 1 ca be show by usig the iclusioexclusio priciple sice oly a fiite umber of g-primes are ivolved. If κ 1 = 1 the the costructio is doe. Otherwise, if κ 1 > 1, we may move a fiite umber of g-primes exceedig R from P 1 so that the coutig fuctio N 2 x) of the g-itegers of the resultig system satisfies N 2 x) = κ 2 x + Ox 1/2 exp{clog x) 2/3 }) with κ 2 < 1 sice p j P 1 p j> R 1 p 1 j ) = 0. Hece we may further assume that N 2 x) κ 2 x C 2 x 1/2 exp{clog x) 2/3 } with 1 A 1 < κ 2 < 1, where A is a iteger satisfyig A > R. Otherwise, 0 < κ 2 1 R + 1) 1. The there is a umber m N satisfyig m R + 1 such that m + 1 < κ 2 m = R +1 1 1 ) 1 1. The ew g-primes R + 1,, m are added to P 2. If the right-had side is a equality the the costructio is doe. Otherwise, m + 1 < κ 3 := κ 2 m = R +1 1 1 ) 1 < 1 ad the coutig fuctio N 3 x) of the ew system P 3 satisfies N 3 x) κ 3 x C 3 x 1/2 exp{clog x) 2/3 } with C 3 = C 2 m = R +1 1 1/2 ) 1. To complete the costructio, we appeal to the followig lemma ad leave its proof to the ext sectio.
464 W.-B. Zhag Lemma 1. Give κ satisfyig 1 A 1 < κ < 1 with a iteger A > R, there is a fiite or ifiite sequece {w } of w N such that w > R, 1 w 1 ) = κ, w 1/2 <, 1 = Olog x). w x Thus we set κ = κ 3 ad apply Lemma 1. Without loss of geerality, we may assume that the sequece {w } is icreasig, i.e., w w +1. Note that 1 w s ) 1 is aalytic for σ > 1/2 ad has o zeros there. We elarge P 3 to cotai {w }. The the resultig g-prime system P R cosists of umbers i {v k } of 1.5) but may ot be a subsequece of {v k }) ad P R [1, R ] cosists exactly of primes i N ot exceedig R sice all ew added w > R. Also, the associated zeta fuctio is give by ζ R s) = ζ 3 s) 1 w s ) 1, which is aalytic for σ > 1/2 except a simple pole at s = 1 with residue κ 3 1 w 1 ) 1 = 1 ad has o zeros o σ > 1/2. Moreover, the coutig fuctio π R x) of g- primes i P R satisfies π R x) = π 3 x) + 1 = lix) + Ox 1/2 ). w x Fially, the coutig fuctio N R x) of g-itegers satisfies 2.1) N R x) x Cx 1/2 exp{clog x) 2/3 }. with C = C 3 1 w 1/2 ) 1. Actually, if {w } is a fiite sequece the 2.1) is plaily true. Otherwise, {w } is ifiite ad hece w. For ay give x 1, o the oe had, if w 1 > x the, 1 1 N R x) κ 3 1 w 1 ) 1 x + C 3 1 w 1/2 ) 1 x 1/2 exp{clog x) 2/3 } =1 x + C 3 =1 =1 1 w 1/2 ) 1 x 1/2 exp{clog x) 2/3 }
Normalizatio of Beurlig primes 465 sice g-itegers i N R ot exceedig x have o g-prime divisors exceedig w 1. O the other had, choosig 1 sufficietly large so that > 1 w 1 x 1 /2, we have { 1 w 1 ) exp 2 } w 1 exp{ x 1 } > 1 > 1 ad hece x Thus, we arrive at 1 ) > 1 1 w 1 ) x1 exp{ x 1 }) 1. 1 1 N R x) κ 3 1 w 1 ) 1 x C 3 1 w 1/2 ) 1 x 1/2 exp{clog x) 2/3 } = =1 =1 = x x 1 ) 1 w 1 ) > 1 1 C 3 1 w 1/2 ) 1 x 1/2 exp{clog x) 2/3 } x C 3 =1 =1 1 w 1/2 ) 1 x 1/2 exp{clog x) 2/3 }. Therefore the system P R has all expected properties ad the costructio is fiished. 3. Proof of Lemma 1 Let a fuctio f be defied by fα) = 11 α ) 1, 1 < α <. The fα) is cotiuous ad strictly decreasig, lim fα) =, ad lim fα) = 1. α 1+ α Hece fα) = 1/κ has a uique solutio α. Note that α > A > R sice 1 α ) 1 = 1 κ < 1 A) 1 1.
466 W.-B. Zhag If α N, put w = α ad the lemma is proved. Otherwise, α / N ad we set a := α α α =: b. Note that a 1 a b b 1 ad a 1 A. Hece ) 1 > fα) = 1 a κ > It follows that ) a b ) 1 < κ ) 1. b ) 1 < 1. b The it ca be show that 3.1) ) ) 1 > 1 2 a b A 2. Actually, the left-had side of 3.1) equals a b exp + a b =1 =1 k 2 1 k a k b k a k b k. Note that a b 1. The secod sum i the expoet is at least k 2 > Hece we see that a 1 1)a k 1 k k 2 ) b 1 + a k 2 a k b = b + + a b k 2 a k b k 1 1 a b a 1. ) 1 exp{ { exp a 1 1 a b 1 } a a 2 1 } a a 2 = 1 a } 1 a )a 2 ) { exp a 1 { = exp > 1 2 A 2. + b k 1 ) > 1 1 } a b a 1 exp { 2 } A 2 >
Normalizatio of Beurlig primes 467 We ow recursively defie a sequece, fiite or ifiite, of triples as follows. First, let {κ m, A m, S m )} κ 1 := κ, A 1 := A, ad S 1 := {b }. The defie A 2 := A 2 1/2 or A ) 2 /2 accordig to A 1 is eve or odd so that A 2 is eve. Note that A 2 satisfies A 2 < κ 1 w S 1 1 w 1 ) 1 1. I geeral, if κ m, A m, S m ) ad A m+1 have bee defied ad if A m+1 < κ m w S m 1 w 1 ) 1 = 1 the k m, A m, S m ) is the last triple of the sequece. I this case, we set {w } = = m j=1 S j ad the lemma is proved. Otherwise, A m+1 < κ m w S m 1 w 1 ) 1 < 1 the a repeat of the same argumet give above with κ = κ m+1 := κ m 1 w 1 ) 1 w S m ad A = A m+1 yields a subset S m+1 of N such that 1 2 A 2 < κ m+1 1 w 1 ) 1 1. m+1 w S m+1 Hece κ m+1, A m+1, S m+1 ) ad A m+2 := A 2 m+1/2 are defied. This procedure yields a fiite or ifiite sequece {κ m, A m, S m )}. Fially let {w } := m S m. Note that A 2 > A 1 > R > 4, ad A 3 = A 2 2/2 > A 3/2 2 ad by iductio A m A 3/2)m 2 2, m 2.
468 W.-B. Zhag Therefore i.e., κ lim m κ m w S 1 1 w 1 ) 1 w S m 1 w 1 ) 1 = 1, Moreover, from the defiitio of S m, we have w 1/2 + w 1/2 + w S 1 w S 2 w S 2 1 w 1 ) 1 = 1 2A 1/2 1 + A 1/2 2 + A 1/2 3 + ) A 1/2 1 + A 1/2 2 + A 1/2 2 ) 3/2 + A 1/2 2 ) 3/2)2 + A 1/2 1 + A 1/2 2 /1 A 1/4 2 ) ad w S 1 w x 1 + w S 2 w x 1 + w S 3 w x log x log A 1 + log x log A 2 + = log x log A 1 + 3 log x log A 2. 1 + This completes the proof of Lemma 1. log x 3/2) log A 2 + log x 3/2) 2 log A 2 + = Refereces [1] Batema, P.T. ad H.G. Diamod, Asymptotic distributio of Beurlig s geeralized prime umbers, i: Studies i Number Theory, Vol. 6, 152 210. Math. Assoc. Am., Pretice-Hall, Eglewood Cliffs 1969). [2] Beurlig, A., Aalyse de la loi asymtotique de la distributio des ombres premiers gééralisés, I, Acta Math., 68 1937), 255 291. [3] Diamod, H.G., H.L. Motgomery ad U. Vorhauer, Beurlig primes with large oscillatio, Math. A., 334 2006), 1 36, MR 2208947 2006j:11131). [4] Ladau, E., Neue Beiträge zur aalytische Zahletheorie, Thales Verlag, Esse, 4 1986), 41 53.
Normalizatio of Beurlig primes 469 [5] de la Vallée Poussi, Ch.J., Sur la foctio ζs) de Riema et le ombre des ombres premier iférieurs à ue limite doée, Vol. 59 o. 1, Mémoires couroés et autres mémoires publiés par l Académie Royle des Scieces, des Lettres et des Beaux-Arts de Belgique, 1899-1900, 74 pp. [6] Zhag, W.-B., Beurlig primes with RH ad Beurlig primes with large oscillatio, Vol. 337, Math. A., 337 2007), 671 704, MR 2274547 2007k:11148). W.-B. Zhag Departmet of Mathematics Uiversity of Illiois Urbaa Illiois 61801 USA Departmet of Applied Mathematics South Chia Uiversity of Techology Guagzhou People s Republic of Chia w-zhag1@math.uiuc.edu