Prime Number Theorem Steven Finch. April 27, 2007

Transcription

1 Prime Number Theorem Steve Fich April 7, 007 Let π(x) = P p x, the umber of primes p ot exceedig x. GaussadLegedre cojectured a asymptotic expressio for π(x). Defie the Möbius mu fuctio μ() = if =, ( ) r if is a product of r distict primes, 0 if is divisible by a square > ; the vo Magoldt fuctio ( l(p) if = p Λ() = m for some prime p ad iteger m, 0 otherwise; ad the Chebyshev fuctios θ(x) = l(p), p x ψ(x) = p m x, m l(p) = x Λ() =l(lcm{,,...,bxc}). Hadamard ad de la Vallée Poussi proved the Gauss-Legedre cojecture, amely, π(x) x, θ(x) x, ψ(x) x l(x) as x. These three formulas are equivalet to each other ad also to μ() =0. The Riema zeta fuctio clearly plays a role here sice, for Re(s) >, s = ζ(s), μ() s = ζ(s). 0 Copyright c 007 by Steve R. Fich. All rights reserved.

2 Prime Number Theorem Of may aspects of the Prime Number Theorem, we focus o the followig error estimates [,, 3, 4]: + + θ(x) x ψ(x) x dx = lim x N p N = γ p N Λ() l(p) p l(n) l(p) p(p ) = , l(n) = γ = where γ is the Euler-Mascheroi costat [5, 6]. The latter implies that Λ() = γ. What ca be said about aalogous expressios coected with the Prime Number Theorem for arithmetic progressios 3k + ad4k +? Nevalia [7, 8] gave a straightforward geeralizatio: μ() = L 3 () = 3 3 π, lim N mod3, lim N mod4, Λ() Λ() which imply that [5] mod3 mod4 μ() = L 4 () = π ; l(n) = γ l(3) 3() Ãπe L0 L 3 () = γ l(3) l γ Γ( 3 )3 Γ( 3 )3 l(n) = 4() Ãπe γ l() L0 L 4 () = γ l() l γ Γ( 3 4 ) Γ( 4 )!!, mod3 Λ() 3 = 3γ + l(3) 3π 6 4l(π)+6l(Γ(/3)), mod4 Λ() 4 = 3γ l() π 3l(π)+4l(Γ(/4)).

3 Prime Number Theorem 3 Here is a more complicated geeralizatio. Defie ( l(p) if = p Λ,3 () = for some prime p mod3 ad iteger m, 0 otherwise, ( l(p) if = p Λ,4 () = for some prime p mod4 ad iteger m, 0 otherwise; θ,3 (x) = p x, p mod3 l(p), θ,4 (x) = p x, p mod4 l(p); ψ,3 (x) = Λ,3 (), ψ,4 (x) = Λ,4 (). x x Just as [] we have [9] Λ() s = ζ0 (s) ζ(s) γ ζ(s) γ, s Λ,3 () s = ζ0 (s) ζ(s) L0 3(s) L 3 (s) l(3) 3 s as s. O the oe had, p mod3 ζ(s) γ L0 3(s) L 3 (s) l(3) 3 s p mod3 l(p) p s l(p) p s Λ,3 () = γ L0 3() L 3 () l(3) p mod3 l(p) p but o the other had, Λ,3 () p N, p mod3 l(p) p l(n)+c,3 + γ + c,3 + + p N, m, p mod3 p mod3 p mod3 l(p) p m l(p) p(p ) l(n) γ l(p) p(p )

4 Prime Number Theorem 4 as N. It follows that + Similarly, + Z θ,3 (x) x N θ,4 (x) x p N, p mod3 l(p) p = γ L0 3() L 3 () l(3) = N p N, p mod4 l(p) p = γ L0 4() L 4 () l() = l(n) = c,3 p mod3 l(n) = c,4 l(p) p l(p) p A simple series acceleratio techique [0] arises from the idetity p(p ) p = p(p ) ; p mod3 p mod4 l(p) p(p ) l(p) p(p ) hece p mod3 hece l(p) p(p ) = = p mod3 p mod3 l(p) p(p ) + l(p) p(p ) + p mod3 p l(p) p l(p) p p mod3 l(p) p l(3) 8 ; hece p mod3 l(p) p + p mod3 c,3 = γ 4log(π)+ 9log(3) 8 l(p) p(p ) = p mod3 +6log(Γ(/3)) + ζ0 () ζ() l(p) p(p ) ζ0 () ζ() l(3) 8 ; p mod3 l(p) p(p ).

5 Prime Number Theorem 5 Similarly, c,4 = γ 3log(π)+ log() 3 More complex acceleratio techiques yield [9] +4log(Γ(/4)) + ζ0 () ζ() p mod4 l(p) p(p ). p mod3 l(p) p = , l(p) p = which permit umerical evaluatios such as ad + + ψ,3 (x) x ψ,4 (x) x N Λ,3 () l(n) = γ L0 3() L 3 () l(3) = ( ), N Λ,4 () l(n) = γ L0 4() L 4 () l() = ( ) p mod3 l(p) p l(p) p p mod3 l(p) p(p ) = , p mod4 l(p) p(p ) = The estimates ad.4... for the theta fuctio itegrals are also foud i [, ]. A parallel aalysis of itegrals ivolvig θ,3 (x) = p x, l(p), θ 3,4 (x) = p x, l(p) p mod3 couldbedoeaswell. Aother type of error estimate was provided by McCurley [3]. The maximum value of θ,3 (x)/x occurs at x = 69 ad, further, θ,3 (x) < x for all x. This result is essetially best possible. By cotrast, the maximum value of θ,3 (x)/x

6 Prime Number Theorem 6 is ot kow! (For x 0 8,itoccursatx = ) It ca be show that θ,3 (x) < x for all x, but improvemet is likely. Sharp aalyses of θ 3,4 (x) ad θ,4 (x) as such seem still to be ope. The maximum value of ψ(x)/x occurs at x = 3 ad ψ(x) < x always []. Motgomery [4] cojectured that limif ψ(x) x x l(l(l(x))) = π, limsup ψ(x) x x l(l(l(x))) = π. Let M(x) = P x μ(x); Odlyzko & te Riele [5] proved that limif M(x) M(x) <.009, limsup >.06. x x The precise growth rate of M(x) has bee the subject of speculatio [6, 7, 8]. Most recetly, Goek ad Ng [9, 0] idepedetly cojectured that limif M(x) x l(l(l(x))) 5/4 = C, limsup M(x) x l(l(l(x))) 5/4 = C for some positive costat C. A proof of this or of Motgomery s cojecture would be sesatioal! 0.. Addedum. The secod-order Ladau-Ramauja costat for coutig itegers of the form a +3b is [] γ L 0 3() L 3 () + l(3) + l(p) = p p mod3 ad the (classical) secod-order Ladau-Ramauja costat for coutig itegers of the form a + b is γ L 0 4() L 4 () + l() + l(p) = p The fact that < resolves a questio raised by Shaks & Schmid [, 3]. Further, the secod-order LR costat correspodig to a +b is [4] γ L 0 8() L 8 () + l() + l(p) = p p 5,7mod8

7 Prime Number Theorem 7 ad the secod-order LR costat correspodig to a b is γ L 0 8() L 8 () + l() + p 3,5mod8 l(p) = p The fact that > > verifies a assertio i []; we used the Selberg-Delage method ad formulas i [5] to deduce the precedig expressios for a ± b. See also [6]. Refereces [] E. Ladau, Hadbuch der Lehre vo der Verteilug der Primzahle, d ed., Chelsea, 953, pp ; MR (6,904d). [] J. B. Rosser ad L. Schoefeld, Approximate formulas for some fuctios of prime umbers, Illiois J. Math. 6 (96) 64-94; MR (5 #39). [3] W. Elliso ad F. Elliso, Prime Numbers, Wiley, 985, p. ; MR (87a:08). [4] N. J. A. Sloae, O-Lie Ecyclopedia of Iteger Sequeces, A00348, A04963, A [5] S. R. Fich, Euler-Mascheroi costat, Mathematical Costats, Cambridge Uiv. Press, 003, pp [6] S. R. Fich, Meissel-Mertes costats, Mathematical Costats, Cambridge Uiv. Press, 003, pp [7] V. Nevalia, O costats coected with the prime umber theorem for arithmetic progressios, Aales Academiae Scietiarum Feicae Ser. A I (973). 539, ; MR (49 #79). [8] S. R. Fich, Quadratic Dirichlet L-series, upublished ote (005). [9] P. Moree, Chebyshev s bias for composite umbers with restricted prime divisors, Math. Comp. 73 (004) ; MR0343 (005b:54). [0]. Gourdo ad P. Sebah, Some costats from umber theory (Numbers, Costats ad Computatio), available olie at [] M. P. Youg, Lower-order terms of the -level desity of families of elliptic curves, It.Math.Res.Notices(005) ; math.nt/ ; MR47004 (006c:076).

8 Prime Number Theorem 8 [] S. J. Miller, Lower order terms i the -level desity for families of holomorphic cuspidal ewforms, ariv: [3] K. S. McCurley, Explicit estimates for θ(x;3,l) ad ψ(x;3,l), Math. Comp. 4 (984) 87 96; MR (85g:085). [4] H. L. Motgomery, The zeta fuctio ad prime umbers, Quee s Papers i Pure ad Applied Mathematics,. 54, Proc. 979 Quee s Number Theory Cof., ed. P. Ribeboim, 980, pp. 3; MR (8k:0047). [5] A. M. Odlyzko ad H. J. J. te Riele, Disproof of the Mertes cojecture, J. Reie Agew. Math. 357 (985) 38 60; MR (86m:070). [6] I. J. Good ad R. F. Churchhouse, The Riema hypothesis ad pseudoradom features of the Möbius sequece, Math. Comp. (968) ; MR04006 (39 #46). [7] B. Saffari, Sur la fausseté delacojecturedemertes,c. R. Acad. Sci. Paris Sér. A-B 7 (970) A097 A0; MR (43 #667). [8] T. Kotik ad J. va de Lue, O the order of the Mertes fuctio, Experimet. Math. 3 (004) ; MR87 (005i:39). [9] N. Ng, The distributio factor of values of the summatory fuctio of the Möbius fuctio, CMS Notes, v. 34 (00). 5, 5 8; available olie at [0] N. Ng, The distributio of the summatory fuctio of the Möbius fuctio, Proc. Lodo Math. Soc. 89 (004) ; MR (005f:5). [] S. R. Fich, Ladau-Ramauja costat, Mathematical Costats, Cambridge Uiv. Press, 003, pp [] D. Shaks ad L. P. Schmid, Variatios o a theorem of Ladau. I, Math. Comp. 0 (966) ; MR00678 (35 #564). [3] P. Moree ad H. J. J. te Riele, The hexagoal versus the square lattice, Math. Comp. 73 (004) ; available olie at MR0343 (005b:55). [4] P. Sebah, Two modulo 8 prime sums, upublished ote (009). [5] S. Fich, Quartic ad octic characters modulo,

9 Prime Number Theorem 9 [6] S. R. Fich, Mertes formula, upublished ote (007).