Summatory function of the number of prime factors

Transcription

1 Summatory function of the number of prime factor Xianchang Meng arxiv: v [math.nt] Jan 08 Abtract We conider the ummatory function of the number of prime factor for integer x over arithmetic progreion. Numerical experiment ugget that ome arithmetic progreion conit more number of prime factor than other. Greg Martin conjectured that the difference of the ummatory function hould attain a contant ign for all ufficiently large x. In thi paper, we provide trong evidence for Greg Martin conjecture. Moreover, we derive a general theorem for arithmetic function from the Selberg cla. Introduction and tatement of reult Let Ωn be the number of prime factor of n counted with multiplicity, and ωn be the number of ditinct prime factor of n. Hardy and Ramanujan [3] howed that x ωn = x log log x + Ax + O,. log log x and n x n x Ωn = x log log x + Bx + O x log log x,. for ome contant A and B. Thu, the average order of Ωn and ωn i log log n. The famou Erdő-Kac Theorem ee [] or [0] ay that the limiting ditribution of ωn log log n log log n i the tandard normal ditribution. 00 Mathematic Subject Claification: M6, N60, M36 Key word: Number of prime factor, Hankel contour, Generalized Riemann Hypothei

2 One may get better error term for the ummatory function in. and. by auming the Riemann Hypothei. Wolke [3] proved that the Riemann Hypothei i true if and only if ωn = x n x 0 j log x/ P j log log x log j x + O x +ɛ,.3 where P j are ome polynomial of degree. Similarly, under the Extended Riemann Hypothei ERH for Dirichlet L-function, we expect the following formula for the correponding ummatory function over arithmetic progreion, n a mod q n x ωn = x φq 0 j log x/ P j log log x log j x + O x +ɛ, a, q =. Thu, when we compare two different arithmetic progreion a b mod q, a, q = and b, q =, under the ERH, we have ωn ωn = O x +ɛ. n a mod q n x n b mod q n x Baed on the above analyi, one may expect that there are roughly equal number of prime factor in different arithmetic progreion. However, Greg Martin noticed that ome arithmetic progreion actually have more number of prime factor by doing numerical calculation. He conjectured that ωn ωn < 0.4 n mod 4 n x n 3 mod 4 n x hold for all ufficiently large x. In thi paper, we give trong evidence for Greg Martin conjecture.4. We conider both cae of Ωn and ωn. Let χ χ 0 be a non-principal Dirichlet character modulo q, and denote ψ f x, χ := n x χnfn, where f = ω or Ω. Theorem. Aume the ERH, the zero of L, χ are imple, and L, χ 0. For any fixed large T 0, we have { } x x ψ ω x, χ = aχ L, χ log x + L, χ L, χ log x x + L ρ, χxiγ log x + iγ + Σx, T 0,.5 0 where aχ = if χ i real, 0 otherwie, and ɛ > 0, lim up Y Y Σe y, T 0 dy T0 ɛ..6

3 Similarly, we have the following reult for ψ Ω x, χ. Theorem. Aume the ERH, the implicity of zero, and L, χ 0. For any fixed large T 0, { ψ Ω x, χ =aχ L + x log x where aχ = if χ i real, 0 otherwie, and ɛ > 0, } x x, χ log x + L, χ L, χ log x L ρ, χxiγ + iγ + Σx, T 0,.7 0 lim up Y Y Σe y, T 0 dy T0 ɛ..8 The Linear Independence Conjecture LI tate that all the poitive imaginary part of zero of L, χ are linearly independent over Q. In [7] and [8], the author ued ERH and LI to tudy the the ditribution of k-free number and ubtle difference of the number of product of k prime among different arithmetic progreion. Let χ 4 be the non-principal Dirichlet character mod 4, i.e. χ 4 = and χ 4 3 =. Let { P ω := N N : ωn } ωn < 0, and P Ω := { N N : n mod 4 n N n mod 4 n N Ωn n 3 mod 4 n N n 3 mod 4 n N Ωn > 0 Since L, χ 4 >0, by Theorem and and further auming LI, we deduce that the logarithmic denitie of the et P ω and P Ω are both equal to. The logarithmic denity of a et S i defined to be δs := lim X log X n X,n S Although we cannot prove the full Greg Martin conjecture.4, we give trong evidence to upport thi conjecture ubject to the ERH and LI. In order to prove the full conjecture, one may need to formulate new idea and introduce more powerful tool. We numerically calculated ψ Ω x, χ 4 and ψ ω x, χ 4 for x 0 8. See Figure. and.. In thee two graph, the blue dot are numerical data and the red line are the function of the main term predicted in our theorem. We ee that our theorem match very well with numerical calculation. Greg Martin provided u more numerical data for different Dirichlet character, including both real and complex character. See Figure.3 and.4. The tructure of thi paper i a follow. In Section, we give the proof of our main theorem. In Section 3, we generalize the method ued in [8] and thi paper to more general arithmetic function in Selberg cla. 3 n. }.

4 Figure.: Graph of ψ Ω x, χ 4 Figure.: Graph of ψ ω x, χ 4 4

5 Figure.3: Real Dirichlet character mod 7 5

6 Figure.4: Complex Dirichlet character mod 7 Proof of Theorem Conider the Dirichlet erie, F ω, χ := n= ωnχn n = L, χf, χ,. 6

7 and where F Ω, χ := n= F, χ = Ωnχn n = L, χ F, χ + F, χ + F 3, χ 3 +,. p prime χp p = log L, χ log L, χ + G,.3 with G abolutely convergent for R σ 0 = In our proof, we borrow the idea developed in [8] to deal with the ingularitie of F ω, χ and F Ω, χ. In [8], we mainly dealt with power of log L, χ. In thi paper, we have to take care of L, χ log L, χ with more careful calculation.. Contour Integral Repreentation By Perron formula [5], Chapter V, Theorem, we have Lemma. For any T, ψ f x, χ = πi c+it where c = +, and f = ω or Ω. log x c it F f, χ x x log x + O +, T Under the ERH, uing the imilar method a in [] Theorem 4.6, one can how that, for any ɛ > 0 and χ χ 0 mod q, there exit a equence of number T = {T n } n=0 atifying n T n n + uch that, Tn ɛ Lσ + it n, χ Tn δ+ɛ, δ < σ <. Let ρ be a zero of L, χ, ρ be the ditance of ρ to the nearet other zero, and D γ := γ T. For each zero ρ, and X > 0, let Hρ, X denote the truncated Hankel contour min T T urrounding the point = ρ with radiu 0 < r ρ min, ρ, Dγ, which include the circle x 3 ρ = r ρ excluding the point = ρ r ρ, and the half-line ρ X, ρ r] traced twice with argument +π and π repectively. Let H, X denote the correponding Hankel contour urrounding = with radiu r 0 =. x Take δ =. By Lemma, we pull the contour to the left to the line R = δ 00 uing the truncated Hankel contour Hρ, δ to avoid the zero of L, χ and uing H, δ to avoid the point =. See Figure.. Then we have the following lemma. Lemma. Aume the ERH, and L, χ 0 χ χ 0. Then for T T, ψ f x, χ = F f, χ x d + aχ F f, χ πi Hρ,δ πi H x,δ x log x + O + x T T + x δ ɛ δ T δ+ɛ,.4 where aχ = if χ i real, 0 otherwie, and f = ω or Ω. 7

8 Figure.: Integration contour Proof. By. and., if χ i not real, = i not a ingularity. Under the ERH, the integral on the horizontal line i bounded by c O T δ+ɛ xσ δ T dσ x = O..5 T δ ɛ And the integral on the vertical line R = δ i bounded by T t + δ+ɛ x δ O dt = O x δ T δ+ɛ..6 t + Then by Lemma, we get the deired formula. T Let H0, X be the truncated Hankel contour urrounding 0 with radiu r. For implicity, we denote [ ] d j Γ j u := dz j Γz. z=u Lemma 3 [6], Lemma 5. For X >, z C and j Z +, we have w z log w j e w dw = j dj + E πi dz j j,z X, Γz H0,X 8

9 where E j,z X eπ Iz π X log t + π j dt. t Rz e t Lemma 4 [8], Lemma. For any integer k and m 0. We have δ 0 log σ iπ k log σ + iπ k σ m x σ dσ m,k log log x k log x m+. Similar to the proof of Lemma 8 in [8], we have the following reult. Lemma 5. Let Ha, δ be the truncated Hankel contour urrounding a complex number a Ra > δ with radiu 0 < r. Then, for any integer k, l 0 x a l log k a x πi Ha,δ = k x a k k log log x k j alog x l+ j Γ j= j l k x a k k log log x k j a log x l+ j Γ j= j l x a log log x k x a δ/ + O k,l + O a Ra δ log x l+3 k,l + xa δ/. a a Proof of Lemma 5. We have the equality = a + a + a a. a With the above equality, we write the integral in the lemma a a l log k a πi a + a a + x =: I a a + I + I 3. Ha,δ For I 3, uing Lemma 4, we get a l log k a a x Ha,δ a δ log σ iπ k log σ + iπ k σ l+ x σ x a r a a σ dσ π + x Ra+r log k π r + π r l+ a Ra r rdα x a δ log σ iπ k log σ + iπ k σ l+ x σ dσ + log + r πk a Ra δ 0 /r l+3 x a log log x k k,l + x a log log x k a Ra δ log x l+3 x l+3 ɛ k,l.7 a Ra δ log x l+3 For I, uing change of variable a log x = w, by Lemma 3, I = w l log w log log x k xa e w πi log x l+ a dw H0,δ log x 9

10 = x a alog x l+ k log log x k πi k + x a alog x l+ = k x a alog x l+ + j= k j= x a alog x l+ k j πi H0,δ log x H0,δ log x w l e w dw log log x k j w l log w j e w dw k log log x k j j Γ j l k k E j, l δ log x log log x k j..8 j j= By Lemma 3, E j, l δ log x t l log t + π j π δ log x e t Hence, we get Ha,δ x a alog x l+ k,l k j= x Ra a log x l+ dt j e δ log x δ log x k E j, l δ log x log log x k j j t l log t j e t/ dt j,l x δ. k x δ log log x k j x a δ/ k,l..9 a j= For I, we have a l log k a a x = πi a a πi Ha,δ a l+ log k a x d..0 a Then, we replace l by l + in the formula of I and multiply by. Thu, we get a I = k x a a log x l+ k j= k log log x k j j Γ j l + O k,l x a δ/ a.. Combining.7,.8,., and.9, we get the concluion of thi lemma.. Proof of Theorem and By. and., we have F ω, χ = L, χ log L, χ L, χ log L, χ + L, χg,. and F Ω, χ = L, χ log L, χ + L, χ log L, χ + L, χg,.3 0

11 where G and G are abolutely convergent for R σ 0 = On Hρ, δ, by integration from 3 in Lemma 8 ee Section.3 below, we have with We write where H ρ = log L, χ = log ρ + H ρ,.4 0< γ γ log ρ + Olog γ..5 L, χ = L ρ, χ ρ + M, ρ ρ,.6 M, ρ := 0 tl ρ + t ρ, χdt..7 Under the ERH, we have M, ρ γ δ+ɛ on Hρ, δ. Now we define a function T x a follow, for T n T atifying [ e n] T n [ e n] +, let T x = T n for e 5 4 n x e 5 4 n+. In particular, we have Thu, by Lemma, for T = T x, ψ ω x, χ = πi Hρ,δ + O x δ 0, and ψ Ω x, χ = + O πi x δ 0. Hρ,δ x 3 T x x 5 6 x e 5 4. L, χ log L, χ x L, χ log L, χ x On each truncated Hankel contour Hρ, δ, d aχ d + aχ πi πi H,δ L, χ log L, χ x.8 H,δ L, χ log L, χ x.9 L, χ log L, χ = L ρ, χ ρ log ρ + M, ρ ρ log ρ + L, χh ρ..0 Since = ρ i not ingularity of L, χh ρ, and by Lemma 5, πi = Hρ,δ πi L, χ log L, χ x Hρ,δ L ρ, χ ρ log ρ + M, ρ ρ log ρ x

12 x = log x L ρ, χx iγ + iγ + O x log 3 + x M, ρ ρ log ρ x πi Hρ,δ.. For the econd integral in.8, if χ i real, = i a pole of L, χ. On the truncated Hankel contour H, δ, we have log L, χ = log + H B,. where H B = O on H, δ. And we write L, χ = L, χ + L, χ + M B,,.3 where, M B, := By.,.3, and Lemma 5, we get 0 tl + t, χ dt = O, on H, δ..4 L, χ log L, χ πi H x,δ = L πi H,δ, χ + L, χ log x M B, log πi H,δ x x x x =L, χ log x + 4L, χ L, χ log x + O log 3 x, log πi x d..5 H,δ M B Then, by Lemma 4, ince r 0 = x, πi M B H,δ δ M B r 0 π r0 0 + x log 3 x + log x x 3, log x σ, σ log σ iπ log σ + iπ x/ σ / σ dσ log + π r 0 dα r 0 x log 3 x..6

13 By.5 and.6, we deduce that L, χ log L, χ πi H x,δ x =L, χ log x + 4L, χ L, χ Thu, by.8,.9,., and.7, we have { x ψ ω x, χ = aχ L, χ log x + L, χ L, χ x L ρ, χx iγ + log x + iγ + πi Hρ,δ + O x log 3 x x x log x + O log 3..7 x } x log x M, ρ ρ log ρ x,.8 and { x ψ Ω x, χ =aχ L, χ log x + L, χ L, χ x L ρ, χx iγ + log x + iγ + πi Hρ,δ x + O log 3 x In the following, for T = T x, define Σx, χ := log x x For fixed large T 0, let Sx, T 0 ; χ := } x log x M, ρ ρ log ρ x..9 x Hρ,δ M, ρ ρ log ρ x d..30 L ρ, χx iγ + iγ L ρ, χxiγ + iγ..3 0 Then we have the following reult. We give their proof in later ubection. Lemma 6. The following lemma i imilar to Lemma in [8]. Lemma 7. For any ɛ > 0, Σe y, χ dy = O..3 Se y, T 0 ; χ dy 3 Y T ɛ 0 + log Y T ɛ 0 + T ɛ 0..33

14 By.8,.9, and Lemma 6 and 7, Theorem and follow. Remark. The reult of Lemma 6 eem good. Actually, we have X Σx, χ dx = ox..34 However, Lemma 7 prevent u from proving a better reult for our theorem..3 Proof of Lemma 6 and 7 Firt, we need the following lemma on the zero of Dirichlet L-function. Let γ be the imaginary part of a zero of L, χ in the critical trip. Lemma 8 [], Lemma.. Let χ be a Dirichlet character modulo q. Let NT, χ denote the number of zero of L, χ with 0 < R < and I < T. Then NT, χ = OT logqt for T. NT, χ NT, χ = OlogqT for T. 3 Uniformly for = σ + it and σ, L, χ L, χ = γ t < Similar to Lemma.4 in [], we have Lemma 9. Aume L, χ 0. For A 0 and 0 δ <, γ, γ A γ γ + Olog q t ρ γ δ γ δ γ γ loga + A + δ. Proof of Lemma 9. We only need to etimate the cae γ γ. By Lemma 8 and partial ummation, the um in the lemma i γ A γ A γ δ γ γ < γ log γ + γ δ γ δ We get the concluion of thi lemma. γ + δ γ δ γ γ γ γ γ γ γ loga A + δ In the following, we ue the above lemma to prove Lemma 6 and 7. Proof of Lemma 6. We rewrite Σx, χ a log x x iγ M, ρ ρ log ρ x ρ d..37 Hρ,δ 4

15 Let Γ ρ repreent the circle in the Hankel contour Hρ, δ. Denote Ex; ρ := M, ρ ρ log ρ x ρ d Hρ,δ δ = M/ σ + iγ, ρσ log σ iπ log σ + iπ r ρ + M, ρ ρ Γρ log ρ x ρ d x σ / σ + iγ dσ =:E h x; ρ + E r x; ρ..38 Since we aume r ρ, and T x x 5 6, x x iγ E r x; ρ E r x; ρ Next, we take care of the following um x iγ E h x; ρ = + γ γ γ, γ T By.38 and uing change of variable, Hence, For Σ x, γ δ ɛ x r ρ log + π r ρ γ γ > γ, γ T xiγ γ E h x; ρ E h x; ρ..39 x ɛ =: Σ x + Σ x..40 E h x; ρ γ δ ɛ y 4 Σ e y dy Σ x = γ γ > γ, γ T For e 5 4 l x e 5 4 l+, T = T x = T l Jx := γ γ > γ, γ T l δ δ x iγ γ r ρ δ 0 σ x σ dσ γ δ ɛ log 3 x y γ.4 dy = O..4 γ δ ɛ x iγ γ E h x; ρ E h x; ρ..43 i a contant, o we define dσ dσ R ρ σ ; xr ρ σ ; x r ρ iγ γ σ + σ,.44 where R ρ σ; x = M/ σ + iγ, ρσ log σ iπ log σ + iπ e 5 4 l+ e 5 4 l γ γ > γ, γ T l x σ / σ+iγ. Thu, x iγ γ E h x; ρ E h x; ρ dx x = Je 5 4 l+ Je 5 4 l..45 5

16 By Lemma 9, Jx γ γ > γ δ ɛ γ δ ɛ γ γ log 6 x log 6 x..46 Hence, for any poitive integer l, Therefore, 5 4 l+ Σ e y dy l 5/4 4 l y 4 Σ e y dy l log Y log5/4 + 4l l l Σ e y dy l log Y log5/4 + 5/4 l = O..48 Combining.39,.4, and.48, we get the deired reult. Proof of Lemma 7. For fixed T 0, let X 0 be the larget x uch that T = T x T 0. Since x 3 T x x 5 6, log X 0 log T 0. Under the ERH, L ρ, χ γ ɛ, by Lemma 8 and 9, T ɛ 0 + T ɛ 0 + Se y, T 0 ; χ dy Y T ɛ 0 log log X 0 log5/4 l log Y log5/4 + log log X 0 log5/4 l log Y log5/4 + + log Y T ɛ 0 + T ɛ 0. log X0 5 4 l+ 5 4 l γ ɛ 0 l 5 4 Thi complete the proof of thi lemma. dy + γ γ T 0 γ, γ T l γ T 0 γ ɛ + + log X 0 T 0 e y γ γ > T 0 γ, γ T l γ, γ T 0 3 General Theorem in the Selberg cla L ρ, χe iγy + iγ dy L ρ, χl ρ, χe iγ γ y + iγ iγ dy γ ɛ γ ɛ γ γ Selberg cla S conit a cla of Dirichlet erie introduced by Selberg []. A Dirichlet erie an F = σ >, 3. n n= i in S provided it atifie the following hypothee. Ramanujan Hypothei: an n ɛ for any ɛ > 0; 6

17 Analyticity: m F i an entire function of finite order for ome nonnegative integer m; 3Functional Equation: there exit poitive real number Q, λ,, λ r, and complex number w, µ,, µ r with Rµ j 0 and w = uch that where Λ F = wλ F, Λ F = F Q r Γλ j + µ j ; 4 Euler Product: for prime power p k there exit complex number bp k atifying log F = bp m p m p m= and bp m p mθ for ome θ <. Let d F be the degree of F S defined by d F = j= r λ j. We aume the correponding Hypothee, Grand Riemann Hypothei GRH: if F S, then F 0 for σ > ; and the Lindelöf Hypothei LH S: ɛ > 0, F t + ɛ for σ. If the Dirichlet erie fn n= can be written a linear combination of the form n log k F for F S and F ha no pole, then we can prove the exitence of a limiting ditribution related to the ummatory function j= S f x := n x fn. 3. It uffice to conider Suppoe let πi a+i n= a i log k F x d. 3.3 n n = log k F, 3.4 Sx := n x n. 3.5 We have the following theorem. Theorem 3. Under the aumption GRH and LH S, Sx = k k x log log x k mk γxiγ log x + iγ + Σx, T 0,

18 where lim up Y Y Σe y, T 0 dy log T 0 k+ T Sketch of the proof of Theorem 3 We have correponding reult for Selberg cla. Lemma 0 [4],.4, or [9], 4. Let F be in Selberg cla. Let N F T denote the number of zero of F with 0 < R < and 0 < I < T. Then N F T = d F π T log T + c F T + Olog T, 3.8 where d F i the degree of F and c F i a poitive contant. Lemma [9], Lemma.. Let F be in the Selberg cla and denote the nontrivial zero of F by ρ = β + iγ. Then, for 5 σ 7, log F = log ρ + Olog t, 3.9 t γ where π < I log ρ π for any t not equal to an ordinate of a zero of F. By Lemma 0, for any integer n 0, there exit n T n n + uch that the ditance of σ + it n to the nearet zero i df for all < σ <. Then, by Lemma, for log n < σ <, log F σ + it n df log n log log n df log T n. 3.0 Denote T := {T n } n=0. Then, uing the ame type of integration contour a Figure., we have Lemma. Aume the GRH, LH S, and F 0. Let ρ be nontrivial zero of F. Then for T T, Sx = log k F x x log x πi Hρ,δ + O xlog T k k + T T By Lemma, on Hρ, δ, we have log k F x πi = + πi Hρ,δ Hρ,δ πi Hρ,δ mγ log ρ k x k j= + x δ log T k+. 3. k mγ log ρ k j H ρ j x d, 3. j 8

19 where mγ i the multiplicity of the zero ρ and mγ log γ by Lemma 0, and H ρ = mγ log ρ + Olog γ on Hρ, δ. 3.3 j= 0< γ γ For the firt integral in 3., by Lemma 5 and Lemma 0, we get mγ log ρ k x πi Hρ,δ k = k x k log log x k j m k γx iγ xlog log x k log x j Γ j 0 + iγ + O k log. x 3.4 For the econd integral in 3., with the ame T x a before, uing the ame proof a Lemma 4 in [8], we have the following lemma. Lemma 3. Let ρ be a zero of F. Aume the function g log γ c on Hρ, δ for ome contant c 0, and H ρ = mγ log ρ + O log γ on Hρ, δ < γ γ For any integer l, n 0, denote Ex; ρ := Then, for T = T x, we have y and Denote Hρ,δ e y S x := k k j= S x; T 0 := Similar to Lemma in [8], we have Lemma 4. and for fixed large T 0, mγ log ρ l H ρ n g x ρ d. e iγy Ee y ; ρ dy = o Y log Y l+n. k log log x k j j Γ j 0 x S e y, T 0 dy Y log T 0 k+ m k γx iγ + iγ 0 m k γx iγ, iγ mk γxiγ iγ S e y dy = oy log Y k, 3.8 T 0 + log Y log T 0 k+3 T 0 + log T 0 k

20 Combining 3. and 3.4 with Lemma 3 and 4, Theorem 3 follow. Acknowledgement. I would like to thank Profeor Greg Martin for propoing me thi project and kindly providing me related numerical data. Reference [] P. Erdő and M. Kac, The Gauian law of error in the theory of additive number theoretic function, Amer. J. Math [] K. Ford, J. Sneed, Chebyhev bia for product of two prime. Experiment. Math., Volume 9, Iue 4 00, [3] G. H. Hardy and S. Ramanujan. The normal number of prime factor of a number n. Quart. J. Math , [4] J. Kaczorowki and A. Perelli, On the tructure of the Selberg cla, VII: < d <, Ann. Math [5] A. A. Karatuba, Baic Analytic Number Theory, Springer-Verlag, 993. [6] Y. K. Lau, J. Wu. Sum of ome multiplicative function over a pecial et of integer. Acta Arithmetica [7] Xianchang Meng, The ditribution of k-free number and the derivative of the Riemann Zeta-function, Math. Proc. Cambridge Philo. Soc., 6 07, no., pp [8] Xianchang Meng, Chebyhev bia for product of k prime, 06, to appear in Algebra and Number Theory, arxiv: [9] L. Pańkowki, J. Steuding, Extreme value of L-function from the Selberg cla. Int. J. of Number Theory, Vol. 9, No [0] A. Rényi and P. Túran, On a theorem of Erdő-Kac, Acta. Arith [] A. Selberg, Old and new conjecture and reult about a cla of Dirichlet erie, in Proc. Amalfi Conf. Analytic Number Theory, Maiori, 989, ed. E. Bombieri et al. Univerità di Salerno, 99, pp [] E. C. Titchmarh, The Theory of the Riemann Zeta-function, nd ed. rev. by D. R. Heath-Brown, Clarendon Pre, Oxford 986 [3] D. Wolke. Über die zahlentheoretiche Funktion ωn. Acta Arith , Centre de Recherche Mathématique, Univerité de Montréal, Montréal, QC, H3C 3J7 Canada meng@crm.umontreal.ca 0