MEAN VALUE ESTIMATES OF z Ω(n) WHEN z 2.

Transcription

1 MEAN VALUE ESTIMATES OF z Ωn WHEN z 2 KIM, SUNGJIN 1 Introdction Let n i m pei i be the prime factorization of n We denote Ωn by i m e i Then, for any fixed complex nmber z, we obtain a completely mltiplicative fnction z Ωn as a fnction of n There are many reslts on the average order of this fnction z Ωn There are remarkable differences in the behavior of z Ωn for different vales of z If we consider main terms only, it is Cx z 1 when z < 2, Cx 2 when z 2, and we have oscillation when z > 2 First two cases are well-known, so in this paper, we will prove pper, lower bond reslts, and oscillatory behavior of z Ωn when z > 2 Frther, we will also prove oscillatory behavior for all z sch that z > 2, and z is not positive real However, we will only prove pper bond reslt for the case z 2, and z 2 We briefly state the known reslts For z < 2, SelbergSee [4], a special case of Theorem 2 proved that: 1, where z Ωn x z 1 f1, z Γz fs, z p 1 + O 1 zp s p s z This has been improved de to Selberg-Delange methodsee [1], Theorem 2, p202: for all δ, 0 < δ < 1, there exist positive constants c 1 c 1 δ, c 2 c 2 δ, sch that, niformly for x 3, N 0, z 2 δ, 2 N z Ωn x z 1 ν k z k + O δr N x k0, where R N x e c1 c N+1, + 2N+1 and νk are fnctions depending only on z When z 2, BatemanSee [2], 3 obtained a reslt: 3 2 Ωn C 0 x 2 + C 1 x + Ox, where C i are constant, and the error term Ox is the best possible Now, we proceed on or reslts Let z 2, and let p 1 2 < < p r z < p r+1 < be prime nmbers We define fnctions A and B which we will se 1

2 2 KIM, SUNGJIN throghot this paper: 4 As 5 Bs p z p z 1 1 p s z p> z Also, we denote d z n by the identity: 6 ζs z p 1 z p s 1 n1 a n n s, 1 zp s p s z 1 p s z d z nn s The Dirichlet series for As, Bs, and ζs z are absoltely convergent respectively on σ >, σ > log p r+1, and σ > 1 Clearly, we have log p r+1 < 1, since 2 z < p r+1 Then, the Dirichlet series F s n1 zωn n s satisfies the identity: n1 7 F s AsBsζs z, where the series is absolte convergent on σ > log z+2kπi n1 b n n s In case of z > 2, we see for all integers k Ths, sing Perron s that As has singlarities on s formlasee [3], Lemma 312, p60 directly on z Ωn is difficlt becase of the resides from too many singlarities Indeed, we derive Theorem 1,2, and Theorem 3 withot sing Perron s formla The first reslt is the pper and lower bond Theorem 1 Let z > 2 be fixed, and x 1 Then there exists a constant B z sch that: 1 8 z x log z z Ωn B z x log z From Theorem 1, we can also derive the oscillatory behavior of z Ωn Theorem 2 Let z > 2 be fixed Then, 9 lim sp x log z z Ωn lim inf log z x x x z Ωn 1 On the other hand, we can extend z to non-real vales Theorem 3 Let z > 2, and z is not a positive real nmber For x 1, we have: 10 Re z Ωn Ω ± x In the remaining case, we have an pper bond Theorem 4 Let z 2, and z 2 For x 3, we have: 11 z Ωn x O log 2 Rez

3 MEAN VALUE ESTIMATES OF z Ωn WHEN z Proof of Theorem 2 Now, we prove Theorem 2 from Theorem 1 Proof of Theorem 2 Theorem 1 implies that, βx x log z fnction of x Consider a bonded seqence {S N } N1, 12 S N β2 N 1 2 N 1 log z n 2 N 1 z Ωn is a bonded z Ωn We can find a sbseqence {S Ni } i1 which converges to K z Then, we have β2 Ni 2 Ni log z z Ωn + z Ni 2 Ni n 2 N i 1 2 N i 1 log z S Ni + 1 Since β2 Ni K z + 1 as i, the difference between lim sp x βx, and lim inf x βx is at least 1 Hence, Theorem 2 is proved 3 Proof of Theorem 1 A simple observation gives the lower bond, 13 z Ωn z e z / z 1 x log z/ 2 e x We remark that a n p e 1 Lemma 1 For x 1, 14 Proof We se indction on r 1 p er r x ze1+ +er, and derive the following lemma a n O x log z +1 1 z 1 z z 1 x log z When r 1, note that 2 e x ze z Let r > 1, and assme the reslt for r 1, namely, e 15 z 1+ +e r 1 C z,r 1 x log z x Then, we have p e 1 1 p er r x p e 1 1 p e r 1 r 1 z e1+ +er p er r x p er r x z er C z,r x log z p e 1 1 p e r 1 r 1 z er C z,r 1 x p er r xp er r log z z e1+ +er 1, where C z,r C z,r 1 e z 1 log p r/ e Cz,r 1 1 z 1 log p r/ 1 This gives the reslt for r, and completes the proof of Lemma 1 Frther, we can write

4 4 KIM, SUNGJIN down Lemma 1 in the form: 16, where 17 C z z z 1 a n C z x log z 2<p z log p z Now, we are ready to prove Theorem 1 Proof of Theorem 1 pper bond By 7, we have zωn vw x b d z va w Then by Lemma 1, b d z va w b d z v vw x v x w x v a w x b d z vc z v v x C z b log z d z v v v log z log z x log z The -sm is convergent, since the Dirichlet series for Bs is absoltely convergent for σ > log z log p r+1 Also, the v-sm is jst ζ log z z Hence, we can write down Theorem 1 in the form: 18 z Ωn B z x log z, where 19 B z C z b log z ζ z log z 4 Proof of Theorem 3 We begin with an oscillation lemma For the proof, see [1], Theorem 8, p112 Lemma 2 Let Gs n1 a nn s be a Dirichlet series with real coefficients having a finite abscissa of convergence Sppose there exists a real nmber σ 0 > 0 sch that Gs has an analytic contination which is reglar at all points of the half line [σ 0, and has a pole on the vertical line σ σ 0 Then the associated smmatory fnction satisfies 20 a n Ω ± x σ0 Proof of Theorem 3 Note that z > 2 and z is not positive real Let F s AsBsζs z F s+f s as before Then Gs 2 has a Dirichlet series n1 RezΩn n s Let z z e iθ, then F has singlarities on the set; { } + i2πk ± θ 21 : k Z

5 Since this set does not contain for the Lemma 2 with σ 0 22 MEAN VALUE ESTIMATES OF z Ωn WHEN z 2 5, the Dirichlet series Gs satisfies all hypotheses Hence, by Lemma 2, Rez Ωn Ω ± x 5 Proof of Theorem 4 Let As, Bs be defined by: 23 As 1 z 1 a n 2 s n s, 24 Bs s z p>2 n1 1 zp s p s z Then we have F s n1 zωn n s AsBsζs z as before The Selberg- Delange methodsee [1], Theorem 5, p191 implies that 25 v x Ths, it follows that z Ωn vw x x/2 x/2 n1 b d z v B1 Γz xz 1 + O x Rez 2 a b v d z w a vw x/ B1 x a Γz b v d z w + We treat the second error term first, a Rez 2 x/2 O x/2 a x/2< x a vw x/ z 1 + O x log 2 Rez b v d z w b n n s Rez 2 + Rez 2 + Ox x/2 < x/2 + Olog O a Rez 2 log 2 Rez Using partial smmation, we find that the first term is small compared to the error term above Let St a t, and note that St is bonded fnction of t x/2 a z 1 x/2 1 t z 1 dst z 1 x/2 x/2 St 1 t + 1 log 1 t Stz 1 x t x/2 1 O1 + O Rez 2 dt 1 t t { O1 + O Rez 1 if Rez 1 O1 + Olog if Rez 1 z 2 dt

6 6 KIM, SUNGJIN Since Rez 1 < 1, we obtain the reslt: 26 z Ωn O x log 2 Rez 6 Frther Remarks In fact, the pper bond in Theorem 4 can be improved to O N xlog Rez N sing a better error term in 25See [1], Theorem 5, p191 There are still some open problems In the Theorem 2, we obtained oscillatory behavior of the fnction βx x log z z Ωn However, we do not know how to obtain lim sp x β z x, and lim inf x β z x explicitly as a fnction of z Also, in the Theorem 4, we only have pper bond reslt, and still do not know what the best possible bond is In case of Theorem 4, the fnction F s n1 zωn n s does not satisfy the hypothesis of Lemma 2, bt the athor conjectres that Re z Ωn Ω ± x holds References 1 G Tenenbam, Introdction to Analytic and Probabilistic Nmber Theory, Cambridge Stdies in Advanced Mathematics 2 P T Bateman, Proof of a conjectre of Grosswald, Dke Math J , E C Titchmarsh, The Theory of the Riemann Zeta-fnction, Second Edition, Oxford 4 A Selberg, Note on a paper by L G Sathe, J Indian Math Soc B , 83 87