χ(p)p (p − 1)(p − χ(p))
×
1+O
1 d2
d2 |n, d2 >z
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· 1+
p|N
χ(p)p (p − 1)(p − χ(p))
1 log z5 + exp − log2 z . 5 log z
−1
258
N. M. TIMOFEEV, M. B. KHRIPUNOVA
Note that χ(p)p 1, (p − 1)(p − χ(p)) z
p|N , z
p|N , p>z
Moreover, we have
d1 ≤z5 n∈A6 (N ,δd1 )
d2 |n, z
N 1 1 R(N , δ). d2 δd1 d22 d1 ≤z5
d2 >z
Therefore, returning to relation (8), we obtain χ(p) X(N , g , δ) ≤ 1− |X1 (N , g , δ)| + R(N , δ), p
(9)
p|N , p>z
where X1 (N , g , δ) =
1
−1
χ(d)
d≤z5 ,
p(d)≤z
χδd (N )
χδd
1 ϕ(δd)
χδd (n)eitg(n) (1 − |t|)e−ita dt.
(10)
n
Let us recall that z5 = exp (log log L)3 , δ ≤ log3 L, L = log N , p(δd) ≤ log22 L = z . δd ≤ log3 L exp (log log L)3 ,
z = log22 L,
We split [−1 ; 1] into two sets U1 and U2 . For t ∈ U1 , there exists a λ(t) = λ , such that |λ| ≤ T = exp (log log L)4 , and a Dirichlet character χm , m = m(t) ≤ T , p(m) ≤ z , such that 1 (1 − Re χm (p)eitg(p) p−iλ ) (log log L)4 . p
(11)
p
We have the set U2 = [−1 ; 1] \ U1 . Applying Lemma 8, we obtain 1 1 itg(n) 4 χδd (n)e dt N z5 exp − (log log L) R(N , δ). ϕ(δd) χ 2 U2 d≤z5
δd
n
Thus, the problem reduces to estimating the integral over U1 . Denote by l = l(t) the modulus of the primitive character generating the Dirichlet character χm occurring in inequality (11). Obviously, χ∗l satisfies (11) as well. Applying Lemmas 9 and 8, we find that all the characters χδd not generated by χ∗l satisfy the inequality itg(n) χ e N L−1/4 . δd n
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THE CONCENTRATION FUNCTION OF ADDITIVE FUNCTIONS
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Therefore, only the characters generated by χ∗l can be retained in the sum over the characters χδd on the right-hand side of (10). Let l = l1 l2 , where δ = l1 δ1 , (δ1 , l2 ) = 1 . In that case, if (n, δd1 ) = 1 , then χδl2 d1 (n) = χδl2 (n) . Thus, we have χ(δl2 d1 ) itg(n) X1 (N , g , δ) = e χδl2 (N ) χδl2 (n) (1 − |t|)e−ita dt ϕ(δl d ) 2 1 U1 d1 ≤z5 /l2 , p(l2 d1 )
n
+ O(R(N , δ)). For t ∈ U1 such that l2 ≥ log3 L , the modulus of the integrand is
N (log L)−1 (log log L)3 R(N , δ) ; ϕ(δ)
therefore, we can assume that l2 ≤ log3 L . Using Lemma 2 and noting the fact that (n, N ) = 1 and (δl2 , nN ) = 1 , we have 1 χ(p)p χ(d1 ) p(p − 1) = 1+ 2 ϕ(δl2 d1 ) ϕ(δl2 ) (p − 1)(p − χ(p)) p − p + χ(p) p≤z
z
d1 ≤ l 5 , 2
p|δl2
p(d1 )
1− × p|nN , p≤z
χ(p)p p2 − p + χ(p)
1 . +O ϕ(δl2 ) log2 L
Returning to relations (9) and (10), we obtain χ(p) |X2 (N , g , δ)| + R(N , δ), 1− X(N , g , δ) p
(12)
p|N
where X2 (N , g , δ) =
U1
1 p(p − 1) , χδl2 (N ) 2 ϕ(δl2 ) p − p + χ(p) p|δl2
χδl2 (n)eitg(n) F (n)(1 − |t|)e−ita dt, (13)
n
s(p) , 1+ √ F (n) = 3 p p|n, p≤z
We have χδl2 (n)F (n)eitg(n) =
d≤log9 L, p(d)≤z
n
√ p 3 p χ(p) . s(p) = − 2 p − p + χ(p)
s(d) χδl2 (d)eitg(d) √ 3 d
χδl2 (n)eitgd (n) + O(log−2 L),
n
where s(d) = p|d s(p) and gd (n) = g(nd) − g(d) is an additive function of n . It follows from inequality (11) that 1 (1 − Re χδl2 (p)eitgd (p) p−iλ ) ≤ (log log L)4 + O(log log L). p
p
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(14)
260
N. M. TIMOFEEV, M. B. KHRIPUNOVA
Applying Abel summation, we obtain eitgd (n) χδl2 (n) = n
eitgd (n) χδl2 (n)n−iλ eiλ log N
n
N/d
− iλ 1
eitgd (n) χδl2 (n)n−iλ eiλ log u
n
du . u
Since |λ| ≤ exp (log log L)4 , it follows from (14) that 1 |1 − χδl2 (p)eitgd (p) p−iλ | ≤ 2 log L (log log L)2 . p
p
Combining this with Lemma 10, we obtain ϕ(N ) N eitgd (n) χδl2 (n) − eitgd (n) χδl2 (n) L−1/5 . N d n
n
Therefore, relations (12) and (13) become χ(p) + 1 |X3 (N , g , δ)| + R(N , δ), 1− X(N , g , δ) p
(15)
p|N
where
X3 (N , g , δ) = U1
1 χδl2 (N )Π(δl2 ) χδl2 (n)eitg(n) F (n)(1 − |t|)e−ita dt. ϕ(δl2 )
(16)
n
Here Π(m) =
p|m
p2
p(p − 1) . − p + χ(p)
To prove Theorem 1, it now suffices to show that X3 (N , g , δ)
N . ϕ(δ) log L
Recalling the definition of F (n) , we can write 1 |X3 (N , g , δ)| ϕ(δ) where
N 1 N √ , gd , δ + , X4 3 d ϕ(δ) log L d
d≤log3 L
X4 N , gd , δ ≤ d
itgd (n) e χδl2 (n) dt.
U1 n
The subsequent arguments almost totally repeat the concluding part of the proof of Theorem 1 in [4]. The presence of the sum over d ≤ log3 L involves an additional difficulty. Consider the case d = 1 . Applying Lemma 8, we obtain 1 1 1 N itg(p) −iλ X4 (N , g , δ) N (1 − Re e , exp − min p χδ (p)) dt + χ ,λ 2 ∆ p log L −1 p
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261
where the minimum is taken over |λ| ≤ log2 L and χ∆ , ∆ ≤ log6 L . Recall that δ ≤ log3 L and l2 ≤ log3 L . We denote M (t) = min
χ∆ ,λ
1 (1 − Re eitg(p) p−iλ χ∆ (p)), p
p
Ek = {t : t ∈ [−1 ; 1], M (t) ≤ k}, Hence X4 (N , g , δ) N
1≤k≤K
k = 1, 2, . . . ,
K = 4[log log L].
k m(Ek ) + N (log L)−1 . exp − 2
Suppose that for all 1 ≤ k ≤ K the following inequality holds: 200k . m(Ek ) ≤ √ log L Then we obtain X4 (N , g , δ) N
k≥1
k k log−1/2 L N log−1/2 L. exp − 2
The additive functions gd (n) and g(n) differ only by p | d ; therefore, applying Lemma 8, we find in this case that for all d ≤ log3 L the following inequality holds: 1 N N 1 1 N itgd (p) −iλ , gd , δ (1 − Re e exp − min p χ∆ (p)) dt + X4 δ d −1 2 χ∆ ,λ p d log L p
Therefore, X3 (N , g , δ)
1 ϕ(δ)
N N τ (d) . log−1/2 L 2 d ϕ(δ) W (N ) d≤log3 L
Suppose that there exists a k , 1 ≤ k ≤ K , for which 200k . m(Ek ) > √ log L The set Ek is symmetric with respect to zero and contains zero; therefore, (see, for example, Lemma 5.3 [1]) each number t ∈ [−1 ; 1] can be expressed as t1 + · · · + tr , where tj ∈ Ek and r = [12/m(Ek )] . Recall that if tj ∈ Ek , then there exists a λi , |λj | ≤ log2 L , and χ∆j , ∆j ≤ log6 L , such that the following inequality is valid: 1 (1 − Re eitj g(p) χ∆j (p)p−iλj ) ≤ k, p
j = 1, . . . , r.
p
Suppose that zj , j = 1, . . . , r , are complex numbers, with |zj | ≤ 1 . Then (see, for example, [4]) by induction we can prove the following inequality: 1 − Re z1 . . . zr ≤ r
2
r j=1
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(1 − Re zj ).
(17)
262
N. M. TIMOFEEV, M. B. KHRIPUNOVA
Let t = t1 + · · · + tr , λ = λ1 + · · · + λr , and χ∆ = χd1 . . . χdr . It follows from the previous two inequalities that for any t ∈ [−1 ; 1] there exists a λ = λ(t) and a χ∆ such that 1 log L log L (1 − Re eitg(p) χ∆ (p)p−iλ ) ≤ r2 k ≤ 132 . 2 p 200 k 102
(18)
p
Note that |λ| ≤ r log2 L ≤ log3 L,
∆(t) ≤ log6r L ≤ log6
√
log L
L,
p(∆) ≤ log6 L.
Let us show that χ∆(t) is the principal character. Consider any t1 , t2 ∈ [−1 ; 1] for which we have t1 + t2 ∈ [−1 ; 1] . We prove that the character χm = χ∆(t1 +t2 ) χ∆(t1 ) χ∆(t1 ) is the principal character. Using inequalities (18) and (17), we obtain 1 log L log L (1 − Re χm (p)p−i(λ(t1 +t2 )−λ(t1 )−λ(t2 )) ) ≤ 9 . p 102 10
p
Let σN = 1 + 1/ log N . Then the left-hand side of the inequality is equal to log ζ(σN ) − log |L(σN + i(λ(t1 + t2 ) − λ(t1 ) − λ(t2 )), χm )| + O(1). If χm is not the principal character, then it is equal to (see, for example, [7]) log L + O(1) + O(log |∆|(|λ(t1 + t2 ) − λ(t1 ) − λ(t2 )| + 2)) = log L + O
log L log log L .
Hence we obtain a contradiction. Thus, χd(t1 +t2 ) χd(t1 ) χd(t2 ) is the principal character. Let H=
s p
,
where
s≤
log L .
p≤log6 L
Then the moduli of all the characters discussed above will divide H . By χ ∆ we denote the character modulo H generated by χ∆ . Then, for all t1 , t2 ∈ [−1 ; 1] , where t1 + t2 ∈ [−1 ; 1] , we ∆(t1 ) · χ ∆(t2 ) , and hence χ ∆(t) = ( χ∆(t/h) )h = χ 0 , where h = ϕ(H) . Thus, we have χ ∆(t1 +t2 ) = χ have proved that χ ∆(t) is the principal character. Since ∆(t) | H and 1 p|H
p
≤ 12 log log log L,
using inequality (18), we find that for any t ∈ [−1 ; 1] there exists a λ = λ(t) , |λ| ≤ log3 L , such that 1 log L (1 − Re eitg(p) p−iλ ) ≤ . p 100 p
It follows from Lemma 9 that the primitive character χ∗l(t) generating the characters χδd is the principal one, i.e., l = l1 · l2 = 1 , and relation (15) takes the form X3 (N , g , δ) =
1 χ (N )Π(δ) ϕ(δ) δ
1
−1 n
eitg(n) F (n)(1 − |t|)e−ita dt + O(R(N , δ)).
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THE CONCENTRATION FUNCTION OF ADDITIVE FUNCTIONS
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Therefore, we obtain 2 N s(d) sin(gd (n) − a)/2 √ + 3 ϕ(d) log3 L d n
1 X3 (N , g , δ) ϕ(δ)
Noting that gd (p) = g(n) only for p | d and using Lemma 11, we can write X3 (N , g , δ)
1 ϕ(δ)
d≤log9 L
1 s(d) N N √ . · 3 dd W (N ) p|d p ϕ(δ) W (N )
As was noted earlier, this estimate yields the assertion of Theorem 1. ACKNOWLEDGMENTS This research was supported by the Russian Foundation for Basic Research under grant no. 0201-00368. REFERENCES 1. I. Ruzsa, “On the concentration of additive functions,” Acta Math. Acad. Sci. Hungar., 36 (1980), 215–232. 2. P. D. T. A. Elliott, “The concentration function of additive functions on shifted primes,” Acta Math., 173 (1994), 1–35. 3. N. M. Timofeev, “The Erd¨ os–Kubilus conjecture on the distribution of values of of additive functions on sequences of shifted primes,” Acta Arithmetica, 58 (1991), no. 2, 113–131. 4. N. M. Timofeev and M. B. Khripunova, “The concentration function of additive functions with nonmultiplicative weight,” Mat. Zametki [Math. Notes], 75 (2004), no. 6, 877–894. 5. N. M. Timofeev, “The Hardy–Littlewood problem for numbers with given number of simple divisors” Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 59 (1995), no. 6, 181–206. 6. G. Tenenbaum, Introduction a ` la th´eorie analytique et probabiliste des nombres, Institut Elie Cartan, 13. Universit´e de Nancy I, 1990. 7. A. A. Karatsuba, Foundations of the Analytic Theory of Numbers [in Russian], Second edition, Nauka, Moscow, 1983. 8. H. Halberstam and H. E. Richert, Sieve Methods, Academic Press, London–New York, 1974. 9. R. Hall and G. Tenenbaum, 90 Divisors, Cambridge University Press, Cambridge, 1988. 10. C. Hooley, Applications of Sieve Methods to the Theory of Numbers, Cambridge Tracts in Mathematics, no. 70, Cambridge University Press, Cambridge–New York–Melbourne, 1976. Vladimir State Pedagogical University E-mail : hripunova@vgpu.vladimir.ru
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