ACTA ARITHMETICA 126.2 (2007)

On the DDT theorem

by G. Bareikis and E. Manstaviciusˇ (Vilnius)

Dedicated to our teacher Professor Jonas Kubilius on his 85th birthday

1. Introduction. We denote the sets of natural, real, complex, and prime numbers by N, R, C, and P respectively. In what follows we assume that p P, d,l,m,n N, and s = σ + iτ C. The letters c and C with or∈ without subscripts∈ denote constants. Either∈ of the notation f = O(g) or f g will mean that f C g for some positive constant C, which may≪ be absolute or depend| | upon ≤ various| | parameters. In such cases we sometimes indicate that by a subscript. Throughout the paper x 3 will be the sequence parameter and in the asymptotic relations it is assumed≥ that x . Let→∞τ(m, v) be the number of natural divisors of m which do not exceed v m, and τ(m) = τ(m,m). In 1979 J.-M. Deshouillers, F. Dress, and G.≤ Tenenbaum [2] (see also [10, Section II.6.2]) obtained the following result. Theorem DDT. Uniformly in u [0, 1], ∈ 1 τ(m,mu) 2 1 (1) = arcsin √u + O . x τ(m) π √log x mX≤x This asymptotic formula was announced already in [3], where some argu- ment relating it to the well known arcsine law in the probabilistic invariance principle was presented. In our opinion, that argument was not very con- vincing. To support that, we observe that the arcsine law in the classical probability theory is in some sense universal while (1) fails to hold if one replaces τ(m) by another multiplicative number-theoretic function whose values on prime numbers differ from two fairly often. This raises a desire to search for another limiting relation generalizing (1). Such an attempt was made by the second author who in [6, Section 8] announced a result

2000 Mathematics Subject Classification: 11K65, 11N60. Key words and phrases: multiplicative function, mean value, natural divisor, beta distribution.

[155] 156 G. Bareikis and E. Manstaviˇcius showing that, instead of the arcsine function, a new limit belonging to the class of beta distributions can appear. This phenomenon was also reported by the first author in the semigroup of polynomials over a finite field [1]. In the present note, we generalize the DDT theorem and slightly sharpen the remainder term. Relation (1) arouses interest from the point of view of arithmetically de- fined random processes. First, observe that (1) will remain true if τ(m,mu) is replaced by τ(m,xu). So, the limit in (1) can be understood as the asymp- totic expectation of the sequence of stochastic processes if a number m x ≤ is taken with probability νx( m )=1/[x]. Later results obtained in [9] dealt with the asymptotic distribution{ } of increments (τ(m,mv) τ(m,mu))/τ(m), − if 0 u v 1. Having all this in mind, the second author [5] asked whether≤ the≤ distributions≤ of the processes τ(m,xu)/τ(m) under the measure νx weakly converge in the Skorokhod space D[0, 1]. This hypothesis together with some generalization was proved in [7]. To formulate the result, we recall some definition. The space D[0, 1] consists of the real-valued functions on [0, 1] which are right-continuous and have left-hand limits. It is assumed that the Skorokhod metric is introduced in it. Write for the Borel σ-algebra in D[0, 1]. For a multiplicative function f : N RD+, we define → T (m,xu) (2) T (m, v) = f(d), X (m, u) = , T (m) = T (m,m), x T (m) d|m,dX≤v where u, v [0, 1]. Let ν X−1 be the induced measure on . ∈ x ◦ x D Theorem MT. If f(p) = κ > 0 for each p P and f(pk) 0 for all p P and k 2, then ν X−1 weakly converges∈ to a limit measure≥ on . ∈ ≥ x ◦ x D In the Addendum [11], G. Tenenbaum generalized this theorem and showed, in addition, that the limit measure is concentrated on the sub- space of continuous functions. A simple example which does not satisfy the conditions of Theorem MT but is covered by the result of [11] is defined via the multiplicative function with f(p)=1+( 1)(p−1)/2 for all odd p P. What is the asymptotic mean value of this particular− process? In the present∈ paper, we give a more general answer.

2. Results. Let f : N R+ be a multiplicative function. Throughout the paper we assume that,→ for some constant C > 0, it satisfies f(pl) C for all p P and l N. We say that f belongs to the class (α), 0 < α≤ < 1, ∈ ∈ M DDT theorem 157 if the function defined by the series 1 1 α , σ > 1, 1 + f(p) − ps Xp for some c > 0 and 0 < δ < 1, has an analytic continuation P (s) into the region c σ σ(t):=1 , t R, ≥ − log( t + 2) ∈ | | where P (s) is holomorphic and P (s) (1 δ)log( t + 2). | | ≤ − | | The well known formula 1 = log ζ(s) + H(s), ps Xp where H(s) is analytic in σ > 1/2, gives an analytic continuation of the sum on the left-hand side into the region σ σ(t), where it is a holomorphic function, say, L(s). Moreover, L(s) loglog(2≥ + t ) there. Thus, if f (α), then, for β := 1 α, the function≪ | | ∈ M − f(p) 1 β 1 + f(p) − ps Xp again has the same properties. This observation will be repeatedly exploited in what follows. Throughout the paper, the dependence of the appearing remainders on the constants α, δ, c and C involved in the definition of the class (α) is allowed. Using (2), we define M 1 T (m,mu) S (u) = , x x T (m) mX≤x and Sx(u, v) = Sx(v) Sx(u) where 0 u v 1. Recall that the distri- bution function of a beta− law has the following≤ ≤ expression:≤ u 1 \ dv B(u; a, b) = , 0 u 1, Γ (a)Γ (b) va(1 v)b ≤ ≤ 0 − where 0 < a = 1 b < 1 and Γ (z) denotes the Euler gamma-function. Set B(u, v; a, b) = B(v−; a, b) B(u; a, b) for 0 u v 1 and a b = min a, b . − ≤ ≤ ≤ ∧ { } Theorem 1. If f (α), then ∈ M 1 1 Sx(u) B(u; α,β) + − ≪ logα x logβ x uniformly in 0 u 1. ≤ ≤ This result follows from the next estimate which is sharper in the central zone. 158 G. Bareikis and E. Manstaviˇcius

Theorem 2. If f (α) and 0 u v 1, then ∈ M ≤ ≤ ≤ 1 1 Sx(u, v) B(u, v; α,β) 1 − ≪ logβ x ∧ (u log x)α  1 1 + 1 . logα x ∧ ((1 u)log x)β  − For the functions discussed in Theorem MT, one can take α = (1+κ)−1. Generalizing the above mentioned example we have the following result. + Corollary. Fix m N and let al R (not all zero) for 1 l

3. Auxiliary lemmas. As usual, let ζ(s) be the Riemann zeta function. First, we present two classical lemmas. Lemma 1 ([10, Section II.5.3]). Let z C, M > 0, 0 ̺ < 1, and F (s) := ∞ a /ns, a 0, be such that the∈ function G (s)≤ := F (s)ζ−z(s) n=1 n n ≥ z can be continuedP as a holomorphic function for σ 1 c1/log( t +2), t R, and in this domain satisfies the bound ≥ − | | ∈ G (s) M(1 + τ )̺. | z | ≤ | | Then, for A > 0 and z A, we have | | ≤ x Gz(1) M an = + O , log1−z x Γ (z) log x nX≤x where the implicit constant of the remainder depends at most on c1, ̺, and A. l Lemma 2 ([4]). Let g be a multiplicative function with 0 g(p ) C1 for all prime numbers p and l N. Then ≤ ≤ ∈ x g(p) g(m) exp , ≪C1 log x  p  mX≤x Xp≤x g(m) g(p) exp . m ≪C1  p  mX≤x Xp≤x DDT theorem 159

Let T (m) be as defined in Section 1. Then T (pk)=1+ f(p) + + f(pk). ··· For the multiplicative function h(m) = f(m) or h(m) = I(m) : 1, we define ≡ ∞ h(n) 1 a h(pi) (3) Q (d; h) = , F (a; h) = 1 x T (nd)  − p T (pi)pi nX≤x Yp Xi=0 where a > 0. Similarly, ∞ ∞ h(pi) h(pi) g(d; h) := g(pl; h), g(pl; h) = ,  T (pi+l)pi   T (pi)pi  pYlkd Xi=0 . Xi=0 g(d) := g(d; f), and g∗(d) := g(d; I). Further, set 1 C r(d) = 1 + 2 , 1 + f(p) pσ0  Yp|d where C2 > 0 and 0 <σ0 < 1 are arbitrary. Lemma 3. If f (α), then, for all d N, ∈ M ∈ 1 1 r(d) (4) Q (d; f) = F (β; f)g(d) + O , x x Γ (β)logα x log x 1 1 1 (5) g(d) = 1 + O . x β  log x Xd≤x Γ (α)F (β; f)log x Moreover, 1 1 (6) r(d) log−β x, f(d)r(d) log−α x. x ≪ x ≪ Xd≤x Xd≤x Proof. Introduce the generating series f(n) (7) φ(s; d, f) = T (nd)ns nX≥1 ∞ ∞ ∞ f(pi) f(pi) −1 f(pi) =  T (pi+k)pis  T (pi)pis  · T (pi)pis pYkkd Xi=0 Xi=0 Yp Xi=0 =: g(s; d, f)φ(s;1, f). Since f(pi)/T (pi+k) f(pi)/(1 + f(pi)) C/(1 + C) < 1 for each k 0 and i 1, the last expansion≤ is valid for σ≤ > 1. ≥ ≥ 160 G. Bareikis and E. Manstaviˇcius

We now examine the factors. First we observe that ∞ f(pi) C (8) v (s; f) := 1 + 1 | p | T (pi)pis ≥ − (1 + C)(pσ 1) Xi=1 − C 1/2 1 > 0 ≥ − 1 + C  for every p P provided that ∈ σ σ = max 3/4, log (1+(C/(1 + C))1/2) . ≥ 0 { 2 } Hence in the same region, if k 1, ≥ ∞ ∞ f(pi) f(pi) −1 g(s; pk, f) = | | T (pi+k)pis T (pi)pis Xi=0 Xi=0

−1 1 C 1 C 1 + + O 1 ≤ 1 + f(p) pσ0 p2σ0  − (1 + C)(pσ0 1) − 1 C 1 1 1 + C + + O . ≤ 1 + f(p)  1 + C pσ0 p2σ0  Consequently, g(s; d, f) = g(s; pk, f) | | | | pYkkd 1 C 1 1 + C + = r(d) ≪ 1 + f(p)  1 + C pσ0  Yp|d if σ σ . ≥ 0 Examine the function ∞ f(p) f(pk) 1 β G(s) := φ(s;1, f)ζ−β(s) = 1 + + 1  T (p)ps T (pk)pks  − ps  Yp Xk=2 f(p) 1 =: exp β H(s).  1 + f(p) − ps  Xp Here H(s) is some product over primes. In the routine way, taking loga- rithms, which is allowed by (8), we verify that the function H(s) is analytic and bounded for σ σ0. Dealing with the sum under the exponent, we ex- ploit the definition of≥ the class (α). Thus, G(s) has an analytic continua- tion into the region σ σ(t), whereM it is holomorphic and G(s) (2+ t )1−δ for some 0 < δ < 1. Together≥ with (7) and (8) this also implies≪ an analytic| | continuation of φ(s; d, f)ζ−β(s) and the estimate φ(s; d, f)ζ−β(s) = g(s; d, f)G(s) r(d)(2 + t )1−δ ≪ | | DDT theorem 161 for σ σ(t). Since g(1; d, f) = g(d), by Lemma 1 we obtain (4) with F (β; f)≥ = G(1). To derive the asymptotic formula (5), we will apply Lemma 1 for ∞ ∞ g(n) 1 α g(pl) G (s) = ζ(s)−α = 1 1 + . α ns  − ps   pls  nX=1 Yp Xl=1 By definition of g(pk), we obtain 1 1 u (s) G (s) = 1 + α + p , α  1 + f(p) − ps p2s  Yp where up(s) C3 uniformly in σ 3/4 and p P. Separating a finite number| of factors,| ≤ with p p , where≥p is sufficiently∈ large, we can ensure ≤ 0 0 that the remaining factors (for p > p0) do not vanish for σ > 1. So expanding their logarithms we obtain the representation 1 1 G (s) = exp α H (s), α  1 + f(p) − ps  1 Xp where H1(s) is an analytic and bounded function in σ 3/4. Again, since f (α), having an analytic expansion of the series under≥ the exponent and∈ Mthe appropriate estimate, we can apply Lemma 1 with M = 1. This yields x 1 g(d) = G (1)+O , β  α log x Xd≤x Γ (α) ln x where Gα(1) > 0. It remains to show that Gα(1)F (β; f) = 1, where F (β; f) has been de- fined in (3). If x(p) := vp(1; f), using α + β = 1, we obtain ∞ ∞ 1 1 1 f(pi) G (1)F (β; f) = 1 1 + x(p) α  − p x(p) pl T (pl+i)pi  Yp Xl=1 Xi=0 ∞ ∞ 1 1 = 1 x(p) + f(pi)  − p T (pj)pj  Yp Xi=0 j=Xi+1

∞ j−1 1 1 = 1 x(p) + f(pi)  − p pjT (pj)  Yp Xj=1 Xi=0 ∞ 1 1 = 1 x(p) + (x(p) 1) =1.  − p pj − −  Yp Xj=1 Thus, relation (5) is proved. 162 G. Bareikis and E. Manstaviˇcius

Since f (α) implies ∈ M r(p) 1 = + O(1) = α log log x + O(1), p p(1 + f(p)) Xp≤x Xp≤x the first estimate in (6) is a corollary of Lemma 2. Similarly, we obtain the second one. Lemma 3 is proved. In a similar manner we prove the next lemma. Lemma 4. If f (α), then, for all d N, ∈ M ∈ 1 1 ∗ r(d) (9) Qx(d; I) = F (α; I)g (d) + O . x Γ (α)logβ x log x Moreover, 1 1 1 (10) g∗(d)f(d) = 1 + O . x Γ (β)F (α; I)logα x log x dX≤x Proof. We now start with 1 φ(s; d,I) = T (nd)ns nX≥1 ∞ ∞ ∞ 1 1 −1 1 =  T (pi+k)pis  T (pi)pis  · T (pi)pis pYkkd Xi=0 Xi=0 Yp Xi=0 = g(s; d,I)φ(s;1,I). For the factor appearing in the denominator, ∞ 1 v (s; I)=1+ , p T (pi)pis Xi=1 as in (8), we have 1 v (s; I) 1 c > 0 | p | ≥ − 3σ 1 ≥ 2 − k for every p 3 if σ 1 c3 with sufficiently small c3 < 1. If f(2 ) 0 ≥ ≥ − −s −1 ≡ for k 1, then v2(s; I) = 1 2 c4 > 0 in the same region. If f(2k) =≥ c > 0 for| some k| 1,| then− | ≥ 5 ≥ 1 1 1 v (s; I) 1 + 1 | 2 | ≥ − 2σ 1 2kσ  − 1 + c  − 5 for σ > 0. Hence v (1,I) c (1 + c )−12−k > 0 which shows that the 2 ≥ 5 5 continuous function v2(s, I) will remain bounded from above by some pos- itive constant if σ | 1 c |with sufficiently small c . In the following, let ≥ − 6 6 DDT theorem 163

σ =1 min c ,c . Now, as in the proof of Lemma 3, we derive the estimate 0 − { 3 6} 1 2 g(s; d,I) 1 + = r(d) | |≪ 1 + f(p) pσ0  Yp|d if σ σ . ≥ 0 Further, expanding the logarithms of non-vanishing factors, we obtain the representation 1 1 φ(s;1,I) = ζα(s)exp α H (s),  1 + f(p) − ps  2 Xp where H2(s) is an analytic and bounded function in σ 3/4. Again, by Theorem 1, having an analytic expansion of the series under≥ the exponent and the appropriate estimate, we can apply Lemma 1 with M = r(d). So, since g(1; d,I) = g∗(d), we obtain (9). Finally, we examine ∞ g∗(n)f(n) W (s) := ζ(s)−β , σ > 1. β ns nX=1 By the definition we obtain g∗(p)f(p) = f(p)/T (p)+O(1/p) and f(p) 1 w (s) W (s) = 1 + β + p , β  1 + f(p) − ps p2s  Yp where w (s) are bounded for σ 3/4. By the same argument as when p ≥ dealing with Gα(s) earlier, we arrive at the formula f(p) 1 W (s) = exp β H (s), β  1 + f(p) − ps  3 Xp where H3(s) is analytic in the region σ 3/4. Since f (α), applying Lemma 1 with z = β and M = 1, we obtain≥ ∈ M x 1 (11) g∗(k)f(k) = W (1)+O , Γ (β)logα x β log x kX≤x where ∞ 1 β f(pi)g(pi) W (1) = 1 > 0. β  − p  pi  Yp Xi=0

Repeating the similar argument above, we obtain Wβ(1)F (α, 1) = 1. This and (11) yield the desired equality (10). Lemma 4 is proved.

4. Proof of Theorem 2. If u 1/log x or 1 1/log x v 1, the assertion of Theorem 2 follows from≤ easy estimates− of the tai≤ls of≤ a beta 164 G. Bareikis and E. Manstaviˇcius distribution. Moreover, we can concentrate on the cases: (A) 1/log x u 1/2 = v and (B) u =1/2 v 1 1/log x. ≤ ≤ ≤ ≤ −

u u (A) For convenience, in the definition of Sx(u) we replace n by x . The error at this step is negligible. Indeed, checking monotonicity of the appropriate factors in sums, we have

1 1 R (u) := f(d) x x T (n) nX≤x Xd|n nu

−α −1 Now, by Lemmas 3 and 4, we obtain Rx(u) u log x uniformly in 1/log x u 1/2. Thus, ≪ ≤ ≤ (12) S (u, 1/2) = S (1/2) S (u)+O(u−α log−1 x), x x − x b b where by Lemma 4,

1 T (n, xu) 1 1 S (u) := = f(d) x x T (m) x T (md) nX≤x dX≤xu mX≤x/d b F (α,I) f(d)g∗(d) f(d)r(d) = + O Γ (α) β  β+1  dX≤xu d log (x/d) dX≤xu d log (x/d) F (α,I) f(d)g∗(d) =: + R (u). Γ (α) β x dX≤xu d log (x/d) b In the same region, summing by parts and using (6) we obtain

xu 1 1 \ dv 1 R (u) + f(d)r(d) . x ≪ β+1 (u log x)α v2  ≪ uα log x log x 1 Xd≤v b DDT theorem 165

Hence Γ (α) 1 1 1 S (u) = f(d)g∗(d) + O F (α,I) x ((1 u)log x)β xu uα log x dX≤xu b − xu \ 1 β + f(d)g∗(d) + dv  2 β 2 β+1  1 Xd≤v v log (x/v) v log (x/v)

xu \ dv 1 = f(d)g∗(d) + O . 2 β uα log x 1 Xd≤v v log (x/v) Further applying (10) and (12) we have

x1/2 1 \ dv S (u, 1/2) = x Γ (α)Γ (β) α β xu v(log v)log (x/v) 1 + O . uα log x Substituting v = xt, we obtain the desired formula for 1/log x u 1/2. ≤ ≤ (B) Now let 1/2 v 1 1/log x and ≤ ≤ − 1 T (n, nv) S (v) = x x T (m) nX≤x 1 1 1 1 =1 f(d) f(d) − x T (n) − x T (n) nX≤x Xd|n nX≤x Xd|n d>xv nv

The first sum can be estimated using Lemma 2. Checking that the function t/logβ t is increasing for t eβ, we obtain ≥ 1 d(v) f(p) (13) Rˇ (u) exp x ≪ x log d(v)  p(1 + f(p)) p≤Xd(v) 1 1 + f(d)(g∗(d) + r(d)) ((1 v)log x)β xv − dX≤xv 1 1 1 + . ≪ xα ((1 v)log x)β logα x ≪ (1 v)β log x − − Here we have also used Lemmas 3 and 4. Dealing with Sˇx(v) we replace d by n/d. Then applying Lemma 3 we obtain 1 f(m) Sˇ (v) = x x T (md) d≤Xx1−v xv

5. Proof of Corollary. We will use the properties of the Dirichlet L(s, χ) functions. Lemma 5 (see [8, Theorems 4.2, 5.7, and 5.8]). Fix m N and let χ be ∈ an arbitrary Dirichlet character modulo m. If c3 = c3(m) > 0 is sufficiently small, in the region σ 1 c /log( t + 2), t R, we have L(s, χ) =0 and ≥ − 3 | | ∈ 6 L(s, χ) log( t + 2). ≪ | | Proof of Corollary. It suffices to show that the function f(m) defined via al, 1 l 1, we have → →∞ χ(p) −1 χ(p) log L(s, χ) = log 1 = + H (s, χ), −  − ps  ps 5 Xp Xp with H5(s, χ) holomorphic in σ > 1/2 and bounded in σ 3/4. By the orthogonality of characters, ≥ 1 1 χ(p) = χ(l) ps ϕ(m) ps p≡l (modX m) Xχ Xp 1 = χ(l)(log L(s, χ) + H (s, χ)). ϕ(m) 5 Xχ Hence 1 1 P (s) = α χ(l)log L(s, χ) + H6(s) ϕ(m) 1 + al −  1≤Xl

Consequently, f(m) (α) and the assertion of the Corollary follow from the theorems. ∈ M

References

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Gintautas Bareikis Eugenijus Manstaviˇcius Department of Mathematics and Informatics Institute of Mathematics and Informatics Vilnius University Akademijos St. 4 Naugarduko St. 24 LT-08663 Vilnius, Lithuania LT-03225 Vilnius, Lithuania E-mail: [email protected] E-mail: [email protected]

Received on 13.4.2006 (5185)