On the DDT theorem. Dedicated to our teacher Professor Jonas Kubilius on his 85th birthday

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1 ACTA ARITHMETICA On the DDT theorem by G Bareikis and E Manstavičius Vilnius Dedicated to our teacher Professor Jonas Kubilius on his 85th birthday Introduction We denote the sets of natural, real, comle, and rime numbers by N, R, C, and P resectively In what follows we assume that P, d, l, m, n N, and s = σ + iτ C The letters c and C with or without subscrits denote constants Either of the notation f = Og or f g will mean that f C g for some ositive constant C, which may be absolute or deend uon various arameters In such cases we sometimes indicate that by a subscrit Throughout the aer 3 will be the sequence arameter and in the asymtotic relations it is assumed that Let τm, v be the number of natural divisors of m which do not eceed v m, and τm = τm, m In 979 J-M Deshouillers, F Dress, and G Tenenbaum [2] see also [0, Section II62] obtained the following result Theorem DDT Uniformly in u [0, ], τm, m u = 2 τm π arcsin u + O log m This asymtotic formula was announced already in [3], where some argument relating it to the well known arcsine law in the robabilistic invariance rincile was resented In our oinion, that argument was not very convincing To suort that, we observe that the arcsine law in the classical robability theory is in some sense universal while fails to hold if one relaces τm by another multilicative number-theoretic function whose values on rime numbers differ from two fairly often This raises a desire to search for another limiting relation generalizing Such an attemt was made by the second author who in [6, Section 8] announced a result 2000 Mathematics Subject Classification: K65, N60 Key words and hrases: multilicative function, mean value, natural divisor, beta distribution [55]

2 56 G Bareikis and E Manstavičius showing that, instead of the arcsine function, a new limit belonging to the class of beta distributions can aear This henomenon was also reorted by the first author in the semigrou of olynomials over a finite field [] In the resent note, we generalize the DDT theorem and slightly sharen the remainder term Relation arouses interest from the oint of view of arithmetically defined random rocesses First, observe that will remain true if τm, m u is relaced by τm, u So, the limit in can be understood as the asymtotic eectation of the sequence of stochastic rocesses if a number m is taken with robability ν {m} = /[] Later results obtained in [9] dealt with the asymtotic distribution of increments τm, m v τm, m u /τm, if 0 u v Having all this in mind, the second author [5] asked whether the distributions of the rocesses τm, u /τm under the measure ν weakly converge in the Skorokhod sace D[0, ] This hyothesis together with some generalization was roved in [7] To formulate the result, we recall some definition The sace D[0, ] consists of the real-valued functions on [0, ] which are right-continuous and have left-hand limits It is assumed that the Skorokhod metric is introduced in it Write D for the Borel σ-algebra in D[0, ] For a multilicative function f : N R +, we define 2 Tm, v = d m, d v fd, X m, u = Tm, u, Tm = Tm, m, Tm where u, v [0, ] Let ν X be the induced measure on D Theorem MT If f = κ > 0 for each P and f k 0 for all P and k 2, then ν X weakly converges to a limit measure on D In the Addendum [], G Tenenbaum generalized this theorem and showed, in addition, that the limit measure is concentrated on the subsace of continuous functions A simle eamle which does not satisfy the conditions of Theorem MT but is covered by the result of [] is defined via the multilicative function with f = + /2 for all odd P What is the asymtotic mean value of this articular rocess? In the resent aer, we give a more general answer 2 Results Let f : N R + be a multilicative function Throughout the aer we assume that, for some constant C > 0, it satisfies f l C for all P and l N We say that f belongs to the class Mα, 0 < α <,

3 DDT theorem 57 if the function defined by the series + f α s, σ >, for some c > 0 and 0 < δ <, has an analytic continuation Ps into the region c σ σt := log t + 2, t R, where Ps is holomorhic and Ps δ log t + 2 The well known formula = log ζs + Hs, s where Hs is analytic in σ > /2, gives an analytic continuation of the sum on the left-hand side into the region σ σt, where it is a holomorhic function, say, Ls Moreover, Ls log log2 + t there Thus, if f Mα, then, for β := α, the function f + f β s again has the same roerties This observation will be reeatedly eloited in what follows Throughout the aer, the deendence of the aearing remainders on the constants α, δ, c and C involved in the definition of the class Mα is allowed Using 2, we define S u = m Tm, m u, Tm and S u, v = S v S u where 0 u v Recall that the distribution function of a beta law has the following eression: Bu; a, b = ΓaΓb u 0 dv v a v b, 0 u, where 0 < a = b < and Γz denotes the Euler gamma-function Set Bu, v; a, b = Bv; a, b Bu; a, b for 0 u v and a b = min{a, b} Theorem If f Mα, then S u Bu; α, β log α + log β uniformly in 0 u This result follows from the net estimate which is sharer in the central zone

4 58 G Bareikis and E Manstavičius Theorem 2 If f Mα and 0 u v, then S u, v Bu, v; α, β log β + log α ulog α u log β For the functions discussed in Theorem MT, one can take α = +κ Generalizing the above mentioned eamle we have the following result Corollary Fi m N and let a l R + not all zero for l < m and l, m = If f : N R + is the multilicative function defined by f = a l for l mod m, where l < m and l, m =, f = 0 otherwise, and 0 f k C for P and k 2, then the assertion of Theorem 2 holds with β := α := ϕm l<m l,m= + a l In the roofs, we aly the analytic aroach roosed in [2] 3 Auiliary lemmas As usual, let ζs be the Riemann zeta function First, we resent two classical lemmas Lemma [0, Section II53] Let z C, M > 0, 0 <, and Fs := n= a n/n s, a n 0, be such that the function G z s := Fsζ z s can be continued as a holomorhic function for σ c /log t +2, t R, and in this domain satisfies the bound n G z s M + τ Then, for A > 0 and z A, we have Gz M a n = log z Γz + O, log where the imlicit constant of the remainder deends at most on c,, and A Lemma 2 [4] Let g be a multilicative function with 0 g l C for all rime numbers and l N Then { } gm C log e g, m m gm m { C e g }

5 Let Tm be as defined in Section Then DDT theorem 59 T k = + f + + f k For the multilicative function hm = fm or hm = Im :, we define 3 Q d; h = hn Tnd, Fa; h = a h i T i i n where a > 0 Similarly, gd; h := g l ; h, g l ; h = l d gd := gd; f, and g d := gd; I Further, set rd = d + f where C 2 > 0 and 0 < σ 0 < are arbitrary 4 5 h i / T i+l i + C 2 σ 0 Lemma 3 If f Mα, then, for all d N, Moreover, 6 7 Q d; f = Γβ log α gd = ΓαFβ; f log β d rd log β, d, Fβ; fgd + O + O rd log log h i T i i,, fdrd log α d Proof Introduce the generating series φs; d, f = fn Tndn s n = k d f i T i+k is =: gs; d, fφs;, f f i T i is f i T i is Since f i /T i+k f i / + f i C/ + C < for each k 0 and i, the last eansion is valid for σ >

6 60 G Bareikis and E Manstavičius We now eamine the factors First we observe that v s; f := + f i T i is C 8 + C σ i= C /2 > 0 + C for every P rovided that σ σ 0 = ma{3/4, log 2 + C/ + C /2 } Hence in the same region, if k, gs; k, f = f i f i T i+k is T i is + f Consequently, + f + C + σ + O 0 C + 2σ 0 C + C σ 0 + O C + C σ 0 2σ 0 gs; d, f = gs; k, f k d + C + C + f + C σ = rd 0 d if σ σ 0 Eamine the function Gs := φs;, fζ β s = { =: e f + f β + f s T s + k=2 } Hs f k T k ks β s Here Hs is some roduct over rimes In the routine way, taking logarithms, which is allowed by 8, we verify that the function Hs is analytic and bounded for σ σ 0 Dealing with the sum under the eonent, we eloit the definition of the class Mα Thus, Gs has an analytic continuation into the region σ σt, where it is holomorhic and Gs 2+ t δ for some 0 < δ < Together with 7 and 8 this also imlies an analytic continuation of φs; d, fζ β s and the estimate φs; d, fζ β s = gs; d, fgs rd2 + t δ

7 DDT theorem 6 for σ σt Since g; d, f = gd, by Lemma we obtain 4 with Fβ; f = G To derive the asymtotic formula 5, we will aly Lemma for G α s = ζs α n= gn n s = s α + By definition of g k, we obtain G α s = + + f α l= s + u s 2s g l ls where u s C 3 uniformly in σ 3/4 and P Searating a finite number of factors, with 0, where 0 is sufficiently large, we can ensure that the remaining factors for > 0 do not vanish for σ > So eanding their logarithms we obtain the reresentation { G α s = e + f α s, } H s, where H s is an analytic and bounded function in σ 3/4 Again, since f Mα, having an analytic eansion of the series under the eonent and the aroriate estimate, we can aly Lemma with M = This yields gd = Γα ln β G α + O, log d where G α > 0 It remains to show that G α Fβ; f =, where Fβ; f has been defined in 3 If := v ; f, using α + β =, we obtain G α Fβ; f = = = = Thus, relation 5 is roved l= l f i j= f i T l+i i j=i+ T j j j j T j f i j= j =

8 62 G Bareikis and E Manstavičius Since f Mα imlies r = + O = α log log + O, + f the first estimate in 6 is a corollary of Lemma 2 Similarly, we obtain the second one Lemma 3 is roved 9 In a similar manner we rove the net lemma Lemma 4 If f Mα, then, for all d N, Moreover, 0 Q d; I = g dfd = d Γα log β Proof We now start with φs; d, I = Tndn s n = k d = gs; d, Iφs;, I rd Fα; Ig d + O log ΓβFα; I log α + O log T i+k is T i is For the factor aearing in the denominator, v s; I = + T i is, as in 8, we have i= v s; I 3 σ c 2 > 0 T i is for every 3 if σ c 3 with sufficiently small c 3 < If f2 k 0 for k, then v 2 s; I = 2 s c 4 > 0 in the same region If f2 k = c 5 > 0 for some k, then v 2 s; I 2 σ + 2 kσ + c 5 for σ > 0 Hence v 2, I c 5 + c 5 2 k > 0 which shows that the continuous function v 2 s, I will remain bounded from above by some ositive constant if σ c 6 with sufficiently small c 6 In the following, let

9 DDT theorem 63 σ 0 = min{c 3, c 6 } Now, as in the roof of Lemma 3, we derive the estimate gs; d, I f σ = rd 0 d if σ σ 0 Further, eanding the logarithms of non-vanishing factors, we obtain the reresentation { φs;, I = ζ α s e + f α s } H 2 s, where H 2 s is an analytic and bounded function in σ 3/4 Again, by Theorem, having an analytic eansion of the series under the eonent and the aroriate estimate, we can aly Lemma with M = rd So, since g; d, I = g d, we obtain 9 Finally, we eamine W β s := ζs β n= g nfn n s, σ > By the definition we obtain g f = f/t + O/ and W β s = f + + f β s + w s 2s, where w s are bounded for σ 3/4 By the same argument as when dealing with G α s earlier, we arrive at the formula { } f W β s = e + f β s H 3 s, where H 3 s is analytic in the region σ 3/4 Since f Mα, alying Lemma with z = β and M =, we obtain g kfk = Γβ log α W β + O, log where k W β = β f i g i i > 0 Reeating the similar argument above, we obtain W β Fα, = This and yield the desired equality 0 Lemma 4 is roved 4 Proof of Theorem 2 If u /log or /log v, the assertion of Theorem 2 follows from easy estimates of the tails of a beta

10 64 G Bareikis and E Manstavičius distribution Moreover, we can concentrate on the cases: A /log u /2 = v and B u = /2 v /log A For convenience, in the definition of S u we relace n u by u The error at this ste is negligible Indeed, checking monotonicity of the aroriate factors in sums, we have R u := n Tn = fd d u + uβ d n n u <d u m /d m<d u/u fd Tmd fdd u/u log β d 3 d u g d + O urd log d u log β fdg d + d u log +β fdrd + u d u Now, by Lemmas 3 and 4, we obtain R u u α log uniformly in /log u /2 Thus, 2 S u, /2 = Ŝ/2 Ŝu + Ou α log, where by Lemma 4, Ŝ u := = =: n Fα, I Γα Fα, I Γα Tn, u Tm d u d u = fd d u fdg d d log β /d + O m /d d u fdg d d log β /d + R u Tmd fdrd d log β+ /d In the same region, summing by arts and using 6 we obtain R u u log β+ ulog α + d v fdrd dv v 2 u α log

11 Hence Γα Fα, I Ŝu = = u log β u + u d v u d v DDT theorem 65 fdg d fdg d Further alying 0 and 2 we have S u, /2 = ΓαΓβ + O fdg d + O d u v 2 log β /v + dv v 2 log β /v + O /2 u u α log dv u α log β v 2 log β+ /v u α log vlog α v log β /v dv Substituting v = t, we obtain the desired formula for /log u /2 Set B Now let /2 v /log and S v = n = Then by Lemma 4, Ř u = Tn, n v Tm n Tn d n d> v =: Šv Řv d dv d dv + fd n dv = e βv/ v e β β + fd Td dv<d v dv<d v fd fd m d v/v Tn Tmd d n n v <d v fd d v/v log d v/v β g rd d + v log d

12 66 G Bareikis and E Manstavičius The first sum can be estimated using Lemma 2 Checking that the function t/log β t is increasing for t e β, we obtain Ř u { dv } log dv e f 3 + f + dv v log β v d v fdg d + rd α + v log β log α v β log Here we have also used Lemmas 3 and 4 Dealing with Šv we relace d by n/d Then alying Lemma 3 we obtain Š v = fm Tmd = d v v <m /d Fβ; f Γβ rd d log α gd + O /d log/d rd gd + O log d v Fβ; f Γβ v log α v d v Fβ; f v = gdd Γβ y log α /y d y rd + d + v rd O d v d v log α+ By Lemmas 2 and 3, the terms containing the function rd contribute the error O v β log, thus, by 3 and Lemma 3 again, S /2, v = Š/2 Šv + O v β log /2 dy = + O ΓαΓβ log y y log β y log α /y /2 + O v v dy y log β y log α+ /y + O Substituting y = t and recalling our notation, we obtain v β log S /2, v = B v, /2; β, α + O v β log Theorem 2 is roved = B/2, v; α, β + O v β log

13 DDT theorem 67 5 Proof of Corollary We will use the roerties of the Dirichlet Ls, χ functions Lemma 5 see [8, Theorems 42, 57, and 58] Fi m N and let χ be an arbitrary Dirichlet character modulo m If c 3 = c 3 m > 0 is sufficiently small, in the region σ c 3 /log t + 2, t R, we have Ls, χ 0 and Ls, χ log t + 2 Proof of Corollary It suffices to show that the function fm defined via a l, l < m and l, m =, belongs to the class Mα, where α is as in the Corollary By classical calculations see [8, Section 43] we elore the sum Ps := + f α s = l<m l,m= α + a l l mod m s +H 4s, where H 4 s is an entire function corresonding to m In the region defined in Lemma 5, we can use the main branch of log Ls, χ defined by Lσ, χ 0 as σ For σ >, we have log Ls, χ = log χ s = χ s + H 5 s, χ, with H 5 s, χ holomorhic in σ > /2 and bounded in σ 3/4 By the orthogonality of characters, s = χl χ ϕm s l mod m χ = χllog Ls, χ + H 5 s, χ ϕm Hence Ps = ϕm = ϕm l<m l,m= l<m l,m= χ α χllog Ls, χ + H 6 s + a l χ α χllog Ls, χ + H 6 s + a l χ χ 0 with some H 6 s holomorhic and bounded in σ 3/4 In the last ste we eloited the choice of α and set χ 0 to be the rincial character By Lemma 5 the last equality gives the desired analytic continuation of Ps into the region σ c 3 /log2+ t where Ps is holomorhic Moreover, by Lemma 5, in the same region we obtain Ps log log3 + t, where the constant in deends also on m and the constants a l

14 68 G Bareikis and E Manstavičius Consequently, fm Mα and the assertion of the Corollary follow from the theorems References [] G Bareikis, Beta distribution in the olynomial semigrou, Ann Univ Sci Budaest Sect Com , [2] J-M Deshouillers, F Dress et G Tenenbaum, Lois de réartition des diviseurs, I, Acta Arith , [3] F Dress, Le théorème DDT, in: Séminaire de Théorie des Nombres , E no 8, CNRS, Talence, 978, 9 [4] H Halberstam and H E Richert, On a result of R R Hall, J Number Theory 979, [5] E Manstavičius, Natural divisors and the brownian motion, J Théor Nombres Bordeau 8 996, 59 7 [6], Functional limit theorems in robabilistic number theory, in: Paul Erdős and his Mathematics, Bolyai Soc Math Stud, G Halász et al eds, 2002, [7] E Manstavičius and N M Timofeev, A functional limit theorem related to natural divisors, Acta Math Hungar , 3 [8] E Prachar, Primzahlverteilung, Grundlehren Math Wiss 9, Sringer, 957 [9] G Tenenbaum, Lois de réartition des diviseurs, IV, Ann Inst Fourier Grenoble , no 3, 5 [0], Introduction to Analytic and Probabilistic Number Theory, Cambridge Univ Press, 995 [], Addendum to the aer of E Manstavičius and N M Timofeev A functional limit theorem related to natural divisors, Acta Math Hungar , 5 22 Gintautas Bareikis Deartment of Mathematics and Informatics Vilnius University Naugarduko St 24 LT Vilnius, Lithuania gintautasbareikis@mafvult Eugenijus Manstavičius Institute of Mathematics and Informatics Akademijos St 4 LT Vilnius, Lithuania eugenijusmanstavicius@mafvult Received on