Notations

Throughout the dissertation we use standard notations. Some of them are fixed below.

Further notations will be explained by their first appearance.

‚ N, Z, R, C denote the set positive integers, integers, real and complex numbers, respectively;

‚ Zn“Z{nZ is the set of residue classes modulo n (nP N);

‚ the prime power factorization of nP Nis n“ś

pp^νppnq, the product being over the primesp, where all but a finite number of the exponents ν_ppnq are zero;

‚ pn₁, . . . n_kqand gcdpn₁, . . . n_kqdenote the greatest common divisor ofn₁, . . . , n_kPN;

‚ rn₁, . . . , n_ksand lcmpn₁, . . . , n_kqdenote the least common multiple ofn₁, . . . , n_kPN;

‚ id is the function idpnq “n (n PN);

‚τpnqis the number of divisors ofn,σpnqis the sum of divisors ofn,σ_spnqis the sum of s-th powers of the divisors of n;

‚ φpnq is Euler’s totient function, φ_spnq “ n^sś

p|np1´1{p^sq is the Jordan function of orders;

‚ µpnq is the M¨obius function, ψpnq is the Dedekind function given by ψpnq “ nś

p|np1`1{pq,κpnq “ś

p|npis the squarefree kernel of n;

‚ ωpnq denotes the number of distinct prime factors of n, Ωpnq “ ř

pν_ppnq is the number of prime power divisors of n;

‚ τ^p2qpnq “ 2^ωpnq is the number of squarefree divisors of n;

‚τ_kpnqis the Piltz divisor function, representing the number of ways n can be written as a product of k factors;

‚ τ^peqpnq “ś

p^ν||nτpνq is the number of exponential divisors of n;

‚ σ^peqpnq “ś

p^ν||n

ř

d|νp^d denotes the sum of exponential divisors of n;

‚ apnq represents the number of non-isomorphic abelian groups of order n;

‚ Ppnq “řn

k“1pk, nq is the gcd-sum function (Pillai’s function);

‚ Ppnqalso denotes the number of unrestricted partitions of n;

‚ c_qpnq “ ř

1ďkďq,pk,qq“1expp2πikn{qq are the Ramanujan sums;

‚ ˚ is the Dirichlet convolution of arithmetic functions;

‚ ζ is the Riemann zeta function;

‚ γ .

“0.577215 is Euler’s constant;

‚ G“ř8 n“0

p´1qⁿ p2n`1q²

“. 0.915956 is the Catalan constant;

‚ ř

p and ś

p are sums and products over the primes;

‚ the O (!), o, Ω and „ notations are used in the usual way, for the first one the implied constant may depend on certain parameters;

Chapter 2

Results for multiplicative functions of one variable

2.1 Average orders

Let τpnq denote the number of divisors of n. Ramanujan [76] stated without proof that the estimate

ÿ

nďx

τpnq² “xpAplogxq³`Bplogxq<sup>2</sup>`Clogx`Dq `Opx^3{5`εq (2.1) holds for any realε ą0, withA “π^´2 and certain constants B, C, D. By using analytic methods, Wilson [122] proved Ramanujan’s claim and generalized it by showing that for any integer rě2 one has

ÿ

nďx

τpnq^r “xP₂^r_´1plogxq `Opx²

r´1 2r`2`ε

q,

whereP₂^r_´1ptq is a polynomial of degree 2^r´1 in t with leading coefficient Cr “ 1

p2^r´1q!

ź

p

ˆ 1´ 1

p

˙2^r 8

ÿ

ν“0

pν`1q^r p^ν .

Note that in the case r “ 2, Wilson’s error term is better than the one stated by Ramanujan.

Now consider τ^peqpnq, denoting the number of exponential divisors of n. See Section 2.4. The function τ^peq is multiplicative and τ^peqpp^νq “ τpνq for every prime power p^ν (ν ě1). Wu [121] showed, improving an earlier result of Subbarao [87], that

ÿ

nďx

τ^peqpnq “A1x`B1x<sup>1{2</sup>`Opx^2{9logxq, (2.2)

where

A₁ :“ ź

p

˜ 1`

8

ÿ

ν“2

τpνq ´τpν´1q p^ν

¸ ,

B₁ :“ź

p

˜ 1`

8

ÿ

ν“5

τpνq ´τpν´1q ´τpν´2q `τpν´3q p^ν{2

¸ .

The error term in (2.2) is strongly related to the error term ∆p1,2;xq on the divisor functionτp1,2;nq “ ř

ab²“n1. It can be sharpened intoOpx1057{4785`εq. See [121, Remark, p. 135].

An asymptotic formula for the function fpnq “ τ^peqpnq^r with any integer r ě 1 fol-lows from the following general result concerning certain multiplicative functionsf such that fpnq depends only on the `-full kernel of n, where ` ě 2 is a fixed integer. Let

∆<sub>k,`</sub>pxq :“ ∆pp1, `, `, . . . , ` looomooon

k´1

q;xq denote the error term of the corresponding generalized divisor problem.

Theorem 2.1.1(T´oth [98], [103, Th. 2]). Letf :NÑCbe a multiplicative arithmetic function. Assume that

i) fppq “ fpp²q “ ¨ ¨ ¨ “fpp^{`´1</sup>q “ 1, fpp<sup>`}q “k for every prime p, where `, k ě2 are fixed integers,

ii) fpp^νq ! 2^ν{p``1q (νÑ 8) uniformly for the primesp.

Then 8

ÿ

n“1

fpnq

n^s “ζpsqζ^k´1p`sqVpsq,

absolutely convergent for<psq ą1, where the Dirichlet seriesVpsqis absolutely convergent for <psq ą1{p``1q.

Furthermore, suppose that ∆_{k,`</sub> !x<sup>αk,`</sup>plogxq<sup>βk,`</sup>, with 1{p``1q ăα<sub>k,`}ă1{`. Then ÿ

nďx

fpnq “ Cr_fx`x<sup>1{`</sup>P_f,k´2plogxq `R_fpxq, (2.3) where P_f,k´2 is a polynomial of degree k´2,

Cr_f :“ź

p

˜ 1`

8

ÿ

ν“`

fpp^νq ´fpp^ν´1q p^ν

¸ ,

and R_fpxq !x^{αk,`</sup>plogxq<sup>βk,`} (is the same).

Remark 2.1.2. For every k, ` ě 2, ∆<sub>k,`</sub>pxq ! x<sup>uk,``ε</sup>, where u<sub>k,`</sub> :“ <sub>3`p2k´1q`</sub><sup>2k´1</sup> P p<sub>``1</sub>¹ ,¹_`q.

See [56, Th. 6.10]. Therefore,R_fpxq !x^uk,``ε is valid, as well.

Applying Theorem 2.1.1 and Remark 2.1.2 to the functionfpnq “ τ^peqpnq^r with` “2, k“2^r, we deduce the following result.

Theorem 2.1.3 (T´oth [98, Eq. (4)]). Let r ě 1 be a fixed integer. The asymptotic formula

ÿ

nďx

τ^peqpnq^r “A_rx`x<sup>1{2</sup>Q<sub>2</sub><sup>r</sup><sub>´2</sub>plogxq `Opx^ur`εq holds for every εą0, where

A_r :“ ź

p

˜ 1`

8

ÿ

a“2

τpaq^r´τpa´1q^r p^a

¸ , Q₂^r_´2 is a polynomial of degree 2^r´2 and u_r :“ ²₂<sup>r`1</sup>r`2^´1`1.

Theorem 2.1.1 applies to other special functions, as well. For example, consider the functionapnq, representing the number of non-isomorphic abelian groups of order n. The function apnq is multiplicative and for every prime power p^ν (ν ě 1), app^νq “ Ppνq is the number of unrestricted partitions ofν. Thus, for every prime p, appq “1,app²q “2, app³q “3, app⁴q “ 5, app⁵q “ 7, etc. An asymptotic formula for the sum ř

nďxapnq was obtained for the first time by Erd˝os and Szekeres [32]. The corresponding error term was investigated by several authors. See, e.g., [49, Ch. 14], [56, Ch. 7] for historical surveys.

It is known that

ÿ

nďx

apnq “ A₁x`A<sub>2</sub>x<sup>1{2</sup>`A₃x^1{3`Rpxq, whereAj :“ś8

k“1,k‰jζpk{jq(j “1,2,3), and the best result for the error term isRpxq ! x^1{4`ε for every εą0, proved by O. Robert and P. Sargos [78].

The asymptotic behavior of the sum ř

nďx1{apnqwas investigated by Nowak [70]. An asymptotic formula for the quadratic moment of the function a, i.e., for ř

nďxapnq² was given by Zhang, L¨u and Zhai [124].

Let ∆_rpxq:“∆pp1,2,2, . . . ,2 loooomoooon

2^r´1

q;xq. For the r-th moment of the functionapnqwe have the next result.

Corollary 2.1.4 (T´oth [103, Th. 1]). Let r ě 2 be a fixed integer. Assume that

∆_rpxq !x^αrplogxq^βr, with 1{3ăα_ră1{2. Then ÿ

nďx

apnq^r “C_rx`x<sup>1{2</sup>S<sub>2</sub><sup>r</sup><sub>´2</sub>plogxq `R_rpxq, where

C_r :“ź

p

˜ 1`

8

ÿ

ν“2

Ppνq^r´Ppν´1q^r p^ν

¸ ,

S₂^r_´2 is a polynomial of degree2^r´2andR_rpxq !x^αrplogxq^βr (is the same). The estimate R_rpxq !x^ur holds true, where u_r is given in Theorem 2.1.3.

Remark 2.1.5. According to a result of Kr¨atzel [57],

∆₂pxq “ ∆pp1,2,2,2q;xq ! x^45{127plogxq⁵, where 45{127 .

“0.354330P p1{3,1{2q, hence the same is the remainder term forř

nďxapnq². This improves R₂pxq ! x^96{245`ε with 96{245 .

“ 0.391836, obtained in [124] by reducing the error term to the Piltz divisor problem concerning ∆₃pxq.

Remark 2.1.6. Referring to our paper [98], Lelechenko [60, Th. 4] pointed out that the error termR_fpxq !x<sup>uk,``ε</sup> given in Remark 2.1.2 can be improved by using another result included in the book by Kr¨atzel [56, Th. 6.8]. Namely, one can take u<sub>k,`</sub> “ <sub>``1´θ</sub>¹

k´1, where θ_t is an exponent such that ∆_tpxq ! x<sup>θt`ε</sup> in the Piltz divisor problem. Since θ<sub>t</sub>pxq ď <sup>t´1</sup><sub>t`2</sub> holds true for t ě 4, see Titchmarsh [94, Th. 12.3], it follows that u_k,` ď

k`1

`pk`1q`3 P p1{p``1q,1{`q for k ě 5. In particular, in the case r ě 3 the error terms of Theorem 2.1.3 and Corollary 2.1.4 can be improved by taking u_r“ ₂r`1<sup>2r`1</sup>`5.

Remark 2.1.7. An elementary proof of the asymptotic formula ÿ

nďx

τpnq² „Axplogxq³ pxÑ 8q,

appears in several places. See, for example, Nathanson [69, Th. 7.8]. Although this and related questions were investigated by several authors, we are not aware even of elementary proofs for the asymptotic formula

ÿ

nďx

τpnq^r „C_rxplogxq^2r´1 pxÑ 8q,

valid for any integerr ě2. In the joint work with Luca [62] we gave a minimal elementary proof of the following more general result:

Let k P N and let f : N Ñ C be a multiplicative function satisfying the following properties:

(i)fppq “k for every primep,

(ii) fpp^νq “ ν^Op1q for every prime p and every integer ν ě 2, where the constant implied by the O symbol is uniform in p.

Then

ÿ

nďx

fpnq

n “ 1

k!C_fplogxq^k`D<sub>f</sub>plogxq<sup>k´1</sup>`Opplogxq^k´2q and

ÿ

nďx

fpnq “ 1

pk´1q!C_fxplogxq^k´1`Opxplogxq^k´2q, where

C_f “ ź

p

ˆ 1´1

p

˙k 8

ÿ

ν“0

fpp^νq p^ν

and D_f is another constant depending on f. In the case fpnq “ τpnq^r with r P N, this applies by selectingk “2^r. Also see Martin [63, Prop. A.3] for a similar result proved by using analytic arguments.

2.2 Alternating sums concerning multiplicative

In document Asymptotic Properties of Multiplicative Arithmetic Functions of One and Several Variables DSc dissertation (Pldal 12-18)