Arithmetic functions associated with exponential divisors

Let d “ ś

pp^νppdq be a divisor of the integer n “ ś

pp^νppnq. Then d is called an exponential divisorof n, if ν_ppdq |ν_ppnq for every primep. Notation: d|en. This concept was introduced by Subbarao [87]. According to the definition, 1 |e 1, but 1 is not an

exponential divisor of n ą1. The smallest exponential divisor of n ą1 is its squarefree kernelκpnq “ś

p|np.

Let τ^peqpnq and σ^peqpnq denote the number and the sum of exponential divisors of n, respectively. The function τ^peq is called the exponential divisor function, already quoted in Section 2.1. The integer n “ ś

pp^νppnq is called exponentially squarefree if all the exponents ν_ppnq ě 1 are squarefree. By convention, 1 is also exponentially squarefree.

Letq^peqdenote the characteristic function of exponentially squarefree integers. Properties of these and related functions were investigated by several authors. See Cao and Zhai [16], Lelechenko [60, 61], P´etermann and Wu [75], Smati and Wu [85], Subbarao [87], Wu [121].

Two integers m, n ą 1 have common exponential divisors if and only if they have the same prime factors and in this case, i.e., for m “ śr

i“1p^a_iⁱ, n “ śr

i“1p^b_iⁱ, a_i, b_i ě 1 (1ďiďr), the greatest common exponential divisor of m and n is

pm, nq_peq :“

r

ź

i“1

p^pa_i ^i,biq.

Herep1,1q_peq “1 by convention and p1, nq_peq does not exist for ną1.

2.4.1 Exponential Euler function

The integers m, n ą 1 are called exponentially coprime, if they have the same prime factors and pa_i, b_iq “ 1 for every 1 ď i ď r, with the notation of above. In this case pm, nq_peq “ κpmq “ κpnq. 1 and 1 are considered to be exponentially coprime. 1 and ną1 are not exponentially coprime.

Forn “śr

i“1p^a_iⁱ ą1 witha_i ě1 (1ďiďr), denote byφ^peqpnqthe number of integers śr

i“1p^c_iⁱ such that 1 ď ci ď ai and pci, aiq “ 1 for 1 ď i ď r, and let φ^peqp1q “ 1. Thus φ^peqpnqcounts the number of divisorsd ofn such that dand n are exponentially coprime.

The function φ^peqpnq is called the exponential Euler function, it is multiplicative and for every prime power p^ν (ν ě1), φ^peqpp^νq “ φpνq, whereφ is Euler’s function.

As another consequence of Theorem 2.1.1 and Remark 2.1.2, by selecting fpnq “ φ^peqpnq^r with ` “3, k “2^r, we have the following result.

Theorem 2.4.1 (T´oth [96, Th. 1], [98, Eq. (6)]). Let rě1 be an integer. Then ÿ

nďx

φ^peqpnq^r “B_rx`x<sup>1{3</sup>T<sub>2</sub><sup>r</sup><sub>´2</sub>plogxq `Opx^tr`εq, (2.14) for every εą0, where

B_r:“ ź

p

˜ 1`

8

ÿ

a“3

φpaq^r´φpa´1q^r p^a

¸ ,

T₂^r_´2 is a polynomial of degree 2^r´2 and t₁ “1{5, t_r:“ ²_3¨2<sup>r`1</sup>r`1^´1 for r ě2.

In the case r “ 1, P´etermann [74, Th. 1] improved the error term in (2.14) into Opx^1{5logxq. Cao and Zhai [17] obtained new results on the four dimensional divisor problem of pa, b, c, cq type, where 1 ď a ď b ă c are fixed integers. As an application, they proved in [17, Th. 3] the following more precise asymptotic formula:

ÿ

nďx

φ^peqpnq “B₁x`B<sub>2</sub>x<sup>1{3</sup>`D₁x^1{5logx`D<sub>2</sub>x<sup>1{5</sup>` `Opx<sup>18{95`ε</sup>q, (2.15) where 18{95 .

“0.189473ă1{5. They also showed that the error term in (2.15) is Ωpx^1{8q. In the case r ě 3 the error term of (2.14) can be improved by takingt_r “ _3p2^2rr<sup>`1</sup>`2q, as shown by Lelechenko [60]. See Remark 2.1.6.

2.4.2 Exponential M¨ obius function

The exponential convolution of the arithmetic functions f and g is defined by pf dgqpnq “ ÿ

b1c1“a1

. . . ÿ

brcr“ar

fpp^b₁¹¨ ¨ ¨p^b_r^rqgpp^c₁¹¨ ¨ ¨p^c_r^rq, wheren “p^a₁¹¨ ¨ ¨p^a_r^r.

The convolution d is commutative, associative and has the identity element µ². Fur-thermore, a function f has an inverse with respect to d if and only if fp1q ‰ 0 and fpp₁¨ ¨ ¨p_sq ‰ 0 for any distinct primes p₁, . . . , p_s. The inverse with respect to d of the constant 1 function is called theexponential M¨obius functionand is denoted byµ^peq. Hence for every ně1,

ÿ

d|en

µ^peqpdq “ µ²pnq.

Hereµ^peqp1q “ 1 and forn “p^a₁¹¨ ¨ ¨p^a_r^r ą1,

µ^peqpnq “µpa₁q ¨ ¨ ¨µpa_rq.

Observe that |µ^peqpnq| “1 or 0, according as n is exponentially squarefree or not. Wu [121, Th. 2] deduced, improving a result by Subbarao [87] that

ÿ

nďx

|µ^peqpnq| “ C1x`Opx^1{4δpxqq, whereδpxqis defined by (1.3) and

C₁ “ź

p

˜ 1`

8

ÿ

a“4

µ²paq ´µ²pa´1q p^a

¸ .

I showed that the corresponding error term can be improved on the assumption of the Riemann hypothesis (RH).

Theorem 2.4.2 (T´oth [97, Th. 3]). If RH is true, then ÿ

nďx

|µ^peqpnq| “C₁x`Opx<sup>1{5`ε</sup>q, for every εą0.

Later on, Cao and Zhai [16, Th. 1] proved the following more precise asymptotic formula: If RH is true, then

ÿ

nďx

|µ^peqpnq| “C₁x`C<sub>2</sub>x<sup>1{5</sup>`Opx^38{193`εq, (2.16) whereC₂ is a computable constant and 38{193 .

“0.196891. Cao and Zhai [16, Th. 2] also proved that the error term in (2.16) is Ωpx^1{8q. Shevelev [80] investigated the asymptotic density of S-exponential numbers defined as the positive integers such that all exponents in their prime power factorization are inS, where S is a fixed subset of N. If S is the set of squarefree numbers, then one reobtains the exponentially squarefree integers.

For the functionµ^peqpnqwe have the next result.

Theorem 2.4.3 (T´oth [97, Th. 2]). (i) The Dirichlet series of µ^peq is of the form

8

ÿ

n“1

µ^peqpnq

n^s “ ζpsq

ζ²p2sqUpsq, <są1, where Upsq:“ř8

n“1 upnq

n^s is absolutely convergent for <są1{5.

(ii)

ÿ

nďx

µ^peqpnq “ Kx`Opx^1{2expp´cplogxq^9{25´δq, (2.17) for any δ ą0, where cą0 is a constant and

K “ź

p

˜ 1`

8

ÿ

a“2

µpaq ´µpa´1q p^a

¸ .

(iii) Assume RH. Let 1{4 ără1{3 be an exponent such that Dpxq:“ř

nďxµ²pnq ´ x{ζp2q “ Opx<sup>r`ε</sup>q for every ε ą 0. Then the error term in (ii) is Opxp2´rq{p5´4rq`ε

q for every εą0.

The best known value – to our knowledge – of r is r “ 17{54 .

“ 0.314814, obtained by Jia [50]. See also Pappalardi [73]. Therefore the error term in (2.17), assuming RH, is Opx^91{202`εqfor every εą0, where 91{202 .

“0.450495.

By using analytic arguments (Perron’s formula and complex integration), Cao and Zhai [16, Th. 1] improved the error term (2.17) under RH intoOpx^37{94`εq, where 37{94 .

“ 0.393617, and showed in [16, Th. 2] that this error term is Ωpx^1{4q.

2.4.3 The function t

^peq

p n q

I introduced and investigated in paper [97] the functions t^peqpnq and κ^peqpnq, de-noting the number of exponentially squarefree exponential divisors of n and the max-imal exponentially squarefree exponential divisor of n, respectively. These are the ex-ponential analogues of the functions representing the number of squarefree divisors of n (i.e. θpnq “ 2^ωpnq) and the maximal squarefree divisor of n (the squarefree kernel κpnq “ś

p|np), respectively.

The functions t^peqpnq and κ^peqpnqare multiplicative and for n “p^a₁¹¨ ¨ ¨p^a_r^r ą1, t^peqpnq “ 2^ωpa1q¨ ¨ ¨2^ωparq,

κ^peqpnq “p^κpa₁ ^1q¨ ¨ ¨p^κpa_r ^rq.

Here I discuss only the function t^peqpnq. Note that for every prime p, t^peqppq “ 1, t^peqpp²q “ t^peqpp³q “ t^peqpp⁴q “ t^peqpp⁵q “ 2, t^peqpp⁶q “4, etc. We have

Theorem 2.4.4 (T´oth [97, Th. 4]). (i) The Dirichlet series of t^peq is of form

8

ÿ

n“1

t^peqpnq

n^s “ζpsqζp2sqVpsq, <s ą1, where Vpsq “ ř8

n“1 vpnq

n^s is absolutely convergent for <s ą1{4.

(ii)

ÿ

nďx

t^peqpnq “C₁x`C<sub>2</sub>x<sup>1{2</sup>`Opx^1{4`εq, (2.18) for every εą0, where C₁, C₂ are constants given by

C₁ :“ ź

p

˜ 1` 1

p² `

8

ÿ

a“6

2^ωpaq´2^ωpa´1q p^a

¸ ,

C₂ :“ζp1{2qź

p

˜ 1`

8

ÿ

a“4

2^ωpaq´2^ωpa´1q´2^ωpa´2q`2^ωpa´3q p^a{2

¸ .

P´etermann [74, Th. 1] improved the error term in (2.18) intoOpx^1{4q. Cao and Zhai [16, Th. 1] obtained that under RH it isOpx3728{15469`εq, where 3728{15469 .

“0.240998ă1{4.

Cao and Zhai [16, Th. 2] also proved that the error term in (2.18) is Ωpx^1{6q.

By generalizing results of R´enyi and Ivi´c, Zurita [128] established asymptotic formulas for the sums

ÿ

nďx Ωpnq´ωpnq“q

fpnq,

where fpnq are certain multiplicative functions, which apply, among others, to the func-tions fpnq “ τ^peqpnq, φ^peqpnq, t^peqpnq, σ^peqpnq{n, Ppnq{n, P^peqpnq{n, apnq. Here P^peqpnq is the exponential analog of the gcd-sum function Ppnq. See Section 2.5.2.

In document Asymptotic Properties of Multiplicative Arithmetic Functions of One and Several Variables DSc dissertation (Pldal 23-28)