On the mean value of the function [([phi](n)/[rho](n)).sup.r].

[section] 1. Introduction

Let n > 1 be an integer. Consider the integer a for which there exist an x such that [a.sup.2]x [equivalent to] a (mod n). Properties of these integer were investigated by J. Morgado [1,2] who called them regular (mod n).

Let [Reg.sub.n] = {a : 1 [less than or equal to] a [less than or equal to] n, a is regular (mod n)} and let [rho](n) denote the number of regular integers a(mod n) such that 1 [less than or equal to] a [less than or equal to] n. This function is multiplicate and [rho]([[rho].sup.v]) = [phi]([p.sup.v]) + 1 = [p.sup.v] - [p.sup.v-1] + 1 for every prime power [p.sup.v](v [greater than or equal to] 1), where [phi] is the Euler function.

Laszlo Toth [3] proved that

[summation over (n[less than or equal to]x)] [phi](n)/[rho](n) = Cx + O([(log x).sup.5/3][(log log x).sup.4/3]), (1)

where C is a constant.

Let r > 1 be a fixed integer. The aim of the short paper is to establish the following asymptotic formula for the mean value of the function [([phi](n)/[rho](n)).sup.r], which generalizes (1).

Theorem. Suppose r > 1 be a fixed integer, then

[Summation over (n[less than or equal to]x)] [([phi](n)/[rho](n)).sup.r] = [A.sub.r]x + O([log.sup.2r] x), (2)

where [A.sub.r] is a constant.

[section] 2. Proof of the theorem

In order to prove our theorem, we need the following lemmas, which can be found in Ivic [4]. Form now on, suppose [zeta](s) denotes the Riemann-zeta function.

Lemma 1. Suppose t [greater than or equal to] 2, then uniformly for a we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Lemma 2. There exists an absolute constant c > 0 such that [zeta](s) [not equal to] 0 for [sigma] > 1 - c/log([absolute value of t]+2).

Proof of the theorem.

Let

f(s) := [[infinity].summation over (n=1)] [[([phi](n)/[[rho](n)).sup.r]]/[[n.sup.s]], Res > 1.

It is easy to see that [([phi](n)/[rho](n)).sup.r] is multiplicative, so by the Euler product formula, for Res > 1 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

It is easy to check that Dirichlet series [[summation].sup.[infinity].sub.n=1] is absolutely convergent for Res [greater than or equal to] - 2/5, so we have

[summation over (n[less than or equal to]x)] [absolute value of g(n)] [much less than] 1. (3)

Let

[[zeta].sup.r](2s + 1)/[[zeta].sup.r](s + 1) = [[infinity].summation over (n=1)] [v.sub.r](n)/[n.sup.s],

then according to the Dirichlet convolution, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (4)

So it is reduced to compute [[summation].sub.l[less than or equal to]x][v.sub.r](l) and [[summation].sub.l[less than or equal to]x][absolute value of [v.sub.r](l)]. Similar to the proof of the prime number theorem, with the help of lemma 1, lemma 2 and perron's formula we get

[summation over (l[less than or equal to]x)] [v.sub.r](l) = C + O([x.sup.-[epsilon]]), (5)

where

C = [Res.sub.s=0] [[zeta].sup.r](2s + 1)/[[zeta].sup.r](s + 1) s

is a constant, [epsilon] is a small positive real number. By the partial summation, we get form (4) that

[summation over (l>x)] [v.sub.r](l)/l [much less than] [x.sup.-1], (6)

[summation over (l>x)] [v.sub.r](l)/l = [[infinity].summation over (t=1)] [v.sub.r](l)/l - [summation over (l>x)] [v.sub.r](l)/l = [C.sub.1] + O([x.sup.-1]). (7)

Now we go on to bound the sum [[summation].sub.l[less than or equal to]x]([absolute value of [v.sub.r](l)]). Since for Res > 1,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], we obtain

[v.sub.r](l) = [summation over (l=[mn.sup.2])] [[mu].sup.r](m)[d.sub.r](n)/mn.

So

[absolute value of [v.sub.r](l)] [less than or equal to] [summation over (l=[mn.sup.2])] [d.sup.r](m)[d.sub.r](n)/mn,

which combining the well-known estimate

[summation over (n[less than or equal to]x)] [d.sub.r](n) [much less than] x [log.sup.r-1]x

gives

[summation over (l[less than or equal to]x)] [absolute value of [v.sub.r](l)] [much less than] x [log.sup.2r]x. (8)

Form (4)-(8), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where

[A.sub.r] = [C.sub.1]x [[infinity].summation over (k=1)] g(k)/k = [[infinity].summation over (l=1)] [v.sub.r](l)/l [[infinity].summation over (k=1)] g(k)/k

is a constant.

This completes the proof of the theorem.

References

[1] J. Morgado, Inteiros regulares modulo n, Gazeta de Matematica (Lisboa), 33(1972), No. 125-128, 1-5.

[2] J. Morgado, A property of the Euler [phi]-function concerning the integers which are regular modela n, Portugal. Math, 33(1974), 185-191.

[3] Laszlo Toth, Regular integers modulo n, Annales Univ. Sci. Budapest, Set. Comp, 29(2008), 263-275.

[4] A. Ivic, The Riemann zeta function, John Wiley & Sons, 1985.

[5] Yanru Dong, On the mean value of the function (^(n))r, Scientia Magna, 5(2009), No. 4, 13-16.

Wenli Clien ([dagger]), Jingmei Wei ([double dagger]) and Yu Huang (#)

([dagger]) ([double dagger]) School of mathematical Sciences, Shandong Normal University, Jinan, 250014

(#) Network and Information Center, Shandong University, Jinan, 250100

E-mail: [e-mail] [e-mail] [e-mail]

(1) This work is supported by N. N. S. F. of China (Grant Nos: 10771127, 11001154) and N. S. F. of Shandong Province (Nos: BS2009SF018, ZR2010AQ009).