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

[section] 1. Introduction

Let n > 1 be an integer. Consider the integers a for which there exists an integer x such that [a.sup.2]x = a(mod n). Properties of these integers 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 [??](n) = #[Reg.sub.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 multiplicative and [??]([p.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] [??](n)/[phi](n)] = Bx + O([log.sup.2] x), (1.1)

where B = [[pi].sup.2]/6 [approximately equal to] 1.6449.

Let r [greater than or equal to] 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 ([??](n)/[phi](n)).sup.r], which generalizes (1.1).

Theorem. Suppose r [greater than or equal to] 1 is a fixed integer, then

[summation over (n [less than or equal to] x] [([??](n)/[phi](n)).sup.r] = [C.sub.r]x + O([log.sup.2r] x), (1.2)

where [C.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]. From now on, suppose [zeta](s) denotes the Riemann-zeta function.

Lemma 1. Suppose t [greater than or equal to] 2, then uniformly for [sigma] 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) := [[summation].sup.[infinity].sub.n=1] [([??](n)/[phi](n)].sup.r], Res > 1. It is easy to see that [([??](n)/[phi](n)).sup.r] is multiplicative, so by the Euler product formula, for Res > 1 we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

It is easy to check that the Dirichlet series [[summation].sup.[infinity].sub.n=1] v(n)/[n.sup.s] is absolutely convergent for Res [greater than or equal to] -1/10. So we have

[summation over (n [less than or equal to] x] [absolute value of v(n)] [much less than] 1, [summation over (n [less than or equal to]x] [absolute value of v(n)][n.sup.1/10] [much less than] [x.sup.[epsilon], (2)

where [epsilon] is a small positive real number.

Let [[zeta].sup.r](s+1)/[[zeta].sup.r](2s+1) = [[summation].sup.[infinity].sub.l=1] [b.sub.r](l)/[l.sup.s], then according to the Dirichlet convolution, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

So the problem now is reduced to compute [[summation].sub.l [less than or equal to] x] [b.sub.r](l)] and [[summation].sub.l [less than or equal to] x] [absolute value of [b.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)] [b.sub.r](l) = 1 + O([e.sup.-C[square root of (log x)]]), (4)

where C > 0 is some positive constant. We omit the proof of (4). By the partial summation, we get from (4) that

[[summation over (l>x)] [b.sub.r](l)/l [much less than] [x.sup.-1], (5)

[[summation over (l [less than or equal to] x] [b.sub.r](l)/l = [[infinity].summation over (l=1)] [b.sub.r](l)/l - [[summation over (l>x] [b.sub.r(l)/l = [C.sub.1] + O([x.sup.-1]). (6)

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

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

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

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

So

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

which combining the well-known estimate

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

gives

[summation over [less than or equal to] x] [absolute value of [b.sub.r](l)] [much less than] [log.sup.2r] x. (7)

From (3)-(7), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

by recalling (2), where [C.sub.r] = [C.sub.1] [[summation].sup.[infinity].sub.k=1] v(k)/k is a constant.

So our proof of the theorem is completed.

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 [psi]-function concerning the integers which are regular modela n, Portugal. Math., 33(1974), 185-191.

[3] Laaszlao Taoth, 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.

(1) This work is supported by National Natural Science Foundation of China (Grant No. 10771127) and Mathematical Tianyuan Foundation (Grant No. 10826028).

Yanru Dong and Jian Wang

School of Mathematical Sciences, Shandong Normal University, Jinan 250014, P.R.China

E-mail: dongyanru5252@163.com wodi111@163.com
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Author:Dong, Yanru; Wang, Jian
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Date:Dec 1, 2009
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