# On the mean value of log [??](n).

[section] 1. Introduction and main results

Let n > 1 be an integer. Consider the integers m for which there exists an integer x such that [m.sup.2]x = m(mod n). Let [??](n) = {m: 1 [less than or equal to] m [less than or equal to] n, m is regular(mod n)}. This function is multiplicative and [??]([p.sup.[gamma]]) = [psi]([p.sup.[gamma]]) + 1 = [p.sup.[gamma]] - [p.sup.[gamma]-1] + 1 for every prime power [p.sup.[gamma]] ([gamma] [greater than or equal to] 1), where [psi](n) is the Euler function (see [1]).

The mean value of the function [??](n) was considered in [2], [4]. One has,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where A = [[PI].sub.p](1 - 1 / [p.sup.2](p+1)) = [zeta](2) [[PI].sub.p] (1 - 1/[p.sup.2] - 1/[p.sup.3] + 1/[p.sup.4]) [approximately equal to] 0.8815 is the so called quadratic class-number constant.

More exactly, V. S. Joshi [2] proved

[summation over (n [less than or equal to] x)] [??](n) = 1/2 A[x.sup.2] + R(x), (1)

where R(x) = O(x [log.sup.3] x). This was improved into R(x) = O(x [log.sup.2] x) in [3], and into R(x) = O(x log x) in [6]. The [OMEGA]-estimate R(x) = [[OMEGA].sub.[+ or -]](x [square root of log log x]) was also proved in [6].

Laszlo Toth [1] proved the following three results:

[summation over (n [less than or equal to] x)] [??](n)/[psi](n) = 3/[[pi].sup.2]x + 0([log.sup.2] x), (2)

[summation over (n [less than or equal to] x)] [psi](n)/[??](n) = Bx + 0([(log x).sup.5/3] [(log log x).sup.4/3]), (3) [??](n)

[summation over (n [less than or equal to] x)] 1/[??](n) = [C.sub.1] log x + [C.sub.2] + O([log.sup.9] x / x), (4)

where [C.sub.1] and [C.sub.2] are constants,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this paper, we shall prove a result about the mean value of log [??](n). Our main result is the following

Theorem. We have

[summation over (n [less than or equal to] x)] log [??](n) = x log x + Ex + O([x.sup.1/2] [log.sup.3/2] x), (5)

where

E = [summation over p] (1 - [p.sup.-1]) [[infinity].summation over [alpha] = 2] [p.sup.-[alpha]] log(1 - [p.sup.-1] + [p.sup.-[alpha]]).

Notations. Throughout this paper, [epsilon] > 0 denotes a small positive constant.

[section] 2. Proof of the theorem

In order to prove our theorem, we need the following lemmas, which can be found in Ivic [5]. From now on, [zeta](s) denotes the Riemann-zeta function.

Lemma 1. Let T [greater than or equal to] 2 be a real number, then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [gamma] is the Euler constant.

Lemma 2. For t [greater than or equal to] [t.sub.0] [greater than or equal to] 2, we have uniformly for [sigma] that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof of Theorem.

Let f (n): = [[??].sup.u](n), where u is a fixed complex number with [absolute value of u] [less than or equal to] 1/4. Then for every prime power [p.sup.[alpha]],

f([p.sup.[alpha]]) = [[??].sup.u]([p.sup.[alpha]]) = [([p.sup.[alpha]] - [p.sup.[alpha]-1] + 1).sup.u] = [p.sup.[alpha]u] [(1 - [p.sup.-1] + [p.sup.-[alpha]]).sup.u]. (6)

Since f(n) is multiplicative, by the Euler product we get for Rs > 1 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Write G(s, u) = [[summation].sup.[infinity].sub.n=1] g(n)/[n.sup.s] (Rs > 1). It is easy to see that this infinite series is absolutely convergent in the range Rs > Ru, which implies that

G(s,u) = O[epsilon](1), Rs [greater than or equal to] Ru + [epsilon]. (9)

Taking T = [x.sup.2]. By Perron's formula we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then by the residue theorem we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Denote s = [sigma] + it, u = Ru + iv, where [absolute value of u] [greater than or equal to] 1/4. Form (9) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

With the partial summation and (14), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where B(t) = [[integral].sup.t.sub.0] [absolute value of [zeta](1/2 + iw)] dw.

With Lemma 1 and Cauchy's inequality, we get (for t [greater than or equal to] 2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

With (15) and (16), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

With Lemma 2, (9) and (13), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since the integrands in the above integrals above are monotone, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

Similarly, we have

[[integral].sub.1] << 1.. (19)

By (10)-(13) and (17)-(19), we obtain

[summation over n [less than or equal to] x ] [[??].sup.u](n) = G(1 + u,u) / 1 + u x [x.sup.1+u] + O ([x.sup.1/2+Ru] [log.sup.3/2] x) (20

By differentiating (20) term by term, we derive

[summation over n [less than or equal to] x ] [[??].sup.u](n) log [??](n) = H'(u)[x.sup.1+u] + H(u)[x.sup.1+u] log x + O([x.sup.1/2+Ru] [log.sup.3/2] x), (21)

where H(u): = G(1+u'u)/1+u.

Letting u = 0 in (21), we get

[summation over n [less than or equal to] x ] log [??](n) = H(0) x log x + H'((0)x + O([x.sup.1/2][log.sup.3/2] x). (22)

Now we evaluate H(0) and H'(0). According to (8), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

which implies immediately that H(0) = 1.

Taking the logarithm derivative from both sides of (23) we get that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

which together with H(1) = 1 gives

H'(0) = [summation over p] (1 - [p.sup.-1]) [[infinity].summation over [alpha]=2] [p.sup.-[alpha]] (log - [p.sup.-1] + [p.sup.-[alpha]]). (25)

Now Theorem follows from (22), (24) and (25).

References

[1] Laszlo Toth (Pecs, Hungary), Regular integers modulo n, Annales Univ. Sci. Budapest., Sect. Comp., 29(2008), 263-275.

[2] V. S. Joshi, Order-free integers (mod m), Number theory (Mysore, 1981), Springer, Lecture Notes in Math., 938(1982), 93-100.

[3] Zhao Hua Yang, A note for order-free integers (mod m), J. China Univ. Sci. Tech. 16(1986), No. 1, 116-118.

[4] S. Finch, Idempotents and nilpotents modulo n, 2006, Preprint available online at http://arxiv.org/abs/math. NT/0605019v1.

[5] A. Ivic, The Riemann Zeta-function, University of Belgrade Yugoslavia, 1985.

[6] J. Herzog, P. R. Smith, Lower bounds for a certain class of error functions, Acta Arith., 60(1992), 289-305.

Lixia Li and Heng Liu

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

E-mail: li09lixia25@163.com sglheng@163.com

Let n > 1 be an integer. Consider the integers m for which there exists an integer x such that [m.sup.2]x = m(mod n). Let [??](n) = {m: 1 [less than or equal to] m [less than or equal to] n, m is regular(mod n)}. This function is multiplicative and [??]([p.sup.[gamma]]) = [psi]([p.sup.[gamma]]) + 1 = [p.sup.[gamma]] - [p.sup.[gamma]-1] + 1 for every prime power [p.sup.[gamma]] ([gamma] [greater than or equal to] 1), where [psi](n) is the Euler function (see [1]).

The mean value of the function [??](n) was considered in [2], [4]. One has,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where A = [[PI].sub.p](1 - 1 / [p.sup.2](p+1)) = [zeta](2) [[PI].sub.p] (1 - 1/[p.sup.2] - 1/[p.sup.3] + 1/[p.sup.4]) [approximately equal to] 0.8815 is the so called quadratic class-number constant.

More exactly, V. S. Joshi [2] proved

[summation over (n [less than or equal to] x)] [??](n) = 1/2 A[x.sup.2] + R(x), (1)

where R(x) = O(x [log.sup.3] x). This was improved into R(x) = O(x [log.sup.2] x) in [3], and into R(x) = O(x log x) in [6]. The [OMEGA]-estimate R(x) = [[OMEGA].sub.[+ or -]](x [square root of log log x]) was also proved in [6].

Laszlo Toth [1] proved the following three results:

[summation over (n [less than or equal to] x)] [??](n)/[psi](n) = 3/[[pi].sup.2]x + 0([log.sup.2] x), (2)

[summation over (n [less than or equal to] x)] [psi](n)/[??](n) = Bx + 0([(log x).sup.5/3] [(log log x).sup.4/3]), (3) [??](n)

[summation over (n [less than or equal to] x)] 1/[??](n) = [C.sub.1] log x + [C.sub.2] + O([log.sup.9] x / x), (4)

where [C.sub.1] and [C.sub.2] are constants,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In this paper, we shall prove a result about the mean value of log [??](n). Our main result is the following

Theorem. We have

[summation over (n [less than or equal to] x)] log [??](n) = x log x + Ex + O([x.sup.1/2] [log.sup.3/2] x), (5)

where

E = [summation over p] (1 - [p.sup.-1]) [[infinity].summation over [alpha] = 2] [p.sup.-[alpha]] log(1 - [p.sup.-1] + [p.sup.-[alpha]]).

Notations. Throughout this paper, [epsilon] > 0 denotes a small positive constant.

[section] 2. Proof of the theorem

In order to prove our theorem, we need the following lemmas, which can be found in Ivic [5]. From now on, [zeta](s) denotes the Riemann-zeta function.

Lemma 1. Let T [greater than or equal to] 2 be a real number, then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [gamma] is the Euler constant.

Lemma 2. For t [greater than or equal to] [t.sub.0] [greater than or equal to] 2, we have uniformly for [sigma] that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Proof of Theorem.

Let f (n): = [[??].sup.u](n), where u is a fixed complex number with [absolute value of u] [less than or equal to] 1/4. Then for every prime power [p.sup.[alpha]],

f([p.sup.[alpha]]) = [[??].sup.u]([p.sup.[alpha]]) = [([p.sup.[alpha]] - [p.sup.[alpha]-1] + 1).sup.u] = [p.sup.[alpha]u] [(1 - [p.sup.-1] + [p.sup.-[alpha]]).sup.u]. (6)

Since f(n) is multiplicative, by the Euler product we get for Rs > 1 that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

Write G(s, u) = [[summation].sup.[infinity].sub.n=1] g(n)/[n.sup.s] (Rs > 1). It is easy to see that this infinite series is absolutely convergent in the range Rs > Ru, which implies that

G(s,u) = O[epsilon](1), Rs [greater than or equal to] Ru + [epsilon]. (9)

Taking T = [x.sup.2]. By Perron's formula we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then by the residue theorem we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

Denote s = [sigma] + it, u = Ru + iv, where [absolute value of u] [greater than or equal to] 1/4. Form (9) we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

With the partial summation and (14), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

where B(t) = [[integral].sup.t.sub.0] [absolute value of [zeta](1/2 + iw)] dw.

With Lemma 1 and Cauchy's inequality, we get (for t [greater than or equal to] 2)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (16)

With (15) and (16), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)

With Lemma 2, (9) and (13), we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Since the integrands in the above integrals above are monotone, we get

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (18)

Similarly, we have

[[integral].sub.1] << 1.. (19)

By (10)-(13) and (17)-(19), we obtain

[summation over n [less than or equal to] x ] [[??].sup.u](n) = G(1 + u,u) / 1 + u x [x.sup.1+u] + O ([x.sup.1/2+Ru] [log.sup.3/2] x) (20

By differentiating (20) term by term, we derive

[summation over n [less than or equal to] x ] [[??].sup.u](n) log [??](n) = H'(u)[x.sup.1+u] + H(u)[x.sup.1+u] log x + O([x.sup.1/2+Ru] [log.sup.3/2] x), (21)

where H(u): = G(1+u'u)/1+u.

Letting u = 0 in (21), we get

[summation over n [less than or equal to] x ] log [??](n) = H(0) x log x + H'((0)x + O([x.sup.1/2][log.sup.3/2] x). (22)

Now we evaluate H(0) and H'(0). According to (8), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (23)

which implies immediately that H(0) = 1.

Taking the logarithm derivative from both sides of (23) we get that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (24)

which together with H(1) = 1 gives

H'(0) = [summation over p] (1 - [p.sup.-1]) [[infinity].summation over [alpha]=2] [p.sup.-[alpha]] (log - [p.sup.-1] + [p.sup.-[alpha]]). (25)

Now Theorem follows from (22), (24) and (25).

References

[1] Laszlo Toth (Pecs, Hungary), Regular integers modulo n, Annales Univ. Sci. Budapest., Sect. Comp., 29(2008), 263-275.

[2] V. S. Joshi, Order-free integers (mod m), Number theory (Mysore, 1981), Springer, Lecture Notes in Math., 938(1982), 93-100.

[3] Zhao Hua Yang, A note for order-free integers (mod m), J. China Univ. Sci. Tech. 16(1986), No. 1, 116-118.

[4] S. Finch, Idempotents and nilpotents modulo n, 2006, Preprint available online at http://arxiv.org/abs/math. NT/0605019v1.

[5] A. Ivic, The Riemann Zeta-function, University of Belgrade Yugoslavia, 1985.

[6] J. Herzog, P. R. Smith, Lower bounds for a certain class of error functions, Acta Arith., 60(1992), 289-305.

Lixia Li and Heng Liu

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

E-mail: li09lixia25@163.com sglheng@163.com

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Author: | Li, Lixia; Liu, Heng |
---|---|

Publication: | Scientia Magna |

Article Type: | Report |

Geographic Code: | 9CHIN |

Date: | Dec 1, 2009 |

Words: | 1213 |

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