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On the Error Term for the Number of Integral Ideals in Galois Extensions

Sanying Shi

School of Mathematics, Hefei University of Technology, Hefei, China

Abstract: Suppose that E is an algebraic number field over the rational field Let a(n) be the number of integral ideals in E with norm n and Δ(x) denote the remainder term in the asymptotic formula of the l-th integral power sum of a(n). In this paper the bound of the average behavior of Δ(x) is given. This result constitutes an improvement upon that of Lü and Wang for the error terms in mean value.

Keywords: dedekind zeta-function, dirichlet series, mean value

Received November 25, 2014; Revised December 05, 2014; Accepted December 18, 2014

Citation: Sanying Shi, On the Error Term for the Number of Integral Ideals in Galois Extensions. Turkish Journal of Analysis and Number Theory, 2 (6). 230-232. January 2014. doi:10.12691/tjant-2-6-8

1. Introduction and the Result

Let be an algebraic number field of degree over the rational field , and be its Dedekind zeta-function. Thus

where runs over all integral ideals of the field , and is the norm of . If denotes the number of integral ideals in with norm , then we have

It is known that is a multiplicative function and satisfies

[picture] | (1)

where is the divisor function.

It is an important problem to study the function . In 1927, Landau [7] first proved that

for any arbitrary algebraic number field of degree , where is the residue of at its simple pole .

It is hard to refine Landau's result. Later, Huxley and Watt [3] and Müller [9] improved the results for the quadratic and cubic fields, respectively.

Until 1993, Nowak [10] obtained the best result

for any arbitrary algebraic number field of degree .

In [1], Chandraseknaran and Good studied the -th integral power sum of in some Galois fields, and they showed that

Theorem 1.0. If is a Galois extension of of degree , then for any and any integer , we have

where , and is a suitable polynomial in of degree .

Recently, Lü and Wang [8] improved the classical result of [1] by replacing with .

Motivated by [2,4,5], the purpose of this paper is to study the remainder term in mean square, and we shall prove the following result.

Theorem 1.1 Subject to assumptions in Theorem 1.0, and define

[picture] | (2)

Then we have

for any given .

Notations. As usual, the Vinogradov symbol means that is positive and the ratio is bounded. The letter denotes an arbitrary small positive number, not the same at each occurrence.

2. Proof of Theorem 1.1

To prove our Theorem, we need the following lemmas.

Lemma 2.1 Let be a Galois extension of degree , and be defined in (1). Define

[picture] | (3)

Then we have

for any integer , where , and denotes a Dirichlet series, which is absolutely and uniformly convergent for .

Proof. This is Lemma 2.1 in [8].

Lemma 2.2. Let be an algebraic number field of degree , then

for and any fixed .

Proof. By Lemma 2.2 in [8] and the Phragmen-Lindelöf principle for a strip (see, e.g. Theorem 5.53 in [6]), Lemma 2.2 follows immediately.

Now we begin to prove our theorem.

Let be a Galois extension of of degree .

Recall denotes the number of integral ideals in with norm , and

Let

From (1), (3) and Perron's formula (see Proposition 5.54 in [6], we get

By the property only has a simple pole at for and Cauchy's residue theorem, we have

where , and is a suitable polynomial in of degree .

From the definition of in （2）, we have

Therefore to prove Theorem 1.1, we shall prove the following results.

[picture] | (4)

and

[picture] | (5)

It is easy to get

[picture] | (6)

Now we consider the integral . We have

Then

[picture] | (7)

To go further, we get

[picture] | (8)

By (7) and (8)

[picture] | (9)

From (9), Lemma 2.1 and 2.3, we have (for )

[picture] | (10)

Finally we estimate trivial bounds of the integrals . By Lemma 2.2, we get

which yields

[picture] | (11)

The inequalities (4), (5) immediately follow from (6), (10) and (11). That is,

Then this completes the proof of Theorem 1.1.

Acknowledgement

This work is supported by The National Science Foundation of China (grant no. 11201107 and 11071186) and by Natural Science Foundation of Anhui province (Grant No. 1208085QA01).

References

[1]   | Chandraseknaran K., Good A., On the number of integral ideals in Galois extensions, Monatsh. Math., 95. 99-109. 1983.
[2]   | Heath-Brown D. R., The number of Abelian groups of order at most x, Journtes Arithmttiques, Luminy 1989.
[3]   | Huxley M. N., Watt N., The number of ideals in a quadratic field II, Israel J. Math. Part A, 120, 125-153. 2000,
[4]   | Ivic A., The number of finite non-isomorphic Abelian groups in mean square, Hardy-Ramanujan J., 9, 17-23. 1986.
[5]   | Ivic A., On the Error Term for the Counting Functions of Finite Abelian Groups, Monatsh. Math. 114, 115-124. 1992.
[6]   | Iwaniec H., Kowalski E., Analytic Number Theory, Amer. Math. Soc. Colloq. Publ., 53, 204-216. 2004.
[7]   | Landau E., Einführung in die elementare and analytische Theorie der algebraischen Zahlen und der Ideale, Teubner, 1927.
[8]   | Lü G., Wang Y., Note on the number of integral ideals in Galois extension, Sci. China Ser. A, 53, 2417-2424. 2010.
[9]   | Müller W., On the distribution of ideals in cubic number fields, Monatsh. Math., 106, 211-219. 1988.
[10]   | Nowak W.G., On the distribution of integral ideals in algebraic number theory fields, Math. Nachr., 161, 59-74. 1993.