Keywords: dedekind zetafunction, dirichlet series, mean value
Turkish Journal of Analysis and Number Theory, 2014 2 (6),
pp 230232.
DOI: 10.12691/tjant268
Received November 05, 2014; Revised December 05, 2014; Accepted December 18, 2014
Copyright © 2013 Science and Education Publishing. All Rights Reserved.
1. Introduction and the Result
Let be an algebraic number field of degree over the rational field , and be its Dedekind zetafunction. 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
 (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
 (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
 (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 PhragmenLindelöf 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.
 (4) 
and
 (5) 
It is easy to get
 (6) 
Now we consider the integral . We have
Then
 (7) 
To go further, we get
 (8) 
By (7) and (8)
 (9) 
From (9), Lemma 2.1 and 2.3, we have (for )
 (10) 
Finally we estimate trivial bounds of the integrals . By Lemma 2.2, we get
which yields
 (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
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