ON THE FIRST POWER MEAN OF L-FUNCTIONS WITH THE WEIGHT OF GENERAL KLOOSTERMAN SUMS

where p is a prime, is any fixed positive number, and (m,n,p) denotes the greatest common divisor of m, n, and p. But for an arbitrary composite number q, we do not know how large |S(m,n,χ,q)| is. In fact the value of |S(m,n,χ,q)| is quite irregular if q is not a prime. The main purpose of this paper is to obtain some good distribution properties of |S(m,n,χ,q)| in some weight mean value problems. For convenience, in this paper we always suppose q ≥ 3 be an integer and L(s,χ) denotes the Dirichlet L-function corresponding to character χmodq. Then we can use the estimates for character sums and the analytic method to prove the following main result.


Introduction.
Let q ≥ 2 be an integer, χ denotes a Dirichlet character modulo q.For any integers m and n, we define the general Kloosterman sums S(m, n, χ, q) as follows: S(m, n, χ, q) = q a=1 χ(a)e ma + n ā q , ( where ā denotes the inverse of a modulo q and e(y) = e 2πiy .This summation is very important, because it is a generalization of the classical Kloosterman sums.Many authors had studied the properties of S(m, n, χ, q).For instance, Chowla [1] and Malyšev [3] obtained a sharper upper bound estimation for S(m, n, χ, q).That is, S(m, n, χ, p) (m,n,p) 1/2 p 1/2+ , ( where p is a prime, is any fixed positive number, and (m,n,p) denotes the greatest common divisor of m, n, and p.But for an arbitrary composite number q, we do not know how large |S(m, n, χ, q)| is.In fact the value of |S(m, n, χ, q)| is quite irregular if q is not a prime.The main purpose of this paper is to obtain some good distribution properties of |S(m, n, χ, q)| in some weight mean value problems.For convenience, in this paper we always suppose q ≥ 3 be an integer and L(s, χ) denotes the Dirichlet L-function corresponding to character χ mod q.Then we can use the estimates for character sums and the analytic method to prove the following main result.
Theorem 1.1.For any integers m and n with (mn, q) = 1, we have the asymptotic formula where is a constant, χ≠χ 0 denotes the summation over all nonprincipal characters modulo q, p denotes the product over all primes p with (p, q) = 1, d(q) is the divisor function, φ(q) is the Euler function, and 2n n = (2n)!/(n!) 2 .
For general integer k ≥ 2, whether there exists an asymptotic formula for is an unsolved problem.
2. Some lemmas.In order to complete the proof of Theorem 1.1, we need the following lemmas.
Lemma 2.1.For any integer q ≥ 3, we have the estimate Proof.Let N = q 3/2 , χ be a nonprincipal character mod q and A(χ, y) = N<n≤y χ(n).Then by Abel identity and Pólya-Vinogradov inequality, we have (2.2) So that On the other hand, let r (n) be a multiplicative function defining by where p is a prime and α is any positive integer.For the function r (n), it is easy to prove that where Note that the triangle inequalities 3), (2.6), Cauchy inequality, and the orthogonality relationship for character sums we have (2.10) This proves Lemma 2.1.
Lemma 2.2.For any integer q ≥ 3, we have the asymptotic formula where is an absolute constant.
Lemma 2.3.Let m, n, and q be integers with q ≥ 3. Then we have the estimates S(m, n, q) = q a=1 e ma + n ā q (m,n,q) 1/2 q 1/2 d(q), (2.14) where a denotes the summation over all a such that (a, q) = 1.

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation