SUMMABILITY METHODS BASED ON THE RIEMANN ZETA FUNCTION

. This paper is a study of summability methods that are based on the Riemann Zeta function. A limitation theorem is proved which gives a necessary condition for a sequence x to be zeta summable. A zeta summability matrix Z associated with a real sequence is introduced; a necessary and sufficient condition on the sequence such that Z maps 11 to 11 is established. Results comparing the strength of the zeta method to that of well-known summability methods are also investigated


0,
The zeta method is a "sequence-to4unction" summability method whose domain consists of those sequence x such that the Dirichlet's sedes ;-1 (xk/k) is convergent for s > 1.
In the second section it is shown that the zeta summability method is regular and totally regular (preserves finite and infinite limits). A limitation theorem is proved which gives a necessary condition for a sequence x to be zeta summable. In section 3 we introduce a zeta summability matrix Z associated with a real sequence t; a necessary and sufficient condition on the sequence such that Z maps 11 into 11 is established. The final section contains results comparing the strength of the zeta method to that of well-known summability methods. For example, the zeta method is stronger than the Cesro method of order but does not include the Ces.ro method of order 2; the zeta method does not include and is not included in the Euler-Knopp method of order for 0 < < 1.
We shall use a variant of this concept, which we shall call a "reflected Stoitz domain of angle o"

S'(a)--{w
Arg(w-1) < and Re(w)>l} This concept is now used to extend the zeta method to one using a complex-valued function limit, and we establish the regularity of this extension. Proof. Since (w) can be expanded in the form (w 1) "1 + P(w 1), where P(w 1) is a power series in (w-1), (Hardy [4], p. 333), we have 30 L.K, CHU 7T + P(a-l) + (w-l/ Since the limit value Iw 11 o 11 -< sec z for w eS*(cz), this proves the assertion. Now we prove Theorem 2.
Proof (of Theorem 2). Let e > 0. Since x converges to L, we can choose N 9-Xk I_ < e/4) N cosa for k>N 1 Next we prove a limitation theorem which asserts that the -summability method cannot sum a sequence that diverges too rapidly. THEOREM 3. If a complex number sequence x is -summable, then for each s > 1, x n o(nS). Moreover, the term o(ns) is the best possible in the sense that the conclusion fails if n s is replaced by any real sequence to such that tn/n s decreases to zero.
Proof. For x-to be i-summable, x must be in the domain of the -summability method. Therefore xn/ n converges for all s > 1, which implies that limn(xn/nS 0. If n s is replaced by n, where tn/nS decreased to 0, then we assert that it will not be true that x n O(tn) whenever x is -summable. This is equivalent to showing that there is a sequence x such that x is -summable and x n O(tn). i.e., x is -summable to 0. But x n = O(tn) because for each n, xn/tn 1.

ZETA SUMMABILITY MATRICES
Definition. Let be a sequence of real numbers such that t(n) > for every n and limnt(n 1.
Then the zeta matrix Z [Znk associated with the sequence is defined by In this section we make use of two well-known theorems in summability theory, which we shall subsequently cite by name only; they are Silverman-Toeplitz Theorem ( [5]and [6]) and the Knopp-Lorentz Theorem [7]. It is an easy calculation to show that Zl satisfies the conditious of the Silverman-Toeplitz Theorem for regularity. Moreover, Z is totally regular because all of its entries are positive real numbers ([3] p. 35). We summarize these observations in the following theorem. THEOREM 4. The zeta matrix Z associated with the sequence is totally regular.
The next result is a characterization of those sequences for which Z is an I-I matrix, i.e., Z maps 11 into THEOREM 5. The matrix Z is an I-I matdx if and only if is in 11.
Proof. Since each row sequence of the matrix Z is decreasing, the set of the sums of column sequences of the matrix Z is bounded by the sum of its first column entries. Therefore by the Knopp-Lorentz Theorem, it is enough to show that the first column sum is finite whenever V (t(n) 1) is convergent. This is a consequence of the inequality which follows immediately from the fact that for s > 1, Hence Z is an I-I matrix.
Conversely, assume Z maps to I1. Since t(n) > and lim n t(n) for every n, we can choose a positive integer N such that 0 < t(n) < for n > N. Suppose is not in 11 then n--N =N 32 L.K. CHU Now ,',', (1/t (t(n))) diverges to infinity because of the inequality 1/ (t (n))) >_ (1 1/t (n)) as in (*). Therefore, by the Knopp-Lorentz Theorem, Z is not an I-I matrix. This completes the proof of the theorem.

INCLUSION THEOREMS.
In this section we compare the strength of the zeta method and the zeta matrix methods to several well-known summability methods. Throughout this section C a denotes the Cesaro summability matrix of order and E the Euler-Knopp summability matrix of order r. LEMMA 2. If x is a sequence that is C1-summable, then x is in the domain of the -summability method, and hence, x is in the domain of every Z method.
Proof. Assume that x is C 1-summable to L: lim n (Xl +-+ Xn)/n L. To get the conclusion it is enough to show that the abscissa of convergence Go of the Direchlet series n.__= xn/n s is less than or equal to 1, where o is given by log xl ro--lira sup log n (Hardy et al [8] or Titschmarch [9]). Since x is c 1-summable to L, there exists a positive integer N such that Proof. This inclusion is equivalent to the regularity of the matrix ZtC1-1 which can be verified by direct calculation using the Silverman-Toeplitz Theorem.
The following example shows that the C method does not include the Z method. By a "continuous parameter sequence-to-function transformation", we mean a summablility method F that is determined as follows by a fuction sequence {fk(z)} --i for a given sequence x form the function Fx(z)--E fk(Z)Xk (*:) k=l if lim z ._) a Fx(z) L, then we say that "x is F-summable to L". For a given function sequence {fk(z)} and a given number sequence t, we can also form an associated matrix F t, which is given by The next lemma, which will be used to compare the C method and the t method, is a comparision of the method if and the associated matrix method F LEMMA 3. Let F be a continuous parameter sequence-to-function transformation as in (**) and define the sequence sets To prove the reverse inclusion, we consider a sequence x which is not in S F. It follows that lim z ._> aFx(z) does not exist. By the sequential criterion for function limits (Almsted [10], p. 73), there is a sequence t\ in T such that limn(Ftx)n does not exist. This implies that x is not in the set Sv, Hence x is not in the set r"i. THEOREM 7. The -summability method is stronger than the C method.
Proof. By Lemma 3, we have S r'lt(TSz,. Since the Z method includes the C method for all in T, we have Sc C ,t? S.,, $.. Now if x is a sequence that is C -summable to L, then x is Z summable to L for all in T. Therefore the sequential criterion for function limits ensures that x is t-summable to L.
Hence, the method includes the C method. It is easy to see that the C method does not include the t method because C method does not include the Z method.
As a consequence of Theorem 6, we can infer that Z includes any method that is included by C 1.
Let H 2 denote the Holder method of order 2. By arguing as in the proof of Theorem 6, we can prove THEOREM 8. If the sequence x is H2-summable to L and x is in the domain of the Z method, then x is Z summable to L.
COROLLARY. If the sequence x is H2-summable to L and x is in the domain of the -summability method, then x is t-summable to L.
The conclusion of the preceding Corollary does not hold if x is not in the domain of the method. This is shown by the following example.
Since the Holder method of order 2 is equivalent to the Cesaro method of order 2 (Hardy [4], po 103), we can immediately get the following theorem. THEOREM 9. If x is a sequence which is C2-summable to L and x is in the domain of the summability method, then x is -summable to L.
It is well known that for each number satisfying 0 < < and any nonzero real number x, Ero By using these facts, we have the following result. THEOREM 10. The method is not included in E for 0 < < 1.
The following example shows that the method does not include E for 0 < < 1.
EXAMPLE. Given between 0 and choose > 0 satisfying < 2 (2 + ). Next define x k (-1 _)k. ACKNOWLEDGEMENT. This work is a portion of the author's doctoral dissertation, written under the supervision of Professor J. A. Fridy at Kent State University, Kent, OH, 1985.

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 • 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