A research on the recursive value and general terms of the analogue Euler zeta function on positive integers
© Kang and Ryoo; licensee Springer 2013
Received: 21 April 2013
Accepted: 3 June 2013
Published: 26 June 2013
Our aim is to find the general term of the analogue Euler zeta function in positive integers by using Fourier series. We also figure out the generalized coefficients of Fourier series and investigate some interesting relation in the integers.
Many mathematicians have studied various kinds of an analogue zeta function such as Dirichlet L-function [1–28]. Friedman and Cohen constructed the p-adic analogue for Hurwitz zeta functions . By using multiple Volkenborn integrals, Tangedal and Young defined a p-adic multiple zeta function and a log gamma function [1–18]. Ryoo, Kim and Kim have defined various analogue zeta functions to combine the Euler numbers and Bernoulli numbers [1–12, 17, 18].
Therefore one of the most important and fascinating functions is the zeta function in mathematics [9, 10]. Bernhard Riemann (1826-1866) found something amazing; namely the Riemann zeta function. He recognized the importance of the function onto the entire complex plane ℂ except .
implies that converges uniformly on .
Some values can be calculated explicitly but , where are still mysterious. The number was demonstrated to be an irrational number by Apery (French mathematician) and can be seen in Hardy, Grosswald, Zhang, Srivastava, and others [20–28]. This Riemann zeta function is the Dirichlet zeta-function, the special case that arises when we take .
where is a special kind of the function . Any function of the form is known as a Dirichlet L-series where s is a real number greater than 1 and χ is called as Dirichlet character.
Mathematicians have studied extensively the Riemann zeta function and the Dirichlet zeta function because these functions play an important role in physics, complex analysis and number theory etc. They also recognized that the discovery of these zeta functions dates back to Euler.
Leonhard Euler (1707-1783) defined the zeta function for any real number greater than 1 by the infinite sum. After Euler defined this function, he showed that it had a deep and profound connection with the pattern of the primes. He also calculated , and has been researched, proved by mathematicians . The values of the Riemann zeta function were computed by the Euler zeta function at even positive integers. The analogue Euler zeta function replaces the Euler zeta function by mathematician’s research. The Euler zeta function was originally constructed by Kim (see ) and Kim gave the values of the Euler zeta function as positive integers (see [, Theorem 3.1]). Kim, Choi, and Kim researched to combine the Euler numbers and Bernoulli numbers in order to get values or a generalized term .
The Euler zeta function is defined as follows.
From Definition 1.1, we define the analogue Euler zeta function as follows.
We easily note that .
In this paper, we find out the generalized coefficients of Fourier series and investigate some interesting relations in the positive integers. We also investigate values and a generalized term of the analogue Euler zeta function in the same way by using the Fourier series.
The Fourier series can be expressed as summation between sine series and cosine series instead of complicated functions.
because , each .
We denote that , , and when .
The paper is organized as follows. In Section 2, we construct generalized coefficients of sine series and cosine series in the positive integers and prove them. We also study some interesting relations about sine series and cosine series in the positive integers. In Section 3, applying these ideas, generalized coefficients will be used to obtain the main results of this paper. We also find the general term of the analogue Euler zeta function.
2 The coefficient’s rule of cosine series and sine series
In this section, we construct the coefficient’s rule of cosine series and sine series. We access some relations about the coefficient sine series and cosine series.
The cosine series for is given by the following theorem.
Proof We shall prove Theorem 2.1 using mathematical induction. We assume that for and l is a positive even integer.
That is, and hold for (m: even) if it holds for . Thus, we complete the proof of the theorem. □
By using the matrix, we can display the coefficient of cosine series in Theorem 2.1 as follows.
From now on, we will see the coefficients of the sine series. The sine series is given by the following theorem.
which is true.
Therefore holds for (m: odd).
Thus, we conclude the proof of the theorem by the principle of mathematical induction. □
By using the matrix, we can arrange the coefficient of sine series in Theorem 2.3 as follows.
From Theorem 2.1 and Theorem 2.3, we get the relation of coefficients between and .
If l is an even integer, then .
3 The analogue Euler zeta function in the integers
In this section, we get the value and generalized term of the analogue Euler zeta function. We derive by using the coefficient of cosine series and obtain using the coefficient of sine series.
From the above, we obtain the following theorem.
By observing Theorem 3.1, we can easily understand the relation between the Euler zeta function and the analogue Euler zeta function.
since is if and 0 if with .
Thus, we have Theorem 3.2.
The authors express their gratitude to the referee for his/her valuable comments. We also thank T. Kim and A. Sankaranarayanan for their great support and encouragement. They kindly offered invaluable lecture and advice about the paper. This work was supported by NRF (National Research Foundation of Korea) Grant funded by the Korean Government (NRF-2013-Fostering Core Leaders of the Future Basic Science Program).
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