Euler Numbers and polynomials associated with zeta functions

In this paper we give some interesting identities between Euler numbers and zeta functions. Finally we will give the new values of Euler zeta function at positive even integers.

Let p be a fixed odd prime. For d(= odd) a fixed positive integer with (p, d) = 1, let (a + dpZ p ), a + dp N Z p = {x ∈ X|x ≡ a (mod dp n )}, where a ∈ Z lies in 0 ≤ a < dp N .
In this paper we prove that µ −q (a + dp N Z p ) = (1 + q) (−1) a q a 1 + q dp N = is distribution on X for q ∈ C p with |1 − q| p < 1. This distribution yields an integral as follows: which has a sense as we see readily that the limit is convergent (see [14]). Let q = 1. Then we have the fermionic p-adic integral on Z p as follows: f (x)(−1) x , cf. [1,5,20,21,14].
In complex plane, the ordinary Bernoulli numbers are a sequence of signed rational numbers that can be defined by the identity These numbers arise in the series expansions of trigonometric functions, and are extremely important in number theory and analysis. From the generating function of Bernoulli numbers, we note that B 0 = 1, , · · · , and B 2k+1 = 0 for k ∈ N. It is well known that Riemann zeta function is defined by We also note that Riemann zeta function is closely related to Bernoulli numbers at positive integer or negative integer in complex plane. Riemann did develop the theory of analytic continuation needed to rigorously define ζ(s) for all s ∈ C − {0}. From this zeta function, he derived the following formula, cf. : Thus, we note that ζ(−n) = 0 if n is even integer and greater than 0. These are called the trivial zeros of the zeta function. In 1859, starting with Euler's factorization of the zeta function ζ(s) = p:prime he derived an explicit formula for the prime numbers in terms of zeros of the zeta function. He also posed the Riemann Hypothesis: if ζ(z) = 0, then either z is a trivial zero or z lies on the critical line Re(z) = 1 2 , cf. [20][21][22][23][24][25][26][27][28][29][30][31][32][33]. It is well known that Thus, [12,[20][21][22][23][24][25][26][27][28][29][30][31][32][33].
From this, we can derive the following famous formula: However, it is not known the values of ζ(2k + 1) for k ∈ N. In the case of k = 1, Apery proved that ζ(3) is irrational number. The constants E * k in the Taylor series expansion where |t| < π, cf. [5,12,20,21,22], are known as the first kind Euler numbers. From the generating function of the first kind Euler numbers, we note that The first few are 1, − 1 2 , 0, 1 4 , · · · , and E * 2k = 0 for k = 1, 2, · · · . The Euler polynomials are also defined by For s ∈ C, Euler zeta function and Hurwitz's type Euler zeta function are defined by [1,11,12,20,21,22].
Thus, we note that Euler zeta functions are entire functions in whole complex s-plane and these zeta functions have the values of Euler numbers or Euler polynomials at negative integers. That is, [1,11,12,20,21,22] .
In this paper, we give some interesting identities between Euler numbers and zeta functions. Finally we will give the values of Euler zeta function at positive even integers. §2. Preliminaries/Euler numbers associated with p-adic fermionic integrals Let f 1 (x) be translation with f 1 (x) = f (x + 1). Then we have If we take f (x) = e (x+y)t , then we can derive the first kind Euler polynomials from the integral equation of I −1 (f ) as follows: That is, For n ∈ N, we have the following integral equation: From this we note that Let f (x) = sin ax (or f (x) = cos ax). By using the fermionic p-adic q-integral on Z p , we see that [15]. and 2 = (cos a + 1) Thus, we obtain , see [15].
From this we note that By the same motivation, we can also observe that , see [15].
These formulae are also treated in the Section 3. Let f (x) = e t(2x+1) . Then we can derive the generating function of the second kind Euler numbers from fermionic p-adic integral equation as follows: Thus, we have (E + 1) n + (E − 1) n = 2δ 0,n , where we have used the symbolic notation E n for E n . The first few are E 0 = 1, E 1 = 0, E 2 = −1, E 3 = 0, E 4 = 5, · · · , E 2k+1 = 0 for k ∈ N. In particular, In the recent, Simsek, Ozden, Cangul, Cenkci, Kurt, etc have studied the various extensions of the first kind Euler numbers by using fernionic p-adic invariant q-integral on Z p , see [ 1,5,20,21,22,27,31]. It seems to be also interesting to study the qextensions of the second kind Euler numbers due to Simsek et al( see [20,21,27]). §3. Some relationships between Euler numbers and zeta functions In this section we also consider Bernoulli and the second Euler numbers in complex plane. The second kind Euler numbers E k are defined by the following expansion: , cf. [12].

2
. Thus, we have From (5), we derive The Fourier series of an odd function on the interval (−p, p) is the sine series: Let us consider f (x) = sin ax on [−π, π]. From (7) and (8), we note that In (9), if we take x = π 2 , then we have (11) sin From (11), we note that In (6), it is easy to see that By (12) and (13), we obtain the following: It is easy to see that By (14) and (15), we obtain the following: Corollary 2. For n ∈ N, we have By simple calculation, we easily see that Thus, we have (16) x tan From (16), we can easily derive By (4), we also see that Thus, we have By (17) and (18), we obtain the following: where E * n are the first kind Euler numbers. It is easy to see that Therefore we obtain the following: Now we try to give the new value of Euler zeta function at positive integers. From the definition of Euler zeta function, we note that By (19), Theorem 3 and Corollary 4, we obtain the following: Remark. We note that ζ(2) = π 2 6 , ζ E (2) = − π 2 6 , ζ(4) = π 4 90 and ζ E (4) = − 7π 4 360 · · · . For q ∈ C with |q| < 1, s ∈ C, q-ζ-function is defined by [ 12,17] .
Note that, ζ q (s) is analytic continuation in C with only one simple pole at s = 1, and where k is a positive integer, cf. [17].