Using the Jordan algebras methods, specially the properties of Peirce decomposition and the Frobenius transformation, we compute the coefficients of the zeta functional equation, in the case of Jordan algebras of type II. As particular cases of our result, we can cite the case of V=M(n, R)...
(1) Titchmarsh points out in his book on the zeta function (section 2.3) that if you blindly apply the Poisson summation formula to the function f(s)=|x|^s, you get the functional equation of the Riemann zeta function immediately, and gives a reference to a paper of Mordell where this...
[2] E.C. Titchmarsh, The Theory of the Riemann Zeta Function. [3] P.G. Rooney, Another proof of the functional equation for the Riemann zeta function. Another Proof of the Functional Equation for the Riemann Zeta Functionwww.sciencedirect.com/science/article/pii/S0022247X84712443编辑...
We further show that the $\mathbb{F}_1$-zeta function $\zeta(s)$ of a split reductive group scheme $G$ of rank $r$ with $N$ positive roots satisfies the functional equation $\zeta(r+N-s) = (-1)^\chi ( \zeta(s) )^{(-1)^r}$. Full-Text Contact Us service@oalib.com ...
These roots are called the trivial zeros of the zeta function. The remaining roots are called the nontrivial zeros or critical roots of the zeta function. Bn+1 n+1 (21) | w | =δ (19) ew ? 1 wn+2 w dw = Bn+1 (20) 6 Riemann Zeta Function The Functional Equation Figure 2: ...
A symmetrical form of this functional equation is given by (14) (Ayoub 1974), which was proved by Riemann for all complex (Riemann 1859). As defined above, the zeta function with a complex number is defined for . However, has a unique analytic continuation to the entire complex plane...
The Hurwitz zeta function can also be given by the functional equation (6) (Apostol 1995, Miller and Adamchik 1999), or the integral (7) If and , then (8) (Hurwitz 1882; Whittaker and Watson 1990, pp. 268-269). The Hurwitz zeta function satisfies (9) for (Apostol 1995...
This function is proportional to a Dedekind zeta function, satisfies a functional equation and has an Euler product formula. It has no zeros with Re z > 1, and it is believed that all its non-trivial zeros lie on the critical line Re z = 1/2 (Potter and Titchmarsh, 1935)....
我们可以对作解析延宕(analyticcontinuation),使得成为定义在整个复数平面上的半纯函数(meromorphicfunction),并且只有在s=1这一点有一个residue为1的simplepole。而且经过解析延宕之后,满足下面的functionalequation: 以上关于的性质,可以在Ahlfors[1]书里找到,这不是这篇文章所要讨论的主题。 黎曼zeta-函数与Bernoulli数...
Furthermore, we extend these results to a one-parameter family of critical polynomials that possess the functional equation pn(s;尾) = 卤 pn (1 鈭s;尾). Normalisation yields the rational function q位 n(s) whose denominator has singularities on the negative real axis. Moreover as s 鈫鈭...