[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...
(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...
In this paper we show that if X is a smooth projective scheme, then its F 1-zeta function satisfies the functional equation ζ X ( n s ) = ( 1 ) χζ X ( s ). We further show that the F 1-zeta function ζ G ( s ) of a split reductive group scheme G of rank r with N...
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 ...
A Milnor-Thurston type dynamical zeta function ζ L ( Z ) is associated with a family of maps of the interval (1, 1). Changing the direction of time produces a new zeta function ζ′ L ( Z ). These zeta functions satisfy a functional equation ζ L ( Z ) ζ′ L ( εZ )= ζ...
豆瓣评分 目前无人评价 目录· ··· Chapter Ⅰ.THE FUNCTION ζ(s) AND THE DIRICHLET SERIESRELATED TO IT Chapter Ⅱ.THE ANALYTIC CHARACTER OF ζ(s) AND THE FUNCTIONAL EQUATION Chapter Ⅲ.THE THEOREM OF HADAMARD AND DE LA VALLEE POUSSIN AND ITS CONSEQUENCES Chapter...
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...