Zeta(n) via hyperbolic functions

关于广义Dirichlet beta与Riemann zeta函数及beta值和zeta值的Ramanujan型公式

On the generalized Dirichlet beta and Riemann zeta functions and Ramanujan-type formulae for beta and zeta values

本文利用双曲正割与余割函数的幂积分定义广义Dirichlet beta和Riemann zeta函数，建立其函数方程，并由奇整数zeta值的Ramanujan恒等式导出新的Ramanujan型公式。

A More Accurate Half-Discrete Hardy-Hilbert-Type Inequality with the Best Possible Constant Factor Related to the Extended Riemann-Zeta Function

By the method of weight coefficients, techniques of real analysis and Hermite-Hadamard's inequality, a half-discrete Hardy-Hilbert-type inequality related to the kernel of the hyperbolic cosecant function with the best possible constant factor expressed in terms of the extended Riemann-zeta function is proved. The more accurate equivalent forms, the operator expressions with the norm, the reverses and some particular cases are also considered.

Determinations of rational Dedekind-zeta invariants of hyperbolic manifolds and Feynman knots and links

We identify 998 closed hyperbolic 3-manifolds whose volumes are rationally related to Dedekind zeta values, with coprime integers $a$ and $b$ giving $a/b vol(M)=(-D)^{3/2}/(2π)^{2n-4} (ζ_K(2))/(2ζ(2))$ for a manifold M whose invariant trace field $K$ has a single complex place, discriminant $D$, degree $n$, and Dedekind zeta value $ζ_K(2)$. The largest numerator of the 998 invariants of Hodgson-Weeks manifolds is, astoundingly, $a=2^4\times23\times37\times691 =9,408,656$; the largest denominator is merely b=9. We also study the rational invariant a/b for single-complex-place cusped manifolds, …

Integrals in Gradshteyn and Ryzhik: Hyperbolic and trigonometric function

The well known table of Gradshteyn and Ryzhik contains indefinite and definite integrals of both elementary and special functions. We give proofs of several entries containing integrands with some combination of hyperbolic and trigonometric functions. In fact, we occasionally present an extension of such entries or else give alternative evaluations. We develop connections with special cases of special functions including the Hurwitz zeta function. Before concluding we mention new integrals coming from the investigation of certain elliptic functions.

关于一类 Berndt 型积分及相关 Barnes 多重 Zeta 函数

On a Class of Berndt-type Integrals and Related Barnes Multiple Zeta Functions

本文利用围道积分、留数定理及 Jacobi 椭圆函数的级数技术，将仅含双曲余弦的 Berndt 型积分化为规范 Ramanujan 型级数，得到 Γ(1/4) 与 π^{-1/2} 的有理多项式显式表达式，并关联 Barnes 多重 zeta 函数。

Some infinite series involving the Riemann zeta function

This paper considers some infinite series involving the Riemann zeta function.