We show the equivalence of the finite expression of Deninger's `R-function' at the rational arguments and the Kronecker limit formula on the line of our past study on the Gauss formula for the digamma function and the Dirichlet class number formula. Here the Gauss formula and the class number formula will be replaced by its analogue for the `R-function' and by the Kronecker limit formula or rather a closed form for the derivative of the Dirichlet L-function respectively. We also make a systematic study of the `$R_k$-function' by appealing to the Lipschitz-Lerch transcendent in which there is the vector space structure built in of these special functions.
We discuss the mean values of the Riemann zeta-function $\zeta(s)$, and analyze upper and lower bounds for $$\int_T^{T+H} \vert\zeta(\frac{1}{2}+it)\vert^{2k}\,dt~~~~~~(k\in\mathbb{N}~{\rm fixed,}~1<\!\!< H \leq T).$$ In particular, the author's new upper bound for the above integral under the Riemann hypothesis is presented.
We study three special Dirichlet series, two of them alternating, related to the Riemann zeta-function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at poles). These values are given in terms of Bernoulli and Euler numbers.