The concept of majorizing sequences is employed to provide a convergence analysis for Newton-like methods in a Banach space. We use Hölder and center-Hölder continuity assumptions on the Fréchet-derivative of the operators involved. This way we show that our convergence conditions are weaker; error bounds on the distances involved finer and the location of the solution more precise than in earlier results.

We prove the Kloosterman-Spectral sum formula for $\mathrm{PSL}_{2}(\mathbb{Z}[i])\backslash \mathrm{PSL}_{2}(\mathbb{C})$, and apply it to derive an explicit spectral expansion for the fourth power moment of the Dedekind zeta-function of the Gaussian number field.

We prove, in standard notation from spectral theory, the asymptotic formula $(B > 0)$ $$\sum_{\kappa_{j}\leq T} \alpha_{j} H_{j}(\frac{1}{2}) = \left(\frac{T}{\pi}\right)^2 - BT \log T + O(T(\log T)^{1/2}),$$ by using an approximate functional equation for $H_{j}(\frac{1}{2})$ and the Bruggeman-Kuznetsov trace formula. We indicate how the error term may be improved to $O(T(\log T)^{\varepsilon})$.

We prove that every function in the extended Selberg class $\mathcal{S}^\sharp$ can be factored into primitive functions. The proof is definitely more involved than in the case of the Selberg class $\mathcal{S}$.

In this paper we shall consider propelties of the eigenfunctions of the Laplace-Beltrami operator $\Delta \rho$ and properties of its gradient for a proper domain $D$ with a conformal metric, which density is equal to the reciprocal value of a defining function $\rho(x)$ for this domain i.e. $ds = \rho^{-1}(x)\vert dx \vert$.

A relationship between square-free divisors of an integer and zeros of the Riemann zeta-function, which is more explicit than the classical formula, is presented and discussed.