In this article, we apply one fixed point theorem in the setting of b-metric-like spaces to prove the existence of solutions for one type of Caputo fractional differential equation as well as the existence of solutions for one integral equation created in mechanical engineering.

In this manuscript, we initiate the concept of rectangular $\alpha$-$G$-admissible mappings with respect to $\beta$ and we consider related type contractions in the setting of $G$-metric spaces. We provide some fixed point results. Also, some examples are given to support the obtained results.

In this paper we study the chebyshev and interpolation error for a real valued generalized biaxisymmetric Potential (GBASP) which is regular in the open hyper sphere about the origin. The lower (p, q) order and lower generalized (p, q) -type have been characterized in terms of these approximation errors.

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Let $K=Q(\sqrt{d})$ be a quadratic field with discriminant $d$. It is shown that $\sum\limits_{(\frac{d}{p})=+1,_{p~ prime}}\frac{1}{p}$ and $\sum\limits_{(\frac{d}{q})=-1,_{q~ prime}}\frac{1}{q}$ are both divergent. Two different approaches are given to show the divergence: one using the Dedekind Zeta function and the other by Tauberian methods. It is shown that these two divergences are equivalent. It is shown that the divergence is equivalent to $L_{d}(1)\neq 0$(de la Vall\'{e}e Poussin's Theorem).We prove that the series $\sum\limits_{(\frac{d}{p})=+1,_{p~ prime}}\frac{1}{p^{s}}$ and $\sum\limits_{(\frac{d}{q})=-1,_{q~ prime}}\frac{1}{q^{s}}$ have singularities on all the imaginary axis(analogue of Landau-Walfisz theorem)