On the divergence of two subseries $\ldots$] {on the divergence of two subseries of $\sum\frac{1}{p}$, and theorems of de La Vall\'{e}e Poussin and Landau-Walfis

Abstract

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)