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Let d be a squarefree integer. We prove thatconverges to zero, where n ′ is the product of prime divisors of n with (The convergence (i) leads to a proof that asymptotically half the squarefree ideals have an even number of prime ideal factors (analogue of Ramanujan's assertion).

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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)

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