On elementary estimates for sum of some functions in certain arithmetic progressions

In this paper we establish, by elementary means, estimates for the sum of some functions in certain arithmetic progressions.Comment: typos correcte

A remark on the strong Goldbach conjecture

Under the assumption that $\sum \limits_{n\leq N}\Upsilon(n)\Upsilon(N-n)>0$, we show that for all even number $N>6$ \begin{align} \sum \limits_{n\leq N}\Upsilon(n)\Upsilon(N-n)=(1+o(1))K\sum \limits_{p|N}\sum \limits_{\substack{n\leq N/p}}\Lambda_{0}(n)\Lambda_{0}(N/p-n)\nonumber \end{align}for some constant $K>0$, and where $\Upsilon$ and $\Lambda_{0}$ denotes the master and the truncated Von mangoldt function, respectively. Using this estimate, we relate the Goldbach problem to the problem of showing that for all $N>6$ $(N\neq 2p)$, If $\sum \limits_{p|N}\sum \limits_{\substack{n\leq N/p}}\Lambda_{0}(n)\Lambda_{0}(N/p-n)>0$, then $\sum \limits_{\substack{n\leq N/p}}\Lambda_{0}(n)\Lambda_{0}(N/p-n)>0$ for each prime $p|N$.Comment: 6 pages; several corrections mad

The master function and applications

In this paper we introduce a function that is neither additive nor multiplicative, and is somewhat akin to the Von Mangoldt function. As an application we show that \begin{align}\sum \limits_{p\leq x/2}\frac{\pi(p)}{p}\geq (1+o(1))\log \log x\nonumber \end{align}as $x\longrightarrow \infty$, and \begin{align}\sum \limits_{p\leq x/2}\theta(x/p)\bigg(\frac{\log x}{\log p}-1\bigg)^{-1}\ll x\log \log x \nonumber \end{align} where $p$ runs over the primes.Comment: 5 page