529,656 research outputs found

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

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    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

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    Under the assumption that βˆ‘n≀NΞ₯(n)Ξ₯(Nβˆ’n)>0\sum \limits_{n\leq N}\Upsilon(n)\Upsilon(N-n)>0, we show that for all even number N>6N>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>0K>0, and where Ξ₯\Upsilon and Ξ›0\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>6N>6 (Nβ‰ 2p)(N\neq 2p), If βˆ‘p∣Nβˆ‘n≀N/pΞ›0(n)Ξ›0(N/pβˆ’n)>0\sum \limits_{p|N}\sum \limits_{\substack{n\leq N/p}}\Lambda_{0}(n)\Lambda_{0}(N/p-n)>0, then βˆ‘n≀N/pΞ›0(n)Ξ›0(N/pβˆ’n)>0\sum \limits_{\substack{n\leq N/p}}\Lambda_{0}(n)\Lambda_{0}(N/p-n)>0 for each prime p∣Np|N.Comment: 6 pages; several corrections mad

    A new upper bound for the prime counting function Ο€(x)\pi(x)

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    In this paper we bring to light an upper bound for the prime counting function Ο€(x)\pi(x) using elementary methods, that holds not only for large positive real numbers but for all positive reals. It puts a threshold on the number of primes p≀xp\leq x for any given xx
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