539,355 research outputs found
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 ,
we show that for all even number \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 , and where and
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 , If , then for each prime .Comment: 6 pages; several corrections mad
A new upper bound for the prime counting function
In this paper we bring to light an upper bound for the prime counting
function 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 for any given
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