Curious conjectures on the distribution of primes among the sums of the first $2n$ primes. (arXiv:1804.04198v1 [math.NT])

Let $p_n$ be $n$th prime, and let $(S_n)_{n=1}^\infty:=(S_n)$ be the sequence of the sums of the first $2n$ consecutive primes, that is, $S_n=\sum_{k=1}^{2n}p_k$ with $n=1,2,\ldots$. Heuristic arguments supported by the corresponding computational results suggest that the primes are distributed among sequence $(S_n)$ in the same way that they are distributed among positive integers. In other words, taking into account the Prime Number Theorem, this assertion is equivalent to \begin{equation*}\begin{split} &\# \{p:\, p\,\,{\rm is\,\,a\,\, prime\,\, and}\,\, p=S_k \,\,{\rm for\,\,some\,\,} k \,\,{\rm with\,\,} 1\le k\le n\} \sim & \# \{p:\, p\,\,{\rm is\,\,a\,\, prime\,\, and}\,\, p=k \,\,{\rm for\,\,some\,\,} k \,\,{\rm with\,\,} 1\le k\le n\}\sim\frac{\log n}{n}\,\, {\rm as}\,\, n\to\infty, \end{split}\end{equation*} where $|S|$ denotes the cardinality of a set $S$. Under the assumption that this assertion is true (Conjecture 3.3), we say that $(S_n)$ satisfies the Restricted P 查看全文>>