Curious conjectures on the distribution of primes among the sums of the first $2n$ primes

TL;DR

This paper explores the distribution of primes among sums of the first 2n primes, proposing conjectures supported by heuristic and computational evidence that suggest primes are distributed similarly in this sequence as among natural numbers.

Contribution

It introduces the concept of the sequence of sums of the first 2n primes and formulates conjectures about prime distribution within this sequence, extending prime number theory insights.

Findings

Proposes the Restricted Prime Number Theorem for the sequence (S_n)

Conjectures primes are distributed among (S_n) similarly to natural numbers

Provides heuristic and computational support for these conjectures

Abstract

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 Prime Number Theorem. Motivated by this, in Sections 1 and 2 we give some definitions, results and examples concerning the generalization of the prime counting function $\pi(x)$ to increasing positive integer sequences. The remainder of the paper (Sections 3-7) is devoted to the study of mentioned sequence $(S_n)$. Namely, we propose several conjectures and we prove their consequences concerning the distribution of primes in the sequence $(S_n)$. These conjectures are mainly motivated by the Prime Number Theorem, some heuristic arguments and related computational results. Several consequences of these conjectures are also established.