We study the gaps between primes in Beatty sequences following the methods in the recent breakthrough by Maynard (2015).

Suppose that $f$ is a primitive Hecke eigenform or a Mass cusp form for ${\Gamma }_0\left(N\right)$ with normalized eigenvalues ${\lambda }_f\left(n\right)$ and let $X>1$ be a real number. We consider the sum $${𝒮}_k\left(X\right):=\sum _{n<X}\sum _{n=n_1,n_2,...,n_k}{\lambda }_f\left(n_1\right){\lambda }_f\left(n_2\right)...{\lambda }_f\left(n_k\right)$$ and show that ${𝒮}_k\left(X\right){\ll }_{f,ϵ}{X}^{1-3/\left(2\right(k+3\left)\right)+ϵ}$ for every $k\ge 1$ and $ϵ>0$. The same problem was considered for the case $N=1$, that is for the full modular group in Lü (2012) and Kanemitsu et al. (2002). We consider the problem in a more general setting and obtain bounds which are better than those obtained by the classical result of Landau (1915) for $k\ge 5$. Since the result is valid for arbitrary...

In a letter written to Landau in 1935, Schur stated that for any integer $t>2$, there are primes $p_1<p_2<\cdots <p_t$ such that $p_1+p_2>p_t$. In this note, we use the Prime Number Theorem and extend Schur’s result to show that for any integers $t\ge k\ge 1$ and real $ϵ>0$, there exist primes $p_1<p_2<\cdots <p_t$ such that $$p_1+p_2+\cdots +p_k>(k-ϵ)p_t.$$