4 problems

Consecutive-prime conjecture. For every fixed $\varepsilon>0$,

Let $k_0$ be a positive integer and let $c>1$ be non-integral. The Piatetski-Shapiro sequence is given by the integer parts $[n^c]$ for positive integers $n$. Consecutive Piatetski…

Let $f$ be a cusp form for the full modular group, let $1<c<8/7$, and let $\mathbbm{P}$ denote the set of primes. Write $\lambda_f(n)$ for the normalized Fourier coefficients of…

For $c>1$, let $[x]$ denote the integral part of $x$, and let … count the integers $n$ whose Piatetski–Shapiro value is prime. Piatetski–Shapiro prime number conjecture. For every…