On sum of Hecke eigenvalue squares over primes in very short intervals

Abstract: Let $\eta>0$ be a fixed positive number, let $N$ be a sufficiently large number. In this paper, we study the second moment of the sum of Hecke eigenvalues over primes in short intervals (whose length is $\eta \log N$) on average (with some weights) over the family of weight $k$ holomorphic Hecke cusp forms. We also generalize the above result to Hecke-Maass cusp forms for $SL(2,\mathbb{Z})$ and $SL(3,\mathbb{Z}).$ By applying the Hardy-Littlewood prime 2-tuples conjecture, we calculate the exact values of the mean values.