Odd moment of a prime cosine sum
= Odd moment of a prime cosine sum
For odd $j$, expanding $(\sum_{p\le X}p^{-\sigma}\cos(t\log p))^j$ gives products with different numbers of <prime factors> on the two sides. <Unique prime factorization> rules out every diagonal equality. The integral over $[T,2T]$ is therefore $O_j(X^j(\sum_{n\le X^j}n^{-\sigma})^2)$.