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Let $D(s)={\sum }_{p\le X}{p}^{-s}$, a Dirichlet polynomial supported on primes. The cosine sum is $(D(\sigma +it)+\overline{D(\sigma +it)})/2$, so its $j$th power is a finite linear combination of $D(\sigma +it{)}^k\overline{D(\sigma +it{)}^{j-k}}$, $0\le k\le j$.

The Dirichlet polynomial $D(s{)}^k$ is supported on integers that are products of exactly $k$ primes, counted with multiplicity. Each coefficient is at most $k!$ by unique prime factorization; for $k=0$ the only coefficient is the one at $1$. As $j$ is odd, $k\ne j-k$, so no integer can appear in both supports. Thus every diagonal term vanishes in the mean value of Dirichlet polynomials from part (a).

Both supports lie in $[1,X^j]$. The first error bound in part (a), with that common length, now gives for each termSumming the finitely many terms proves the odd moment of a prime cosine sum estimate:The argument holds for all real $\sigma$ and all $T\ge 0$; no cancellation estimate involving $T$ is needed once the diagonal is absent.