# Analytical Theory of Numbers – Integral Product Estimation of Terms \$ cos (t log p) \$

I would like to prove the following proposition of A. Harper's document "Strong conditional upper bound for moments of Riemann Zeta's function"

Proposal.
Let $$T$$ to be tall and leave $$n = p_1 ^ { alpha_1} cdots p_r ^ { alpha_r}$$, or $$p_i$$ are distinct prime numbers and $$alpha_i in mathbb {N}$$ for everyone $$i$$. then
$$int_T ^ {2T} prod_ {i = 1} ^ r cos (t log p_i) ^ alpha_i dt = Tf (n) + mathcal {O} (n)$$
or $$f (n) = 0$$ if one of the exhibitors $$alpha_i$$ is strange, and if not
$$f (n) = prod_ {i = 1} ^ r frac {1} {2 ^ {{alpha_i}} frac { alpha_i!} {(( alpha_i / 2)!) ^ 2}$$

He says that there is a slight variant of Lemma 4 from Radzwill's article on Selberg's central limit theorem, noting that the error is estimated there. rather generously and that it can actually be taken into account. $$mathcal {O} (n)$$.
The proof continues as follows: consider the prime numbers $$q_1, dots q_k leq x$$ with $$q_1 cdots q_k = p_1 ^ { alpha_1} cdots p_r ^ { alpha_r} = n$$. Since $$cos (t) = frac {e ^ {it} + e ^ {- it}} {2}$$ we have
$$cos (t log p_i) ^ { alpha_i} = frac {1} {2 ^ {{alpha_i}} (e ^ {it log p_i} + e ^ {- it log p_i}) ^ { alpha_i} = frac {1} {2 ^ { alpha_i}} { alpha_i choose alpha_i / 2} + sum _ { substack {0 leq ell leq alpha_i \ ell neq alpha_i / 2}} frac {1} {2 ^ { alpha_i}} { alpha_i choose ell} e ^ {i ( alpha_i-2 ell) t log p_i}$$
Therefore
$$int_T ^ {2T} prod_ {i = 1} ^ r cos (t log p_i) ^ alpha_i dt = int_T ^ {2T} prod_ {i = 1} ^ r left ( frac {1 } {2 ^ { alpha_i}} { alpha_i choose alpha_i / 2} + sum _ { substack {0 leq ell leq alpha_i \ ell neq alpha_i / 2}} frac {1} {2 ^ { alpha_i}} { alpha_i choose ell} e ^ {i ( alpha_i-2 ell) t log p_i} right)$$
which equals
$$Tf (n) + sum _ { substack {0 leq ell_i leq _ { alpha_i} \ text {for all} 1 leq i leq r \ text {not all} ell_i = alpha_i / 2}} int_T ^ {2T} prod_ {i = 1} ^ r frac {1} {2 ^ { alpha_i}} { alpha_i choose ell} e ^ {i ( alpha_i 2 ell) t log p_i} dt$$
He then says that the number of terms of the sum is $$leq ( alpha_1 + 1) cdots ( alpha_r + 1) leq 2 ^ { alpha_1} cdots 2 ^ { alpha_r} = 2 ^ k$$ or $$k = alpha_1 + cdots alpha_r$$. Moreover, each of these terms is of the form
$$int_T ^ {2T} gamma e ^ {it ( beta_1 log p_1 + cdots beta_r log p_r)} dt$$
with $$| gamma | 1$$ and integers $$| beta_i | leq alpha_i$$. He then claims that $$beta_1 log p_1 + cdots beta_r log p_r gg x ^ {- k}$$ and therefore, using this $$int e ^ {cit} = – frac {ie ^ {cit}} {c} + constant$$ and that $$| gamma | 1$$ we get that
$$int_T ^ {2T} gamma e ^ {it ( beta_1 log p_1 + cdots beta_r log p_r)} dt ll x ^ k$$
so the sum actually contributes to the more $$mathcal {O} (2 ^ kx ^ k)$$.
My question is: how can I go from $$mathcal {O} (2 ^ kx ^ k)$$ at $$mathcal {O} (n)$$?
I guess the link $$( alpha_1 + 1) cdots ( alpha_r + 1) leq 2 ^ { alpha_1} cdots 2 ^ { alpha_r} = 2 ^ k$$ could be improved. I do not know how to show the link
$$beta_1 log p_1 + cdots beta_r log p_r gg x ^ {- k}$$
maybe that can be improved too.
The only other limit used is $$| gamma | 1$$ or $$gamma$$ is the product of form factors $$frac {1} {2 ^ { alpha_i}} { alpha_i choose ell}$$. Using Stirling's approximation for the factorial that I've
$${ alpha_i choose ell} leq { alpha_i choose alpha_i / 2} sim frac { sqrt {2}} { sqrt { pi alpha_i}}$$