Let

$$

H (s) = frac {1} {2} s (1-s) pi ^ {- s / 2} Gamma left ( frac {s} {2} right).

$$

Using Stirling's approximation for Gamma function, I would like to prove that

$$

frac {H (1/2 + it) overline {H} (1/2 + it + iu)} { left | H (1/2 + it) overline {H} (1/2 + it + iu) right |} = left ( frac {2 pi} {t} right) ^ {iu / 2} left (1 + mathcal {O} left ( frac {u ^ 2 + 1} {T} right) right)

$$

or $ T <t <2T $ and $ | u | leq Delta $. Do you have an idea of how to show it?

I guess I should use Stirling's approximation

$$

ln Gamma (s) = (s-1/2) ln ss + frac {1} {2} ln 2 pi + sum_ {m = 1} ^ { infty} frac {B_ { 2m}} {2m (2m-1) s ^ {2m-1}}

$$

What I thought I could work is: since we normalize our function to estimate, we can use the fact that

$$

z = | z | e ^ {i cdot arg (z)}

$$

so what we want to estimate is basically

$$

e ^ {i cdot arg (H (1 / + it) overline {H} (1/2 + it + iu))}}.

$$

To do this, we use it $ arg (z) = Im ( log z) $, So $ arg ( Gamma (s)) = Im ( ln Gamma (s)) $ for which we use Stirling's approximation.

The contribution of the non-gamma factor is easier to estimate and should be

$$

( pi) ^ {iu / 2}

$$

All that remains is to estimate the gamma contribution.

To this end, we use Stirling's approximation (in an answer to this question, there are many useful approximations) to get

$$

arg left ( Gamma ( frac {1} {4} + i frac {t} {2})) right) = Im left[left(frac{1}{4}+ifrac{t}{2}-frac{1}{2}right)ln(frac{1}{4}+ifrac{t}{2})-frac{1}{4}-ifrac{t}{2}+frac{1}{2}ln(2pi)+mathcal{O}left(frac{1}{t}right)right]

$$

So

$$

arg left ( Gamma ( frac {1} {4} + i frac {t} {2})) right) = left[frac{tln(1/16+t^2/4)}{4}-frac{1}{4}arctan(2t)-frac{t}{2}+mathcal{O}(1/t)right]

$$

If I only use the first term of such an expansion, I get

$$

e ^ {i cdot arg left ( Gamma ( frac {1} {4} + i frac {t} {2})) right)} sim left ( frac {1} {16} + frac {t ^ 2} {4} right) ^ {it / 4}

$$

and similarly

$$

e ^ {i cdot arg left ( Gamma ( frac {1} {4} -i frac {t} {2} -i frac {u} {2} {}} right)} sim left ( frac {1} {16} + left ( frac {t} {2} + frac {u} {2} right) ^ 2 right) ^ {- i (t + u) / 4}

$$

So, I think that it remains to be proven, that is that

$$

left ( frac {1} {16} + frac {t ^ 2} {4} right) ^ {it / 4} cdot left ( frac {1} {16} {16} + left ( frac {t} {2} + frac {u} {2} right) ^ 2 right) ^ {- i (t + u) / 4} = left ( frac {2} {t} right) ^ {iu / 2} left (1 + mathcal {O} left ( frac {u ^ 2 + 1} {T} right) right)

$$

and that all the additional terms in the expansion series of $ ln Gamma (s) $ also go into the error term.

Thank you in advance for any help!