# Questions tagged [analytic-number-theory]

Questions on the use of the methods of real/complex analysis in the study of number theory.

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### Let $g(k)$ be the greatest odd divisor of $k$ show that $0< \sum_{k=1}^n \frac {g(k)}{k} - \frac {2n}{3} \lt \frac 23$

Prove that for all positive intergers $n$, $$0< \sum_{k=1}^n \frac {g(k)}{k} - \frac {2n}{3} < \frac {2}{3}$$ Where $g(k)$ denotes the greatest odd divisor of $k$. Here's my try: ...
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### Structure of the group of arithmetic functions

This question was originally posted in Elements of finite order in the group of arithmetic functions under Dirichlet convolution. and it goes as follows: Let G be the group consisting of all ...
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### How to prove that below quantity is purely imaginary?

How do I prove that the following quantity is purely imaginary: $$\sum_{0\leq l_1<l_2<l_3<l_4\leq q-1} e^{-2\pi i \frac{(l_1^2-l_2^2+l_3^2-l_4^2)}{q} }$$ where $q$ is an odd number?
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### Fourier transform of the critical line of zeta?

Is there a known expression for the (distributional) Fourier transform of the Riemann zeta function, taken along the critical line? I'd love to say that it's a weighted sum of delta distributions, ...
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