# 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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### Change of order of summation.

I feel like an idiot for asking this, so bear my stupidity. I have the sum $\sum_{n\leq N} \sum_{p | n ; \ p \ prime} 1$, and I want to change the order of summation of these two sums I think it ...
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### Prove that von Mangoldt function satisfies $\sum_{n \le x} \Lambda(n) \lfloor{\frac{x}{n}}\rfloor= x \ln(x)-x+O(\ln x)$

The picture above is what I have, which has an error that is too big.
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### Regarding definitions of modular functions and modular forms

I am self studying Analytic number theory from Apostol Dirichlet series and modular functions in number theory and I have a doubt regarding defintions of modular forms and modular functions. My ...
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### Elementary number theory in modular functions

I am self studying Apostol Dirichlet series and Modular functions in number theory and I am struck on Ch -2 Problem 17. It is a problem of elementary number theory but I am not able to think about it. ...
On Wikipedia, the following formula is given: $$\frac {F^\prime(s)}{F(s)} = - \sum_{n=1}^\infty \frac{f(n)\Lambda(n)}{n^s}$$ I understand the proof for the zeta-function, but not for the Dirichlet ...