# Tagged Questions

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

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### Is it something new?

$W(n)$ is the function that counts number of distinct prime divisors of $n$. I have been able to prove for any $m$ consecutive integers starting with $1+a$ with the condition $a\leq (m^2-4m)/4$ , ...
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### We know the asymptotic density of primes. What about the asymptotic density of numbers with n prime factors? [duplicate]

Question in title. When I say n prime factors, I don't mean n distinct prime factors.
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### Simultaneous degree two residues

Given integer $q$ is there many pairs of $x,y\in\Bbb Z$ with $q^{5/6}<x,y<2q^{5/6}$ such that $$q^{4/6}\quad<\quad x^2\bmod q,\quad xy \bmod q,\quad y^2\bmod q\quad<\quad 2q^{4/6}$$ holds?...
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### If $\forall \varepsilon>0\;\;\;\;a(n)\leq C(\varepsilon)\; n^{r+\varepsilon}$ then $\sum_{n=1}^{\infty}\frac{a(n)}{n^s}$ converge

Let $\{a(n)\}_{n\in\mathbb{N}}$ be a sequence of real numbers. I tried to prove that if $$\forall \varepsilon>0\;\;\;\;a(n)\leq C(\varepsilon)\; n^{r+\varepsilon}$$where $C(\varepsilon)$ is a ...
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### Vaughan's identity, a didactic example

I know that Vaughan's identity is one of the methods used in analityc number theory, I would like see an example, I say a simple example of application of this theorem for encourage to study the ...
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### Poisson summation in analytic number theory, an example

I've read a theorem of a course of analytical methods in theory of numbers, and I want to ask to clarify it, since I know this theorem from other context. The theorem is Poisson summation formula (I ...
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### Can you estimate the difference of primes between numerator and denominator?

Let $p_n$ the nth twin prime, it is $p_n$ is a prime number and $2+p_n$ is also a prime. It is well know that Brun's theorem states (unconditionally) that \mathcal{B}=\sum_{n\geq 1}\left(\frac{1}{...
### Number of ways, modulo a prime $p$, to write $n$ as a sum $n = x_1^k + x_2^k + \cdots + x_s^k$
Removing the restriction on $p$, this is known as Waring's problem. The circle method has been used successfully to tackle this. Using analysis, nice estimates can be given. I wonder what analytic ...
Let $N_k$ denote the $k-th$ primorial number. That is, the product of the first $k$ primes and $\phi(n)$ be the Euler totient function. How can one show that there exists a constant $\theta>1$ ...