# 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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### Condition for ({$n\alpha$},{$n\beta$}) being dense on [0,1]$\times$[0,1] [duplicate]

What is the condition for ({$n\alpha$},{$n\beta$}) being dense in [0,1]$\times$[0,1]? (Here {a} represents the fraction part of a). If the qustion changes to ({$n\alpha$},{$m\beta$}), then it is ...
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### Inequation in paper from Terence Tao on the Collatz Conjecture

I try to understand Terence Tao's paper on the Collatz Conjecture [1909.03562], but got stuck on page 25. We have $n$ copies of a geometric random variable of mean $2$, denoted by $a_i$ and $a_{[i,j]}$...
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### Why is the following equation true?

This is a fairly simple question but it seems really confusing for me. So basically I am reading the proof of the composition formula of Hecke operators: $$T(m)T(n)=\sum_{d\mid (m,n)}d^{k-1}T(mn/d^2)$$...
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### Asymptotic formula for Prime Numbers

My question is: Obtain an asymptotic estimate for the sum $$S(x)=\sum_{x<p \leq 2 x} \frac{1}{p}$$ with relative error $1 / \log x$ (i.e., an estimate of the form $$S(x)=f(x)(1+O(1 / \log x))$$ ...
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### odd prime factors of the form $10k+1$ of a number of the form $20n-4$

Let, $N= (19 \cdot 29 \cdot 59...\cdot q)^{2} - 5.,$ where the primes numbers of the form $10n+9$ has been considered in the product. Then $N$ is of the form $20n-4$ and has odd prime factors in the ...
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