# Tagged Questions

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### Positive Integer points of $f(x)=\frac{1}{c-\frac{1}{x}}$, where c is fixed

So I am looking for the integer solutions of $f(x)=\frac{1}{c-\frac{1}{x}}$ for fixed $c\in \mathbb{Q}$ i.e. points $(x,f(x))\in \mathbb{N}\times \mathbb{N}$. (The c equals $\frac{4}{n}-\frac{1}{k}$ ...
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### Generalizations of results on the sum of divisors function over $\mathbb{Q}$ to number fields

Consider the sum of divisor function $$\sigma_0(n) = \sum_{d\mid n} 1.$$ This is known to satisfy $\sum_{n\leq x} \sigma_0(n) = (x\log x)+2\gamma x+\mathcal{O}(\sqrt{x})$. If, instead, we shift the ...
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### Analytic number theory books after Apostol

I am planning to learn some classical results on analytic number theory. I have read Apostol's Introduction to Analytic Number Theory, but nothing about algebraic number theory. Can anyone recommend ...
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### On prime factors of $n^2+1$

It is a well-known conjecture that there are infinitely many primes of the form $n^2+1$. However, there are weaker results that one can prove. For example, There are infinitely many positive ...
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### What motivated Rademacher's contour along the Ford circles when he used the circle method?

After Ramanujan and Hardy found the infinite sum representation of the partition function $p(n)$, Rademacher went about simplifying their proof; the form generally seen involves integrating ...
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### What is known about meromorphic functions agreeing with $\pi(n)$?

Let $f$ be a meromorphic function in some region containing the positive real axis such that $f(n) = \pi(n)$ for all but finitely many positive integers $n$, where $\pi(n)$ is the number of primes ...
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### Two Dirichlet's series related to the Divisor Summatory Function and to the Riemann's zeta-function, $\zeta(s)$

Considering the $\textit{Divisor Summatory Function}$, $D(n)$, defined as $$D(n) = \sum_{k=1}^{n}d(k) ,$$ where $$d(n) = \sum_{k|n}^{n}1.$$ One can observe the following pattern in the values of ...
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### Analytic number theory primer — sequences and series

For a book like Titchmarsh, or Iwaniec and Kowalski, it seems a thorough knowledge of asymptotics is a prerequisite. What are good books for training oneself in such manipulation of asymptotics, ...
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### Convergence of prime series

Where can I read about convergence of series constituted of prime number such as the following: $$\sum_p \frac{1}{p (\log{p})^\alpha}\;?$$ How does convergence depend on $\alpha$?
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### Zeta functions in Chebychev's Prime Number theory

In two papers from 1848 and 1850, the Russian mathematician Pafnuty L'vovich Chebyshev attempted to prove the asymptotic law of distribution of prime numbers. His work is notable for the use of the ...
### Ramanujan's First Letter to Hardy and the Number of $3$-Smooth Integers
A positive integer is $B$-smooth if and only if all of its prime divisors are less than or equal to a positive real $B$. For example, the $3$-smooth integers are of the form $2^{a} 3^{b}$ with ...