# Tagged Questions

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### Squares modulo 2^n

How many squares are there modulo $2^n$? If we would deal with $p^n$, where p an odd prime, then we could use Hensel's Lemma, which clearly doesn't work with $2^n$.
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### Error term of the prime number theorem in arithmetic progressions

It is known that if $(a, q)$ and $q\le (\ln x)^N$, then the following is true $$\sum_{k\le x, k\equiv a\pmod{q}}\Lambda(k) = \frac{x}{\phi(q)} + O(x\exp(-C\sqrt{\ln x}))$$ where $C$ depends only on ...
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### Prime Counting: Relationship between Chebyshev's function and the Prime counting function

How do I show that if $\psi(x)=x+O(x^{1/2}\log^2(x))$ then $\pi(x)=\int_2^x \frac{dt}{logt} + O(x^{1/2}\log x)$ Where $\psi(x)$ is Chebyshev's second function and $\pi(x)$ is the prime counting ...
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### Numerical verification of the ternary Goldbach conjecture

In his proof of the ternary Goldbach conjecture, H.A. Helfgott says that it has been verified that every odd number less than $N_0 = 10^{30}$ is the sum of at most 3 primes. How would one verify this ...
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### Proof of convergence of $L'\left(1,\chi\right)$

can someone give me a good reference for a clear proof of the convergence of $L'\left(1,\chi\right)$, $\chi$ real-valued, non-principal Dirichlet character? Thanks in advance.
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### Questions about the proof that every odd integer is the sum of 5 primes

In http://arxiv.org/pdf/1201.6656.pdf, Tao proved that all odd numbers greater than 1 are the sum of 1, 3, or 5 primes. In page 36-37, he uses the fact that for all $x > 1.1\times10^{10}$, there ...
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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 ...
### On asymptotics for the number of ways to write $2n$ as the sum of two primes
Let $g(n)$ be the number of ways to write $2n$ as the sum of two primes. Define $G(n) = g(2) + \cdot \cdot \cdot + g(n)$. Are there any conjectured or known asymptotics or bounds for $G(n)$? ...