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Let $k$ be a number field and $V$ a non-trivial irreducible Artin representation over $k$. Consider the associated Artin $L$-function with corresponding Euler product decomposition $L(V,s)= \prod_v L_ …

Yes. See the paper: Browning, T. D.; Vishe, P. Cubic hypersurfaces and a version of the circle method for number fields, Duke Math. J. 163 (2014), no. 10, 1825–1883, doi:10.1215/00127094-2738530, arX …

Mertens' Theorem states that $$\sum_{p \leq x}\frac{1}{p} = \log \log x + M + O(1/\log x).$$ This is weaker than the prime number theorem; in fact according to the Wikipedia page, the prime number the …

The "best" way to deal with quadratic forms using the circle method is via Heath-Brown's delta symbol method. You can read about this in detail in the paper: Heath-Brown - A New Form of the Circle Met …

Actually the usual Chebotarev density theorem in the Galois case can also be applied to the non-Galois case. For example, consider a non-Galois cubic extension $K=\mathbb{Q}[x]/(f)$. I claim that the …

(Upgrading comments to answer.) Counting the number of $D$ for which the equation has a rational solution is a fairly classical problem. This is asymptotic to $c_nD/(\log D)^{1/2}$ for some $c_n > 0$. …

Let $F_n$ be the Fibonacci sequence and $\chi$ a non-principal primitive Dirichlet character. Does there exist $n$ such that $\chi(F_n) \neq 0,1$? One way to prove this would be to obtain non-trivial …

Let $\alpha$ be irrational. A famous theorem of Vinogradov says that $\{ \alpha p\}$ is equidistributed in $[0,1]$ as $p$ runs over all primes. Let $a,q$ be natural numbers with $\gcd(a,q) = 1$. Then …

In D. J. Newman's paper A simple analytic proof of the prime number theorem published in The American Mathematical Monthly 87 (1980) 693-696, Newman proved a result of this type which may also be …

Serre deals with problems of this type in the paper: Serre -Divisibilité de certaines fonctions arithmétiques The fact you want should follow from the results in Sections 1 and 2. Alternatively, th …

Yes this is the fact that for a non-trivial irreducible Artin character, the associated Artin L-function is holomorphic and non-zero on $\rm{re}\, s \geq 1$. For the trivial character, one just obtai …

The first problem is completely solved in the paper: Tim Browning - The divisor problem for binary cubic forms. J. Théorie Nombres Bordeaux 23 (2011), 579-602. The method is to change the order of …

Recall that an integer $n$ is called squareful if for every prime $p$ with $p \mid n$, we also have $p^2 \mid n$. Any squareful number can be written uniquely as $n= x^2 y^3$ where $y$ is squarefree. …

This should be asymptotic to an expression of the form $$\frac{cX^2}{\log X}$$ as $X \to \infty$, for some $c > 0$ (this should be interpreted as $(X/\sqrt{\log X}\,)^2$). Proving an upper bound of th …

This result has been generalised a fair bit. The main generalisation is that you can replace congruence conditions by so-called "Frobenian conditions", namely conditions of the type which arise in the …