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This is not a very detailed answer, but I can give you an idea how to solve these problems. For the first equality you can deal with the coprimality condition using Mobius inversion. Then it is simpl …

This problem has been considered by a few authors. Versions for $a$ and $b$ rational were considered by Hooley and Guo (independently). See Hooley - On ternary quadratic forms that represent zero. …

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 …

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 …

If you havn't done so already, you might find it useful to look at the proof of theorem 12.2 on the divisor problem in Titchmarsh - The theory of the Riemann zeta function. Here, he goes through a det …

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. …

For a multiplicative function $f$ and $x>0$ let $$S_f(x)= \sum_{n \leq x} f(n).$$ Studying sums of this type is a favourite pastime of analytic number theorists. I'm trying to understand what kind of …

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 …

Yes there is quite a bit in the literature on this problem. Apparently it was first solved by Bernays in his 1912 PhD thesis under Landau. The density is as expected (namely proportional to $x/\sqrt{\ …

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 …

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 …

The circle method is a very powerful and versatile tool which can be made to work in a wide range of situations. It certainly works over number fields, see for example Skinner - Forms over number fie …

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 …

It is also possible to handle this problem using the theory of frobenian functions, due to Serre. You can find the relevant definitions and proofs in Serre's well-written book "Lectures on $N_X(p)$" ( …

(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 $c_n > 0$. …