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In some respects the theory of divergent series is still a very important part of number theory. A large part of number theory concerns the study of Dirichlet series $$f(s) = \sum_{n=1}^\infty \frac …

This is false. Take $X$ to be a smooth plane conic without a rational point. Consider the surface $$S = X \times X.$$ Over the algebraic closure this becomes isomorphic to $\mathbb{P}^1 \times \mathbb …

Shparlinski has done a lot of work on problems like this, in the more general setting of linear recurrence sequences. (Your sequence of Mersenne numbers is such a sequence). I'd recommend looking at C …

Let $\pi:X \to S$ be a morphism of schemes (I can assume that $\pi$ is sufficiently nice, e.g. proper and flat, but certainly not smooth). Does there exist a scheme $I_{X/S}$ which parametrises th …

Recall that the radical of an integer $n$ is defined to be $\operatorname{rad}(n) = \prod_{p \mid n } p$. For a paper, I need the result that $$\sum_{n \leq x} \frac{1}{\operatorname{rad}(n)} \ll_\v …

I am looking for a reference for the following fact. Let $k$ be a number field and let $S$ be a finite set of places of $k$ of even cardinality. Then there exists a unique conic $C$ over $k$ such …

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 …

In a famous paper Hartshorne - Varieties of small codimension, Hartshorne formulates a conjecture, which roughly says that varieties of small codimension in projective space are complete intersectio …

It is difficult to get far in the modern theory without some algebraic geometry. This is the approach taken in the book: Bjorn Poonen, Rational points on varieties, Graduate Studies in Mathematics 18 …

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 …

A conjecture of Hartshorne states that any vector bundle of a small rank on a projective space of large dimension is split (i.e. isomorphic to a direct sum of line bundles). I would like to know the c …

As explained in the comments, there is no such variety. This is an application of a general set of techniques called "spreading out". You can find a very nice treatment of this in Chapter 3 of the bo …

Here is an example which illustrates that in general things are incredibly hard. Consider the equation: $$x_1^3 + x_2^3 + x_3^3 = h.$$ Here are two famous open problems: For which $h$ does this eq …

These are called Dirichlet polynomials. They arise in many places in analytic number theory. For example, in approximate functional equations of $L$-functions.