Analytic number theory is often quite useful in questions on rational points on varieties. For example, using the circle method, weak approximation gives formulas of the type
What kind of asymptotic formulae can we expect when weak approximation fails?
We have the following ``theorem'': There is a modification of the Hardy-Littlewood formula in which
Theorem. [H-B, Moroz] Let be coprime, with . Then the surface
To prove this, there are two ingredients. First, a result of Satgé: has a rational point if is prime, (proved by Heegner point construction). Second, takes infinitely many prime values .
In the other direction, analytic number theorists are often interested in rational points on varieties. We take the counting function
This is known for any , and conjectured to be true for any , so it is reasonable to expect.
What about for ? We can show for . This variety has lines in trivial planes of the type , and no other lines if ; what other quadric or low degree curves can be found?
Proposition. [Green 1975] Any curve of genus 0 or in has constant for some , as soon as .
(Here .) This involves meromorphic functions and Nevanlinna theory.
Proposition. [Davenport 1963] Any cubic form in variables has a nontrivial integer zero.
This applies to an arbitrary cubic form; there is a better result for smooth forms with . Define a matrix with entries
Vinogradov's mean value theorem refers to the counting function of the variety defined by the equations
One can easily show for all ; if then is a permutation of , . In fact, for (Vaughan, Wooley).
For , we have .
Applications: Exponential sums, zero-free region for and the error term in the prime number theorem. So this question has several far-reaching implications!
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