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Treat an explicit but messy function abstractly

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Quick description

Sometimes, it is not a blessing to have an explicit description of an integrand, because the explicit structure is not easy to exploit. This is particularly common in analytic number theory, when the integrand involves an explicit but analytically complicated object (such as the prime numbers). In such cases, it can clarify things to try abstraction: replace an explicit but messy function by an abstract function f(x), and write down all the potentially useful facts about f that one can read off from the explicit description. Then, forget about that explicit description, and focus on whether one can get a decent bound for the original expression just using the abstract facts about f.


Example 1

(Need to come up with a good example of this. The Green-Tao theorem on long arithmetic progressions in primes is one, but is certainly not the most elementary such example! Any ideas?)

General discussion


Trigonometric functions

I think there should exist some examples involving trigonometric (or inverse trigonometric) functions, where we forget about the actual evaluations of the functions but we use their main properties, like periodicity, a sine is a cosine with a phase change, bounds between -1 and 1, sin^2(x)+cos^2(x)=1, dsin(x)/dx=cos(x), etc.

More generally, I think we almost always do this abstraction: for us, mathematical objects are mostly what their main properties say they are; for example, we just associate every function with their properties, not with their sets of values (e.g., exponentials are the functions that verify the rules of exponentiation, are always positive, eigenfunctions for the derivative operator, etc). And when we define a family of "well-known" functions (for example, "elemental" functions and operations to represent an undefined integral), we are just restricting ourselves to functions whose properties we know well. I think this main idea could be the subject of another, even more abstract article about "Forget the actual object and stand by its properties" (not just for functions).

(Actually, I also want to write an article some day titled "Just check the definition", because sometimes it shows more fruitful to just stick to the definition of the object in question rather than trying characterizations and well-known properties of it. If you read this and feel like writing that article, feel free to do so!).