The motivation behind this question is somewhat similar to that of the tricky project launched by Gowers et al, but is certainly a specialization. My work tends to rely on both exact formulae and analytic techniques, most notably from complex analysis and Fourier analysis. It appears that search engines and fora like this is a great way to discover useful exact formulae that at least partially fit the bill, but analytic techniques, especially those magic ones like stationary phase approximation, are harder to come by, perhaps due to their subtle and amorphous nature. Thus there seems a genuine need to enumerate some of the most exemplary articles that exploit these techniques, from which I can draw inspirations.

Note that I am specifically interested in getting bounds, as opposed to exact formulae. So for instance Riemann's use of complex analytic continuation to derive the functional equation would not count, even though one could argue exact formulae are a special case of bounds. An example that helped me move forward is Nazarov's extension of Remez's inequality. Another is Lucia's solution to this problem of mine, as well as the paper he cited.

Not to give the wrong impression, I do think most ground-breaking results, including bounds, are proved by exploiting idiosyncratic properties pertaining to the problem itself. These could be some exact formulae, some symmetries, or some geometric interpretation. However it has become clear to me that analytic number theorists often have an edge in situations involving bounds of algebraic formulae.

The motivation behind this question is somewhat similar to that of the tricky project launched by Gowers et al, but is certainly a specialization. My work tends to rely on both exact formulae and analytic techniques, most notably from complex analysis and Fourier analysis. It appears that search engines and fora like this is a great way to discover useful exact formulae that at least partially fit the bill, but analytic techniques, especially those magic ones like stationary phase approximation, are harder to come by, perhaps due to their subtle and amorphous nature. Thus there seems a genuine need to enumerate some of the most exemplary articles that exploit these techniques, from which I can draw inspirations.

Note that I am specifically interested in getting bounds, as opposed to exact formulae. So for instance Riemann's use of complex analytic continuation to derive the functional equation would not count, even though one could argue exact formulae are a special case of bounds. An example that helped me move forward is Nazarov's extension of Remez's inequality. Another is Lucia's solution to this problem of mine, as well as the paper he cited.

Not to give the wrong impression, I do think most ground-breaking results, including bounds, are proved by exploiting idiosyncratic properties pertaining to the problem itself. These could be some exact formulae, some symmetries, or some geometric interpretation. However it has become clear to me that analytic number theorists often have an edge in situations involving bounds of algebraic formulae.