When reading the literature, innumerable examples of Voronoi formulas are found, that is, formulas that take a sum over Fourier coefficients, twisted by some character and controlled by some suitable test function, and spit a different sum over the same coefficients Fourier, twisted by some different characters, and this time controlled by some integral transformation of the test function.

The reason why one wants to do this in practice is that the second sum is somehow better, of course, which in my experience (certainly limited) tends to be reduced to the duration of the second sum that has changed significantly to best.

I will give an example (from Xiaoqing Li Bounds for GL (3) × GL (2) L-functions and GL (3) L-functions, because that is what I have in front of me).

In this case we have the formula GL (3) Voronoi

$$

sum_ {n> 0} A (m, n) e Bigl ( frac {n bar d} {c} Bigr) psi (n) sim sum_ {n_1 mid cm} sum_ {n_2 > 0} frac {A (n_2, n_1)} {n_1 n_2} S (md, n_2; mc n_1 ^ {- 1}) Psi Bigl ( frac {n_2 n_1 ^ 2} {c ^ 3 m} Great R),

$$

where $ psi $ It is a soft and compact test function, $ Psi $ as suggested above is an integral transformation of it, $ A (m, n) $ are Fourier coefficients of (in this case) a form of Maass SL (3), $ (d, c) = 1 $Y $ d bar d equiv 1 pmod {c} $.

(I have omitted many details here, but I think the details are not relevant to my question).

Doing so essentially transforms the $ n $-sum in the $ n_2 $-sum, where, as is evident in the formula, the $ n_2 $-sum has a very different argument in its test function.

What happens in practice now is that, once we reach a point where applying the Voronoi formula is appropriate, we transform the sum and study the integral transformation, mainly by stationary phase analysis to find the length of the new $ n_2 $-sum is.

In the particular example we have at hand, this, after identifying the stationary phase and playing, takes us from a $ n $-sum in $ N leq m ^ 2 n leq 2 N $that is to say $ n sim frac {N} {m ^ 2} $at one $ n_2 $-sum in

$$

frac {2} {3} frac {N ^ {1/2}} {n_1 ^ 2} leq n_2 leq 2 frac {N ^ {1/2}} {n_1 ^ 2},

$$

that is to say., $ n_2 sim frac {N ^ {1/2}} {n_1 ^ 2} $, which means that the arguments in the test function are now sized $ frac {N ^ {1/2}} {c ^ 3 m} $.

I can follow the movements of performing this stationary phase analysis, etc., but my question is this: **Is there any intuition about how and how these Voronoi formulas alter the lengths of the sums?**