On the representation of numbers in the form ax2+by2+cz2+dt2

Farey arcs and series

“…To avoid this and at the same to take into account new achievements of modular theory concerning quadratic forms it would be much more natural to separate this question from the circle method. This has really happened, as the example of papers [4] and [5] shows. After Eichler's paper [5] the development of the theory of quadratic forms took a different direction.…”

“…In conclusion let us note that if we compare papers [4] and [5], where one and the same problem is solved by different methods, then we will clearly see the connection between the circle method and modular theory. where The modular point of view explains why the solution of the problem of equalizing Farey's arcs is so complicated and not uniquely defined in the circle method.…”

“…After that we had often discussed this problem. It was clear that the problem of constructing Farey's arcs could be solved in defferent ways, and the question was to find a simple and natural way of representing all Farey's arcs.This problem was solved for the first time by Kloosterman in paper [4], where his famous sums were introduced. However, Kloosterman's method of constructing Farey's arcs is rather complicated, and the reason for this is the uniform representation of the arcs for all points of the Faxey's series.…”

New equidistribution estimates of Zhang type

“…For our purpose of establishing some non-trivial Type I and Type II estimates for a given choice of σ (and in particular for σ slightly above 1{6) and for sufficiently small , δ, it is not necessary to have the full square root cancellation in (4.2), and any power savings of the form p 1´c for some fixed c ą 0 would suffice (with the same dependency on P and Q); indeed, one obtains a non-trivial estimate for a given value of σ once one invokes the q-van der Corput method a sufficient number of times, and once the gains from the cancellation are greater than the losses coming from completion of sums, with the latter becoming negligible in the limit , δ Ñ 0. Such a power saving (with c " 1{4) was obtained for the Kloosterman sum (4.3) by Kloosterman [34] using an elementary dilation argument (see also [41] for a generalization), but this argument does not appear to be available for estimates such as (4.4).…”

Partition Functions for Supersymmetric Black Holes

“…(4.7). The (scalar) Kloosterman sum K c (m, n) [131] is originally defined as 26) wheredd = 1 mod c [129]. This "scalar" Kloosterman sum appears in the expansion when the unitary prefactor is equal to one and the modular vector has length one.…”