Consideration of a singular series of the asymptotic formula of Kloosterman’s problem

Abstract

The representation problem of a natural number 𝑛 in the diagonal quadratic form with four variables 𝑎𝑥2 + 𝑏𝑦2 + 𝑐𝑧2 + 𝑑𝑡2, where 𝑎, 𝑏, 𝑐, 𝑑 are given positive integers, is considered in this paper. The question is posed to define under what conditions on the coefficients 𝑎, 𝑏, 𝑐, 𝑑 such representation does not exist for a given 𝑛. These conditions, which obtained based on the theory of congruences or without proof, are given in the Kloosterman’s work (1926).
Kloosterman also has obtained an asymptotic formula for the number of solutions to the equation 𝑛 = 𝑎𝑥2 + 𝑏𝑦2 + 𝑐𝑧2 + 𝑑𝑡2. The main term of this formula is a series +Σ︀∞𝑞=1Φ(𝑞) of a multiplicative function Φ(𝑞) containing the one-dimensional Gaussian sums with coefficients 𝑎,
𝑏, 𝑐, 𝑑. Our work is related to the study of the representation of this special series as a product over primes Π︀ 𝑝|𝑞 (1 + Φ(𝑝) + Φ(𝑝2) + · · · ).
Previously, the authors have been considered the case when 𝑝 ̸= 2. Conditions for the coefficients 𝑎, 𝑏, 𝑐, 𝑑, 𝑛 under which the equation 𝑛 = 𝑎𝑥2 + 𝑏𝑦2 + 𝑐𝑧2 + 𝑑𝑡2 has no solutions have been proved with using exact formulas for the one-dimensional Gaussian sums, Ramanujan sum and the generalized Ramanujan sum from the power of a prime.
The case for 𝑝 = 2 and 𝑛 odd is considering in this paper. Taking into account formulas for the one-dimensional Gaussian sums from the power of two, the some not previously studied sums that are close to the Kloosterman sum, are appeared. For such sums from the power of
two, we obtained the exact values. This allowed us to give a complete proof of the conditions on the coefficients 𝑎, 𝑏, 𝑐, 𝑑, at least two of which are even. Under these conditions an odd natural number cannot be represented by a diagonal quadratic form with four variables. Note
that some of these conditions are new and are not mentioned in Kloosterman’s work.