##
**Bilinear forms with Kloosterman fractions.**
*(English)*
Zbl 0873.11050

The Kloosterman sum with character \(\chi \pmod c\) is
\[
S_\chi (a,b;c) = \sum_{y \pmod c} \overline \chi(y)e \left({a \overline y+by \over c} \right)
\]
where \(e(t)= e^{2\pi it}\) and \(\overline yy \equiv 1 \pmod c\). If \(\chi\) is the principal character this is the classical Kloosterman sum and if \(\chi(y) =({y \over c})\) is the Jacobi symbol then this is the Salié sum. In applications one usually needs estimates for incomplete sums where the variable runs over a segment of an arithmetic progression. A standard method of bounding such a sum is to express it in terms of the complete sums \(S_\chi (a,b;c)\) by a Fourier technique and then apply Weil’s bound for the latter. When the segment is too small this method gives nothing. Fortunately, for many applications it suffices to bound these sums on average over the modulus \(c\). In this paper the authors consider general bilinear forms of the type
\[
{\mathcal B} (M,N)= {\underset {(m,n)=1} {\sum\sum}} \alpha_m \beta_ne \left(a {\overline m \over n} \right),
\]
where \(a\) is a possibly large positive integer and \(\alpha_m\), \(\beta_n\) are arbitrary complex numbers for \(M <m \leq 2M\), \(N<n \leq 2N\), respectively and \(\overline mm \equiv 1 \pmod n\). A trivial bound for this is \(|{\mathcal B} (M,N) |\leq 2 |\alpha ||\beta|(MN)^{1 \over 2}\) where \(|\cdot |\) denotes the \(\ell_2\)-norm and the authors’ goal is to improve this. Their first result is:

Theorem 1. For any positive integer \(a\) we have \[ {\mathcal B} (M,N) \ll |\alpha ||\beta |\left \{(M+N)^{1\over 2} +\left(1 +{a \over MN} \right)^{1 \over 2} \min (M,N) \right\} (MN)^\varepsilon \] where the implied constant depends only on \(\varepsilon\).

Theorem 1 provides nontrivial bounds except in the case that one of the ranges \(M,N\) is much larger than the other and in the case when they are nearly equal. In the former case one has essentially a single variable and one cannot in general obtain nontrivial bounds since the unknown coefficients may be chosen to counteract the change in argument of the exponential. By contrast, the case when \(M\) and \(N\) are nearly equal is important for applications and here one expects cancellation. The second theorem provides such a bound.

Theorem 2. For any positive integer \(a\), \[ {\mathcal B} (M,N) \ll|\alpha ||\beta |(a+MN)^{3 \over 8} (M + N)^{{11 \over 48} + \varepsilon} \] the implied constant depending only on \(\varepsilon\).

The combination of Theorem 1 and Theorem 2 gives a nontrivial estimate whenever \(N> M^\varepsilon\) and \(M> N^\varepsilon\). The authors apply the above to bound the Salié sum on average.

Theorem 4. For positive integers \(a\), \(r\) with \(8\mid r\) and \(b\) with \((b,r) =1\) and \(\chi(y) =({y\over c})\), \[ \sum_{\substack{ c\leq x,\;(c,a)=1\\ c \equiv b \pmod r}} c^{-{1 \over 2}} S_\chi (a,a;c) =24 \pi^{-2} \varepsilon_b \delta(a) \psi (ar)a E(x/2a) +O \bigl(r^{1 \over 5} (a+x)^{{47 \over 118} x^{35 \over 59} +\varepsilon} \bigr) \] where the implied constant depends only on \(\varepsilon\), \(\delta (a)\) is the characteristic function on the squares, \(\varepsilon_b\) is 1 or \(i\) according as \(b\) is one or three modulo four, \[ \psi(q) =\prod_{p |q} \left(1 +{1 \over p} \right)^{-1}, \quad E(T) =\int^T_0 e(-1/t)dt. \] Here the main point is that one obtains a useful result with \(a\) somewhat larger than \(x\).

Several other applications are mentioned. One is to the estimation of the Fourier coefficients of forms automorphic with respect to a theta multiplier and thereby a simplification of the solution to the Linnik problem of equidistribution of integer points on the sphere. Another is to the problem of improving the convexity bounds for the \(L\)-functions attached to the class group characters of an imaginary quadratic field, and more generally to Artin \(L\)-functions of degree two. Some further applications deal with the number of solutions of determinant equations, to new mean value theorems for Dirichlet \(L\)-functions and Dirichlet polynomials and to sums occurring in the distribution of divisor functions and of primes in arithmetic progressions.

The main difficulty in the work is the proof of Theorem 2 when \(M,N\) are close to the same size. The authors use their amplification technique, developed in a number of earlier works in connection with sums of “characters” of various types. Therein they replace, by positivity, their character sum by a carefully weighted sum over a family of characters. The main innovation here is the use of this idea to amplify the contribution coming from a principal character. This seems at first glance a foolish enterprise which, if it worked at all, could be applied to almost any sum by attaching to it a principal character.

The improvements, although by a fixed power, are all quantitatively small. The authors point out a number of places where further refinements can be made, but which sould still lead to bounds far short of what is expected to be true.

Theorem 1. For any positive integer \(a\) we have \[ {\mathcal B} (M,N) \ll |\alpha ||\beta |\left \{(M+N)^{1\over 2} +\left(1 +{a \over MN} \right)^{1 \over 2} \min (M,N) \right\} (MN)^\varepsilon \] where the implied constant depends only on \(\varepsilon\).

Theorem 1 provides nontrivial bounds except in the case that one of the ranges \(M,N\) is much larger than the other and in the case when they are nearly equal. In the former case one has essentially a single variable and one cannot in general obtain nontrivial bounds since the unknown coefficients may be chosen to counteract the change in argument of the exponential. By contrast, the case when \(M\) and \(N\) are nearly equal is important for applications and here one expects cancellation. The second theorem provides such a bound.

Theorem 2. For any positive integer \(a\), \[ {\mathcal B} (M,N) \ll|\alpha ||\beta |(a+MN)^{3 \over 8} (M + N)^{{11 \over 48} + \varepsilon} \] the implied constant depending only on \(\varepsilon\).

The combination of Theorem 1 and Theorem 2 gives a nontrivial estimate whenever \(N> M^\varepsilon\) and \(M> N^\varepsilon\). The authors apply the above to bound the Salié sum on average.

Theorem 4. For positive integers \(a\), \(r\) with \(8\mid r\) and \(b\) with \((b,r) =1\) and \(\chi(y) =({y\over c})\), \[ \sum_{\substack{ c\leq x,\;(c,a)=1\\ c \equiv b \pmod r}} c^{-{1 \over 2}} S_\chi (a,a;c) =24 \pi^{-2} \varepsilon_b \delta(a) \psi (ar)a E(x/2a) +O \bigl(r^{1 \over 5} (a+x)^{{47 \over 118} x^{35 \over 59} +\varepsilon} \bigr) \] where the implied constant depends only on \(\varepsilon\), \(\delta (a)\) is the characteristic function on the squares, \(\varepsilon_b\) is 1 or \(i\) according as \(b\) is one or three modulo four, \[ \psi(q) =\prod_{p |q} \left(1 +{1 \over p} \right)^{-1}, \quad E(T) =\int^T_0 e(-1/t)dt. \] Here the main point is that one obtains a useful result with \(a\) somewhat larger than \(x\).

Several other applications are mentioned. One is to the estimation of the Fourier coefficients of forms automorphic with respect to a theta multiplier and thereby a simplification of the solution to the Linnik problem of equidistribution of integer points on the sphere. Another is to the problem of improving the convexity bounds for the \(L\)-functions attached to the class group characters of an imaginary quadratic field, and more generally to Artin \(L\)-functions of degree two. Some further applications deal with the number of solutions of determinant equations, to new mean value theorems for Dirichlet \(L\)-functions and Dirichlet polynomials and to sums occurring in the distribution of divisor functions and of primes in arithmetic progressions.

The main difficulty in the work is the proof of Theorem 2 when \(M,N\) are close to the same size. The authors use their amplification technique, developed in a number of earlier works in connection with sums of “characters” of various types. Therein they replace, by positivity, their character sum by a carefully weighted sum over a family of characters. The main innovation here is the use of this idea to amplify the contribution coming from a principal character. This seems at first glance a foolish enterprise which, if it worked at all, could be applied to almost any sum by attaching to it a principal character.

The improvements, although by a fixed power, are all quantitatively small. The authors point out a number of places where further refinements can be made, but which sould still lead to bounds far short of what is expected to be true.

Reviewer: J.Friedlander (Toronto)

### MSC:

11L05 | Gauss and Kloosterman sums; generalizations |

11E39 | Bilinear and Hermitian forms |

11F30 | Fourier coefficients of automorphic forms |

11M06 | \(\zeta (s)\) and \(L(s, \chi)\) |

11N37 | Asymptotic results on arithmetic functions |

11N25 | Distribution of integers with specified multiplicative constraints |