On the Generalization of Lehmer Problem and High-Dimension Kloosterman Sums

On the Generalization of Lehmer Problem and High-Dimension Kloosterman Sums

For any fixed integer k ≥ 2 and integer r with (r, p) = 1, it is clear that there exist k integers 1 ≤ a i ≤ p - 1(i = 1, 2, …, k) such that a 1 a 2 ⋯ a k ≡ r mod p. Let N(k, r; p) denote the number of all (a 1, a 2, ⋯ a k ) such that a 1 a 2 ⋯ a k ≡ r mod p and 2†(a 1 + a 2 + ⋯ + a k ). In this paper, we will use the analytic method and the estimate for high-dimension Kloosterman sums to study...

On zeros of Kloosterman sums

On the Distribution of Kloosterman Sums

For a prime p, we consider Kloosterman sums Kp(a) = ∑ x∈F p exp(2πi(x + ax)/p), a ∈ Fp, over a finite field of p elements. It is well known that due to results of Deligne, Katz and Sarnak, the distribution of the sums Kp(a) when a runs through Fp is in accordance with the Sato–Tate conjecture. Here we show that the same holds where a runs through the sums a = u+v for u ∈ U , v ∈ V for any two s...

A Note on Kloosterman Sums

has played an increasingly important role in the analytic theory of numbers. The dash ' beside the summation symbol indicates that the letter of summation runs only through a reduced residue system with respect to the modulus. The number h is defined as any solution of the congruence hh = l (mod k), and n denotes an arbitrary integer. It was shown by Salie almost fifteen years ago that Ak(n) ma...

Legendre Sums, Soto-andrade Sums and Kloosterman Sums

The three sums named in the title are all known to appear in connection with the complex representation theory of GL(2, q). The first two are incarnations of certain spherical vectors, whereas the third is a matrix coefficient for a parabolic basis. In this work, Legendre and Soto-Andrade sums are shown to occur in a second way, as parabolic ClebschGordan coefficients for the tensor product of ...