Sign changes of Kloosterman sums with almost prime moduli

Kloosterman Sums and Traces of Singular Moduli

where q = e. Let d ≡ 0, 3 (mod 4) be a positive integer, so that −d is a negative discriminant. Denote by Qd the set of positive definite integral binary quadratic forms Q(x, y) = ax + bxy + cy = [a, b, c] with discriminant −d = b − 4ac, including imprimitive forms (if such exist). We let αQ be the unique complex number in the upper half plane H which is a root of Q(x, 1) = 0. Values of j at th...

Kloosterman sums for prime powers in P-adic fields

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Sums of almost equal prime squares

In this short note, we prove that almost all integers N satisfying N ≡ 3 (mod 24) and 5 -N or N ≡ 4 (mod 24) is the sum of three or four almost equal prime squares, respectively: N = p21 + · · ·+ p 2 j with |pi − (N/j) 1/2| ≤ N1/2−9/80+ε for j = 3 or 4 and 1 ≤ i ≤ j.

On zeros of Kloosterman sums

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...