Incomplete Kloosterman Sums and Multiplicative Inverses in Short Intervals
نویسندگان
چکیده
We investigate the solubility of the congruence xy ≡ 1 (mod p), where p is a prime and x, y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums.
منابع مشابه
Explicit values of multi-dimensional Kloosterman sums for prime powers, II
For any integer m > 1 fix ζm = exp(2πi/m), and let Z ∗ m denote the group of reduced residues modulo m. Let q = pα, a power of a prime p. The hyper-Kloosterman sums of dimension n > 0 are defined for q by R(d, q) = ∑ x1,...,xn∈Z∗ q ζ x1+···+xn+d(x1···xn) q (d ∈ Zq), where x−1 denotes the multiplicative inverse of x modulo q. Salie evaluated R(d, q) in the classical setting n = 1 for even q, and...
متن کاملDistribution of Modular Inverses and Multiples of Small Integers and the Sato–Tate Conjecture on Average
We show that, for sufficiently large integers m and X, for almost all a = 1,. .. , m the ratios a/x and the products ax, where |x| X, are very uniformly distributed in the residue ring modulo m. This extends some recent results of Garaev and Karatsuba. We apply this result to show that on average over r and s, ranging over relatively short intervals, the distribution of Kloosterman sums K r,s (...
متن کاملDistribution of Inverses and Multiples of Small Integers and the Sato–Tate Conjecture on Average
We show that, for sufficiently large integers m and X, for almost all a = 1,. .. , m the ratios a/x and the products ax, where |x| X, are very uniformly distributed in the residue ring modulo m. This extends some recent results of Garaev and Karatsuba. We apply this result to show that on average over r and s, ranging over relatively short intervals, the distribution of Kloosterman sums K r,s (...
متن کاملComputing the inverses, their power sums, and extrema for Euler's totient and other multiplicative functions
Wepropose a generic dynamic programming algorithm for computing the inverses of a multiplicative function. We illustrate our algorithm with Euler’s totient function and the sum of k-th powers of divisors. Our approach can be further adapted for computing certain functions of the inverses, such as their quantity, the smallest/largest inverse, which may be computed faster than the inverses themse...
متن کاملKloosterman sums and primitive elements in Galois fields
1. Introduction. Let F q denote the finite (Galois) field of order q, a power of a prime p. The multiplicative group F
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2012