A determinant related to the Jacobi symbol

A prime sensitive Hankel determinant of Jacobi symbol enumerators

Abstract We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes iff n is composite. If the dimension is a prime p, then the determinant evaluates to a polynomial of degree p−1 which is the product of a power of p and the generating polynomial of the partial sums of Legendr...

The Jacobi Symbol

  is only defined when the bottom is an odd prime. You can extend the definition to allow an odd positive number on the bottom using the Jacobi symbol. Most of the properties of Legendre symbols go through for Jacobi symbols, which makes Jacobi symbols very convenient for computation. We’ll see, however, that there is a price to pay for the greater generality: Euler’s formula no longer works,...

Efficient Algorithms for Computing the Jacobi Symbol

We present two new algorithms for computing the Jacobi Symbol: the right-shift and left-shift k-ary algorithms. For inputs of at most n bits in length, both algorithms take O(n 2 = log n) time and O(n) space. This is asymptotically faster than the traditional algorithm, which is based in Euclid's algorithm for computing greatest common divisors. In practice, we found our new algorithms to be ab...

Kronecker-Jacobi symbol and Quadratic Reciprocity

Probability laws related to the Jacobi theta

This paper reviews known results which connect Riemann's integral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these prob...