272 research outputs found
Further results on the distinctness of binary sequences derived from primitive sequences modulo square-free odd integers
This paper studies the distinctness of primitive sequences over Z/(M) modulo 2, where M is an odd integer that is composite and square-free, and Z/(M) is the integer residue ring modulo M. A new sufficient condition is given for ensuring that primitive sequences generated by a primitive polynomial f(x) over Z/(M) are pairwise distinct modulo 2. Such result improves a recent result obtained in our previous paper [27] and consequently the set of primitive sequences over Z/(M) that can be proven to be distinct modulo 2 is greatly enlarged
A new result on the distinctness of primitive sequences over Z(pq) modulo 2
Let Z/(pq) be the integer residue ring modulo pq with odd prime numbers p and q. This paper studies the distinctness problem of modulo 2 reductions of two primitive sequences over Z/(pq), which has been studied by H.J. Chen and W.F. Qi in 2009. First, it is shown that almost every element in Z/(pq) occurs in a primitive sequence of order n > 2 over Z/(pq). Then based on this element distribution property of primitive sequences over Z/(pq), previous results are greatly improved and the set of primitive sequences over Z/(pq) that are known to be distinct modulo 2 is further enlarged
On the distinctness of binary sequences derived from primitive sequences modulo square-free odd integers
Let M be a square-free odd integer and Z/(M) the integer residue ring modulo M. This paper studies the distinctness of primitive sequences over Z/(M) modulo 2. Recently, for the case of M = pq, a product of two distinct prime numbers p and q, the problem has been almost completely solved. As for the case that M is a product of more prime numbers, the problem has been quite resistant to proof. In this paper, a partial proof is given by showing that a class of primitive sequences of order 2k+1 over Z/(M) is distinct modulo 2. Besides as an independent interest, the paper also involves two distribution properties of primitive sequences over Z/(M), which related closely to our main results
The Langlands-Kottwitz approach for the modular curve
We show how the Langlands-Kottwitz method can be used to determine the local
factors of the Hasse-Weil zeta-function of the modular curve at places of bad
reduction. On the way, we prove a conjecture of Haines and Kottwitz in this
special case.Comment: 39 page
Families of L-functions and their Symmetry
In [90] the first-named author gave a working definition of a family of
automorphic L-functions. Since then there have been a number of works [33],
[107], [67] [47], [66] and especially [98] by the second and third-named
authors which make it possible to give a conjectural answer for the symmetry
type of a family and in particular the universality class predicted in [64] for
the distribution of the zeros near s=1/2. In this note we carry this out after
introducing some basic invariants associated to a family
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