scholarly journals On the distribution of consecutive square-free primitive roots modulo p

2015 ◽  
Vol 65 (2) ◽  
pp. 555-564
Author(s):  
Huaning Liu ◽  
Hui Dong
Keyword(s):  
Modulo P ◽  
Primitive Roots

scholarly journals Note on Artin's Conjecture on Primitive Roots

2022 ◽  
Vol Volume 44 - Special... ◽  
Author(s):  
Sankar Sitaraman
Keyword(s):  
Primitive Root ◽  
Artin's Conjecture ◽  
Multiplicative Order ◽  
Primitive Root Modulo ◽  
Modulo P ◽  
Primitive Roots

E. Artin conjectured that any integer $a > 1$ which is not a perfect square is a primitive root modulo $p$ for infinitely many primes $ p.$ Let $f_a(p)$ be the multiplicative order of the non-square integer $a$ modulo the prime $p.$ M. R. Murty and S. Srinivasan \cite{Murty-Srinivasan} showed that if $\displaystyle \sum_{p < x} \frac 1 {f_a(p)} = O(x^{1/4})$ then Artin's conjecture is true for $a.$ We relate the Murty-Srinivasan condition to sums involving the cyclotomic periods from the subfields of $\mathbb Q(e^{2\pi i /p})$ corresponding to the subgroups $<a> \subseteq \mathbb F_p^*.$

scholarly journals PRIMITIVE ROOTS IN QUADRATIC FIELDS

2006 ◽  
Vol 02 (01) ◽  
pp. 7-23 ◽  
Author(s):  
JOSEPH COHEN
Keyword(s):  
Riemann Hypothesis ◽  
Primitive Root ◽  
Maximal Order ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Generalized Riemann Hypothesis ◽  
Artin's Conjecture ◽  
Modulo P ◽  
Primitive Roots ◽  
Real Quadratic

We consider an analogue of Artin's primitive root conjecture for units in real quadratic fields. Given such a nontrivial unit, for a rational prime p which is inert in the field the maximal order of the unit modulo p is p + 1. An extension of Artin's conjecture is that there are infinitely many such inert primes for which this order is maximal. This is known at present only under the Generalized Riemann Hypothesis. Unconditionally, we show that for any choice of 7 units in different real quadratic fields satisfying a certain simple restriction, there is at least one of the units which satisfies the above version of Artin's conjecture.

scholarly journals On additive decompositions of the set of primitive roots modulo p

2011 ◽  
Vol 169 (3-4) ◽  
pp. 317-328 ◽  
Author(s):  
Cécile Dartyge ◽  
András Sárközy
Keyword(s):  
Modulo P ◽  
Primitive Roots

The distribution of primitive roots in finite fields

1990 ◽  
Vol 45 (1) ◽  
pp. 223-224 ◽  
Author(s):  
G I Perel'muter ◽  
I E Shparlinskii
Keyword(s):  
Finite Fields ◽  
Primitive Roots

GENERALIZING THE TITCHMARSH DIVISOR PROBLEM

2012 ◽  
Vol 08 (03) ◽  
pp. 613-629 ◽  
Author(s):  
ADAM TYLER FELIX
Keyword(s):  
Natural Number ◽  
Asymptotic Relation ◽  
Divisor Problem ◽  
Implied Constant ◽  
Artin's Conjecture ◽  
Primitive Roots

Let a be a natural number different from 0. In 1963, Linnik proved the following unconditional result about the Titchmarsh divisor problem [Formula: see text] where c is a constant dependent on a. Titchmarsh proved the above result assuming GRH for Dirichlet L-functions in 1931. We establish the following asymptotic relation: [Formula: see text] where Ck is a constant dependent on k and a, and the implied constant is dependent on k. We also apply it a question related to Artin's conjecture for primitive roots.

scholarly journals Solutions of KZ differential equations modulo p

2018 ◽  
Vol 48 (3) ◽  
pp. 655-683 ◽  
Author(s):  
Vadim Schechtman ◽  
Alexander Varchenko
Keyword(s):  
Differential Equations ◽  
Modulo P
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