Explicit upper bound on the least primitive root modulo p2

AbstractWe estimate double sums $$\begin{eqnarray}S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})=\mathop{\sum }\limits_{x\in {\mathcal{I}}}\mathop{\sum }\limits_{{\it\lambda}\in {\mathcal{G}}}{\it\chi}(x+a{\it\lambda}),\quad 1\leq a<p-1,\end{eqnarray}$$ with a multiplicative character ${\it\chi}$ modulo $p$ where ${\mathcal{I}}=\{1,\dots ,H\}$ and ${\mathcal{G}}$ is a subgroup of order $T$ of the multiplicative group of the finite field of $p$ elements. A nontrivial upper bound on $S_{{\it\chi}}(a,{\mathcal{I}},{\mathcal{G}})$ can be derived from the Burgess bound if $H\geq p^{1/4+{\it\varepsilon}}$ and from some standard elementary arguments if $T\geq p^{1/2+{\it\varepsilon}}$, where ${\it\varepsilon}>0$ is arbitrary. We obtain a nontrivial estimate in a wider range of parameters $H$ and $T$. We also estimate double sums $$\begin{eqnarray}T_{{\it\chi}}(a,{\mathcal{G}})=\mathop{\sum }\limits_{{\it\lambda},{\it\mu}\in {\mathcal{G}}}{\it\chi}(a+{\it\lambda}+{\it\mu}),\quad 1\leq a<p-1,\end{eqnarray}$$ and give an application to primitive roots modulo $p$ with three nonzero binary digits.

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Character sums for primitive root densities

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004114000450 ◽

2014 ◽

Vol 157 (3) ◽

pp. 489-511 ◽

Cited By ~ 6

Author(s):

H. W. LENSTRA ◽

P. STEVENHAGEN ◽

P. MOREE

Keyword(s):

Galois Representations ◽

Multiplicative Group ◽

Primitive Root ◽

Infinite Product ◽

Character Sums ◽

Correction Factors ◽

Cyclic Reduction ◽

Local Factors ◽

Artin's Conjecture ◽

Primitive Root Modulo

AbstractIt follows from the work of Artin and Hooley that, under assumption of the generalised Riemann hypothesis, the density of the set of primes q for which a given non-zero rational number r is a primitive root modulo q can be written as an infinite product ∏p δp of local factors δp reflecting the degree of the splitting field of Xp - r at the primes p, multiplied by a somewhat complicated factor that corrects for the ‘entanglement’ of these splitting fields.We show how the correction factors arising in Artin's original primitive root problem and several of its generalisations can be interpreted as character sums describing the nature of the entanglement. The resulting description in terms of local contributions is so transparent that it greatly facilitates explicit computations, and naturally leads to non-vanishing criteria for the correction factors.The method not only applies in the setting of Galois representations of the multiplicative group underlying Artin's conjecture, but also in the GL2-setting arising for elliptic curves. As an application, we compute the density of the set of primes of cyclic reduction for Serre curves.

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Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004115000808 ◽

2016 ◽

Vol 160 (3) ◽

pp. 477-494 ◽

Cited By ~ 11

Author(s):

J. CILLERUELO ◽

M. Z. GARAEV

Keyword(s):

Upper Bound ◽

Absolute Constant ◽

Primitive Root ◽

Multiple Roots ◽

Number Of Solutions ◽

Residue Classes ◽

Primitive Root Modulo ◽

Image Position ◽

Almost All ◽

Positive Integers

AbstractIn this paper we obtain new upper bound estimates for the number of solutions of the congruence $$\begin{equation} x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in \mathcal{U}, \end{equation}$$ for certain ranges of H and |${\mathcal U}$|, where ${\mathcal U}$ is a subset of the field of residue classes modulo p having small multiplicative doubling. We then use these estimates to show that the number of solutions of the congruence $$\begin{equation} x^n\equiv \lambda\pmod p; \quad x\in \mathbb{N}, \quad L<x<L+p/n, \end{equation}$$ is at most $p^{\frac{1}{3}-c}$ uniformly over positive integers n, λ and L, for some absolute constant c > 0. This implies, in particular, that if f(x) ∈ $\mathbb{Z}$[x] is a fixed polynomial without multiple roots in $\mathbb{C}$, then the congruence xf(x) ≡ 1 (mod p), x ∈ $\mathbb{N}$, x ⩽ p, has at most $p^{\frac{1}{3}-c}$ solutions as p → ∞, improving some recent results of Kurlberg, Luca and Shparlinski and of Balog, Broughan and Shparlinski. We use our results to show that almost all the residue classes modulo p can be represented in the form xgy (mod p) with positive integers x < p5/8+ϵ and y < p3/8. Here g denotes a primitive root modulo p. We also prove that almost all the residue classes modulo p can be represented in the form xyzgt (mod p) with positive integers x, y, z, t < p1/4+ϵ.

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Note on Artin's Conjecture on Primitive Roots

Hardy-Ramanujan Journal ◽

10.46298/hrj.2022.7664 ◽

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 $<a> \subseteq \mathbb F_p^*.$

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Square-full primitive roots

International Journal of Number Theory ◽

10.1142/s1793042118500604 ◽

2018 ◽

Vol 14 (04) ◽

pp. 1013-1021 ◽

Cited By ~ 4

Author(s):

Marc Munsch ◽

Tim Trudgian

Keyword(s):

Primitive Root ◽

Least Square ◽

Character Sum ◽

Primitive Root Modulo ◽

Primitive Roots

We use character sum estimates to give some bounds on the least square-full primitive root modulo a prime. In particular, we show that there is a square-full primitive root mod [Formula: see text] less than [Formula: see text].

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On Character Sums and Primitive Roots†

Proceedings of the London Mathematical Society ◽

10.1112/plms/s3-12.1.179 ◽

1962 ◽

Vol s3-12 (1) ◽

pp. 179-192 ◽

Cited By ~ 95

Author(s):

D. A. Burgess

Keyword(s):

Character Sums ◽

Primitive Roots

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The Jones polynomial: quantum algorithms and applications in quantum complexity theory

Quantum Information and Computation ◽

10.26421/qic8.1-2-10 ◽

2008 ◽

Vol 8 (1&2) ◽

pp. 147-180

Author(s):

P. Wocjan ◽

J. Yard

Keyword(s):

Quantum Computation ◽

Braid Group ◽

Quantum Algorithms ◽

Primitive Root ◽

Jones Polynomial ◽

Tutte Polynomial ◽

Roots Of Unity ◽

Complete Problems ◽

Primitive Roots ◽

Quantum Complexity

We analyze relationships between quantum computation and a family of generalizations of the Jones polynomial. Extending recent work by Aharonov et al., we give efficient quantum circuits for implementing the unitary Jones-Wenzl representations of the braid group. We use these to provide new quantum algorithms for approximately evaluating a family of specializations of the HOMFLYPT two-variable polynomial of trace closures of braids. We also give algorithms for approximating the Jones polynomial of a general class of closures of braids at roots of unity. Next we provide a self-contained proof of a result of Freedman et al.\ that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. Our proof encodes two-qubit unitaries into the rectangular representation of the eight-strand braid group. We then give QCMA-complete and PSPACE-complete problems which are based on braids. We conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a \#P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the Tutte polynomial and graph coloring.

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