Distribution of exponential functions modulo a prime power

Shparlinski, Igor E.

doi:10.1016/j.jnt.2014.04.010

Cited by 3 publications

(2 citation statements)

References 9 publications

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“…This lemma is only a special case of a result proved by Korobov [9,Theorem 3]. The reader is also referred to Shparlinski's paper [12] for a convenient reference to this result. In particular, our formulation of Lemma 3 follows the exposition in [12].…”

Section: Proof Of Theoremmentioning

confidence: 85%

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Coefficients of Unitary Cyclotomic Polynomials of Order Three

Bachman¹

2021

Preprint

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A unitary cyclotomic polynomial of order three is a polynomial of the form, where P , Q and R are powers of three distinct primes p, q and r. Fixing any such prime triple generates a family of these polynomials corresponding to all possible choices of P = p a , Q = q b and R = r c . We study the coefficients of polynomials in such a family. In particular, we show that the coefficients of polynomials in every such family cover all of Z.

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Section: Proof Of Theoremmentioning

confidence: 85%

“…The reader is also referred to Shparlinski's paper [12] for a convenient reference to this result. In particular, our formulation of Lemma 3 follows the exposition in [12]. Lemma 3.…”

Section: Proof Of Theoremmentioning

confidence: 99%

Coefficients of Unitary Cyclotomic Polynomials of Order Three

Bachman¹

2021

Preprint

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Double exponential sums with exponential functions

Shparlinski

Yau

2017

Int. J. Number Theory

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On an exponential D. H. Lehmer problem

Xu,

Wang,

Zhu

2024

era

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<abstract>For an odd prime $ p $ and a positive integer $ \alpha $, let $ g $ be of multiplicative order $ \tau $ modulo $ q $ and $ q = p^{\alpha} $. Denote by $ N(h, g, q) $ the number of $ a $ such that $ h\nmid (a+(g^{a})_{q}) $ for any $ 1\leq a\leq \tau $ and a fixed integer $ h\geq 2 $ with $ (h, q) = 1 $. The main purpose of this paper is to give a sharp asymptotic formula for <disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ N(k, h, g, q) = \mathop{\sum\limits_{\begin{smallmatrix} a = 1\\ h\nmid (a+(g^{a})_{q}) \end{smallmatrix}}^{\tau}}\left|a-(g^{a})_{q}\right| ^{2k} $\end{document} </tex-math></disp-formula> where $ k $ is any nonnegative integer and $ (a)_{q} $ denotes the smallest positive residue of $ a $ modulo $ q $. In addition, we know that $ N(h, g, q) = N(0, h, g, q) $.</abstract>

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Distribution of exponential functions modulo a prime power

Cited by 3 publications

References 9 publications

Coefficients of Unitary Cyclotomic Polynomials of Order Three

Coefficients of Unitary Cyclotomic Polynomials of Order Three

Double exponential sums with exponential functions

On an exponential D. H. Lehmer problem

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