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Search: MSC category 11L07 ( Estimates on exponential sums )

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1. CJM Online first

Drappeau, Sary
Sommes friables d'exponentielles et applications
An integer is said to be $y$-friable if its greatest prime factor is less than $y$. In this paper, we obtain estimates for exponential sums over $y$-friable numbers up to $x$ which are non-trivial when $y \geq \exp\{c \sqrt{\log x} \log \log x\}$. As a consequence, we obtain an asymptotic formula for the number of $y$-friable solutions to the equation $a+b=c$ which is valid unconditionnally under the same assumption. We use a contour integration argument based on the saddle point method, as developped in the context of friable numbers by Hildebrand and Tenenbaum, and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers.

Keywords:théorie analytique des nombres, entiers friables, méthode du col
Categories:12N25, 11L07

2. CJM 2009 (vol 62 pp. 582)

Konyagin, Sergei V.; Pomerance, Carl; Shparlinski, Igor E.
On the Distribution of Pseudopowers
An x-pseudopower to base g is a positive integer that is not a power of g, yet is so modulo p for all primes $ple x$. We improve an upper bound for the least such number, due to E.~Bach, R.~Lukes, J.~Shallit, and H.~C.~Williams. The method is based on a combination of some bounds of exponential sums with new results about the average behaviour of the multiplicative order of g modulo prime numbers.

Categories:11A07, 11L07, 11N36

3. CJM 2009 (vol 61 pp. 481)

Banks, William D.; Garaev, Moubariz Z.; Luca, Florian; Shparlinski, Igor E.
Uniform Distribution of Fractional Parts Related to Pseudoprimes
We estimate exponential sums with the Fermat-like quotients $$ f_g(n) = \frac{g^{n-1} - 1}{n} \quad\text{and}\quad h_g(n)=\frac{g^{n-1}-1}{P(n)}, $$ where $g$ and $n$ are positive integers, $n$ is composite, and $P(n)$ is the largest prime factor of $n$. Clearly, both $f_g(n)$ and $h_g(n)$ are integers if $n$ is a Fermat pseudoprime to base $g$, and if $n$ is a Carmichael number, this is true for all $g$ coprime to $n$. Nevertheless, our bounds imply that the fractional parts $\{f_g(n)\}$ and $\{h_g(n)\}$ are uniformly distributed, on average over~$g$ for $f_g(n)$, and individually for $h_g(n)$. We also obtain similar results with the functions ${\widetilde f}_g(n) = gf_g(n)$ and ${\widetilde h}_g(n) = gh_g(n)$.

Categories:11L07, 11N37, 11N60

4. CJM 2009 (vol 61 pp. 336)

Garaev, M. Z.
The Large Sieve Inequality for the Exponential Sequence $\lambda^{[O(n^{15/14+o(1)})]}$ Modulo Primes
Let $\lambda$ be a fixed integer exceeding $1$ and $s_n$ any strictly increasing sequence of positive integers satisfying $s_n\le n^{15/14+o(1)}.$ In this paper we give a version of the large sieve inequality for the sequence $\lambda^{s_n}.$ In particular, we obtain nontrivial estimates of the associated trigonometric sums ``on average" and establish equidistribution properties of the numbers $\lambda^{s_n} , n\le p(\log p)^{2+\varepsilon}$, modulo $p$ for most primes $p.$

Keywords:Large sieve, exponential sums
Categories:11L07, 11N36

5. CJM 2005 (vol 57 pp. 338)

Lange, Tanja; Shparlinski, Igor E.
Certain Exponential Sums and Random Walks on Elliptic Curves
For a given elliptic curve $\E$, we obtain an upper bound on the discrepancy of sets of multiples $z_sG$ where $z_s$ runs through a sequence $\cZ=\(z_1, \dots, z_T\)$ such that $k z_1,\dots, kz_T $ is a permutation of $z_1, \dots, z_T$, both sequences taken modulo $t$, for sufficiently many distinct values of $k$ modulo $t$. We apply this result to studying an analogue of the power generator over an elliptic curve. These results are elliptic curve analogues of those obtained for multiplicative groups of finite fields and residue rings.

Categories:11L07, 11T23, 11T71, 14H52, 94A60

6. CJM 2001 (vol 53 pp. 414)

Rivat, Joël; Sargos, Patrick
Nombres premiers de la forme $\floor{n^c}$
For $c>1$ we denote by $\pi_c(x)$ the number of integers $n \leq x$ such that $\floor{n^c}$ is prime. In 1953, Piatetski-Shapiro has proved that $\pi_c(x) \sim \frac{x}{c\log x}$, $x \rightarrow +\infty$ holds for $c<12/11$. Many authors have extended this range, which measures our progress in exponential sums techniques. In this article we obtain $c < 1.16117\dots\;$.

Categories:11L07, 11L20, 11N05

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