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

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1. CMB Online first

Liu, H. Q.
The Dirichlet divisor problem of arithmetic progressions
We design an elementary method to study the problem, getting an asymptotic formula which is better than Hooley's and Heath-Brown's results for certain cases.

Keywords:Dirichlet divisor problem, arithmetic progression
Categories:11L07, 11B83

2. CMB 2010 (vol 53 pp. 654)

Elliott, P. D. T. A.
Variations on a Paper of Erdős and Heilbronn
It is shown that an old direct argument of Erdős and Heilbronn may be elaborated to yield a result of the current inverse type.

Categories:11L07, 11P70

3. CMB 2001 (vol 44 pp. 87)

Lieman, Daniel; Shparlinski, Igor
On a New Exponential Sum
Let $p$ be prime and let $\vartheta\in\Z^*_p$ be of multiplicative order $t$ modulo $p$. We consider exponential sums of the form $$ S(a) = \sum_{x =1}^{t} \exp(2\pi i a \vartheta^{x^2}/p) $$ and prove that for any $\varepsilon > 0$ $$ \max_{\gcd(a,p) = 1} |S(a)| = O( t^{5/6 + \varepsilon}p^{1/8}) . $$

Categories:11L07, 11T23, 11B50, 11K31, 11K38

4. CMB 1998 (vol 41 pp. 187)

Loh, W. K. A.
Exponential sums on reduced residue systems
The aim of this article is to obtain an upper bound for the exponential sums $\sum e(f(x)/q)$, where the summation runs from $x=1$ to $x=q$ with $(x,q)=1$ and $e(\alpha)$ denotes $\exp(2\pi i\alpha)$. We shall show that the upper bound depends only on the values of $q$ and $s$, %% where $s$ is the number of terms in the polynomial $f(x)$.


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