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Exponential Sums with a Difference
Proceedings of the London Mathematical Society, 1990This paper extends work on exponential sums by \textit{E. Bombieri} and \textit{H. Iwaniec} [Ann. Sc. Norm. Super. Pisa Cl. Sci., IV. Ser. 13, 449-472 (1986; Zbl 0615.10047)], \textit{H. Iwaniec} and \textit{C. J. Mozzochi} [J. Number Theory 29, No.1, 60-93 (1988; Zbl 0644.10031)] and \textit{M. N. Huxley} [Proc. Lond. Math. Soc., III. Ser.
Heath-Brown, D. R., Huxley, M. N.
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Journal of the London Mathematical Society, 1938
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Optimization with sums of exponentials and applications
2014 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2014We present a method for optimization with sums of exponentials subject to positivity constraints and apply it to the modeling of empirical probability distribution functions and to the design of IIR filters with non-negative impulse response. Our approach uses exponents in a sparse arithmetic progression and hence is able to transform the positivity ...
Bogdan Dumitrescu, Bogdan C. Sicleru
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Analysis, 1996
The classical Kuzmin-Landau and van der Corput inequalities imply estimates for the exponential sum \(\sum_{a\leq n\leq b} e(f(n))\) once certain bounds for the derivatives \(f'(t)\) and \(f''(t)\) are known. These are generalized to double exponential sums \(\sum_{(n_1,n_2) \in D} e(f(n_1,n_2))\), where \(D\) is a rectangle, or a more general domain ...
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The classical Kuzmin-Landau and van der Corput inequalities imply estimates for the exponential sum \(\sum_{a\leq n\leq b} e(f(n))\) once certain bounds for the derivatives \(f'(t)\) and \(f''(t)\) are known. These are generalized to double exponential sums \(\sum_{(n_1,n_2) \in D} e(f(n_1,n_2))\), where \(D\) is a rectangle, or a more general domain ...
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, 2018
We obtain new bounds of exponential sums modulo a prime $p$ with binomials $ax^k + bx^n$. In particular, for $k=1$, we improve the bound of Karatsuba (1967) from $O(n^{1/4} p^{3/4})$ to $O\left(p^{3/4} + n^{1/3}p^{2/3}\right)$ for any $n$, and then use ...
I. Shparlinski, J. Voloch
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We obtain new bounds of exponential sums modulo a prime $p$ with binomials $ax^k + bx^n$. In particular, for $k=1$, we improve the bound of Karatsuba (1967) from $O(n^{1/4} p^{3/4})$ to $O\left(p^{3/4} + n^{1/3}p^{2/3}\right)$ for any $n$, and then use ...
I. Shparlinski, J. Voloch
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On Cubic Exponential Sums and Gauss Sums
Journal of Mathematical Sciences, 2018zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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Journal of Mathematical Sciences, 2006
Let \(p> 2\) be a prime and \(k\geq 2\), \(n\geq 2\). This paper concerns the exponential sum \[ \sum\limits^{p^n}_{x=1} e^{2\pi i (ax^k + bx) p^{-n}} \] where \(a, b\) are integers. This sum occurs in studies of Waring's problem. The author proves for this sum the bound \[ p^{(1-V)/2} p^{n/2} (b, p^n)^{1/2}\quad\text{for}\quad n\equiv 1\pmod k \] and ...
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Let \(p> 2\) be a prime and \(k\geq 2\), \(n\geq 2\). This paper concerns the exponential sum \[ \sum\limits^{p^n}_{x=1} e^{2\pi i (ax^k + bx) p^{-n}} \] where \(a, b\) are integers. This sum occurs in studies of Waring's problem. The author proves for this sum the bound \[ p^{(1-V)/2} p^{n/2} (b, p^n)^{1/2}\quad\text{for}\quad n\equiv 1\pmod k \] and ...
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ON A CERTAIN GENERAL EXPONENTIAL SUM
International Journal of Number Theory, 2005In this paper we study the general exponential sum related to multiplicative functions f(n) with |f(n)| ≤ 1, namely we study the sum [Formula: see text] and obtain a non-trivial upper bound when α is a certain type of rational number.
Maier, H., Sankaranarayanan, A.
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Exponential sums with multiplicative coefficients without the Ramanujan conjecture
, 2020Yujiao Jiang, G. Lü, Zhiwei Wang
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