Recent zbMATH articles in MSC 11L https://zbmath.org/atom/cc/11L 2021-07-26T21:45:41.944397Z Werkzeug On subset sums of pseudo-recursive sequences https://zbmath.org/1463.11023 2021-07-26T21:45:41.944397Z "Bakos, Bence" https://zbmath.org/authors/?q=ai:bakos.bence "Hegyvári, Norbert" https://zbmath.org/authors/?q=ai:hegyvari.norbert "Pálfy, Máté" https://zbmath.org/authors/?q=ai:palfy.mate "Yan, Xiao-Hui" https://zbmath.org/authors/?q=ai:yan.xiaohui Summary: Let $$a_0 = a \in \mathbb{N}$$, $$\{M_i\}^{\infty}_{i=1}$$ be an infinite set of integers and $$\{b_1, b_2, \ldots, b_k\}$$ be a finite set of integers. We say that $$\{a_i\}^{\infty}_{i=0}$$ is a pseudo-recursive sequence if $$a_{n+1} = M_{n+1}a_n + b_{j_{n+1}} (b_{j_{n+1}} \in \{b_1, b_2, \ldots b_k\})$$ holds. In the first part of the paper, we investigate the subset sum of a generalized version of $$A_{\infty} := \{a_n = \lfloor 2^n\alpha \rfloor : n = 0, 1, 2, \ldots \}$$, which is a special pseudorecursive sequence. In the second part, we use $$A_{\alpha}$$ for an encryption algorithm. On generalizations of $$p$$-sets and their applications https://zbmath.org/1463.11127 2021-07-26T21:45:41.944397Z "Zhou, Heng" https://zbmath.org/authors/?q=ai:zhou.heng "Xu, Zhiqiang" https://zbmath.org/authors/?q=ai:xu.zhiqiang Summary: The $$p$$-set, which is in a simple analytic form, is well distributed in unit cubes. The well-known Weil's exponential sum theorem presents an upper bound of the exponential sum over the $$p$$-set. Based on the result, one shows that the $$p$$-set performs well in numerical integration, in compressed sensing as well as in uncertainty quantification. However, $$p$$-set is somewhat rigid since the cardinality of the $$p$$-set is a prime $$p$$ and the set only depends on the prime number $$p$$. The purpose of this paper is to present generalizations of $$p$$-sets, say $$\mathcal{P}_{d, p}^{a, \varepsilon}$$, which is more flexible. Particularly, when a prime number $$p$$ is given, we have many different choices of the new $$p$$-sets. Under the assumption that Goldbach conjecture holds, for any even number $$m$$, we present a point set, say $${\mathcal{L}_{p, q}}$$ with cardinality $$m-1$$ by combining two different new $$p$$-sets, which overcomes a major bottleneck of the $$p$$-set. We also present the upper bounds of the exponential sums over $$\mathcal{P}_{d, p}^{a, \varepsilon}$$ and $${\mathcal{L}_{p, q}}$$, which imply these sets have many potential applications. Jordan's totient function and trigonometric sums https://zbmath.org/1463.11132 2021-07-26T21:45:41.944397Z "Chu, Wenchang" https://zbmath.org/authors/?q=ai:chu.wenchang Summary: Two classes of trigonometric sums about integer powers of secant function are evaluated that are closely related to Jordan's totient function. On a trigonometric sum of Hardy and Littlewood https://zbmath.org/1463.11133 2021-07-26T21:45:41.944397Z "He, Yuan" https://zbmath.org/authors/?q=ai:he.yuan|he.yuan.1 Summary: In this paper, we perform a further investigation for a finite trigonometric sum considered by Hardy and Littlewood. By making use of some properties for the Chebyshev polynomials and Möbius function, we establish an interesting identity for the finite trigonometric sum of Hardy and Littlewood, by virtue of which an explicit asymptotic formula is also derived. On the fourth power mean of one kind of four-term exponential sums https://zbmath.org/1463.11134 2021-07-26T21:45:41.944397Z "Liu, Xinyu" https://zbmath.org/authors/?q=ai:liu.xinyu "Chen, Zhuoyu" https://zbmath.org/authors/?q=ai:chen.zhuoyu Summary: The main purpose of this paper is using the properties of the trigonometric sums and the number of the congruence equation to study the computational problem of the one kind fourth power mean involving the four-term exponential sums, and give two interesting computational formulae for it. The hybrid power mean of character sums of polynomials and two-term exponential sums https://zbmath.org/1463.11135 2021-07-26T21:45:41.944397Z "Qi, Lan" https://zbmath.org/authors/?q=ai:qi.lan "Chen, Zhuoyu" https://zbmath.org/authors/?q=ai:chen.zhuoyu Summary: In this paper, we discuss the computational problem of a hybrid power mean involving character sums of polynomials and two-term cubic exponential sums by using analytic methods and the properties of two-term exponential sums and Dirichlet characters. Meanwhile, we obtain a sharp asymptotic formula. On the power mean of a sum analogous to the Kloosterman sum https://zbmath.org/1463.11136 2021-07-26T21:45:41.944397Z "Chern, Shane" https://zbmath.org/authors/?q=ai:chern.shane Let $$p>2$$ be a prime and let $$e(y)=e^{2\pi iy}$$. As an analogue of the generalized Kloosterman sum $\sum_{a=1}^{p-1} \chi (a)\; e\left( \frac {ma+n\overline{a}}{p}\right)$ \textit{X. Lv} and \textit{W. Zhang} [Lith. Math. J. 57, 359--366 (2017; Zbl 1373.11060)] considered the following new sum $\sum_{a=1}^{p-1} \chi (ma+n\overline{a})\; e\left( \frac {ka}{p}\right)$ where $$\chi$$ is a Dirichlet character mod $$p$$, $$m, n$$ and $$k$$ are integers, $$\overline{a}$$ is the multiplicative inverse of $$a$$ mod $$p$$. In the paper under review the author proves that, for $$n$$ and $$k$$ coprime to $$p$$, $\sum_{\chi \mod p}\left\vert\sum_{m=1}^{p-1}\left\vert\sum_{a=1}^{p-1} \chi(ma+n\overline{a})\;e\left(\frac {ka}{p}\right )\right\vert^2\right \vert^2 = (p-1)(p^4-7p^3+17p^2-5p-25).$ For $$n$$ and $$k$$ coprime to $$p>3$$, the author also gives $\sum_{\chi \mod p}\sum_{m=1}^{p-1}\left\vert\sum_{a=1}^{p-1} \chi(ma+n\overline{a})\;e\left(\frac {ka}{p}\right )\right\vert^4$ in terms of Legendre symbols. Exponential sums over primes in arithmetic progressions https://zbmath.org/1463.11137 2021-07-26T21:45:41.944397Z "Liu, Huake" https://zbmath.org/authors/?q=ai:liu.huake "Wang, Tianqin" https://zbmath.org/authors/?q=ai:wang.tianqin Summary: This paper uses elementary method to study the problem of exponential sums, and gives an estimate for nonlinear exponential sums over primes in arithmetic progressions, which could be useful in most numerical problems when the circle method is applied. Goldbach partitions and norms of cusp forms https://zbmath.org/1463.11138 2021-07-26T21:45:41.944397Z "Davis, Simon" https://zbmath.org/authors/?q=ai:davis.simon-brian Summary: An integral formula for the Goldbach partitions requires uniform convergence of a complex exponential sum. The dependence of the coefficients of the series is found to be bounded by that of cusp forms. Norms may be defined for these forms on a fundamental domain of a modular group. The relation with the integral formula is found to be sufficient to establish the consistency of the interchange of the integral and the sum, which must remain valid as the even integer $$N$$ tends to infinity. On the mean value of $$\frac{L'}{L} (1, \chi)$$ with the weight of character sums in a short interval https://zbmath.org/1463.11139 2021-07-26T21:45:41.944397Z "Tian, Qing" https://zbmath.org/authors/?q=ai:tian.qing Summary: Let $$\chi$$ be a Dirichlet character modulo $$q > 2$$, and $$L (s, \chi)$$ denote the Dirichlet $$L$$-function corresponding to $$\chi$$. The main purpose of this paper is using the estimate for character sums and the analytic method to study the mean value properties of $$\frac{L'}{L} (1, \chi)$$ with the weight of character sums in a short interval, and to give an interesting mean value formula for it. On instances of Fox's integral equation connection to the Riemann zeta function. https://zbmath.org/1463.42012 2021-07-26T21:45:41.944397Z "Patkowski, Alexander E." https://zbmath.org/authors/?q=ai:patkowski.alexander-eric Author's abstract: We consider some applications of the singular integral equation of the second kind of Fox. Some new solutions to Fox's integral equation are discussed in relation to number theory.