共查询到20条相似文献,搜索用时 31 毫秒
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Periodica Mathematica Hungarica - For any integer l and any positive integer n, let $$ \sigma _{l}(n)=\sum _{d\mid n}d^{l}$$ . In 1936, Erd?s proved that the set of positive integers n with... 相似文献
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The Ramanujan Journal - In this note, using the well-known series representation for the Clausen function, we also provide some new representations of Apery’s constant $$\zeta (3)$$ . In... 相似文献
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We aim at evaluating the following class of series involving the product of the tail of two consecutive zeta function valueswhere \({k\geq 2}\) is an integer. We show that the series can be expressed in terms of Riemann zeta function values and a special integral involving a polylogarithm function.
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$$\sum\limits_{n=1}^{\infty}\left(\zeta(k)-1-\frac{1}{2^k}-\cdots-\frac{1}{n^k}\right)\left(\zeta(k+1)-1-\frac{1}{2^{k+1}}-\cdots-\frac{1}{n^{k+1}}\right),$$
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The Ramanujan Journal - Let $$k\geqslant 2$$ be a fixed natural number and $$d_k(n)$$ denote the number of ways n can be written as a product of k positive integers. We use $$\Delta _k(x)$$ to... 相似文献
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We investigate the compatibility of \(I_0\) with various combinatorial principles at \(\lambda ^+\), which include the existence of \(\lambda ^+\)-Aronszajn trees, square principles at \(\lambda \), the existence of good scales at \(\lambda \), stationary reflections for subsets of \(\lambda ^{+}\), diamond principles at \(\lambda \) and the singular cardinal hypothesis at \(\lambda \). We also discuss whether these principles can hold in \(L(V_{\lambda +1})\). 相似文献
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In this paper, we construct combinatorial bases of Feigin–Stoyanovsky’s type subspaces of standard modules for level k affine Lie algebra \(C_\ell ^{(1)}\). We prove spanning by using annihilating field \(x_\theta (z)^{k+1}\) of standard modules. In the proof of linear independence, we use simple currents and intertwining operators whose existence is given by fusion rules. 相似文献
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The Ramanujan Journal - Recently, Andrews, Dixit, and Yee defined two partition functions $$p_{\omega }(n)$$ and $$p_{\nu }(n)$$ that are related with Ramanujan’s mock theta functions... 相似文献
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大家知道1·2+2·3+3·4+…n(n+1)的求和可利用通项公式来求,即: 1·2+2·3+3·4+…+n(n+1)=(1~2+2~2+3~2+…+n~2)+(1+2+3+… +n)=(1/6)n(n+1)(2n+1)+(1/2)n(1+n)-(1/3)n(n+1)(n+2) 但是用这种方法求和涉及到数列1~2,2~2,3~2…n~2的求和,如果给出累进数列的每项乘积因子则又涉及数列{n~3},{n~4},…的求和,所以利用通项求常见累进数列 相似文献
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If \(q\ge 2\) is an integer, we denote by \(S_q(n)\) the sum of the digits in base q of the positive integer n and by \(v_q(n)\) its q-adic valuation. The goal of this work is to study exponential sums of the form \(\displaystyle \sum \nolimits _{n\le x}\exp \big (2i\pi \big (\frac{l}{m} S_q(n)+\frac{k}{m'}S_q(n+1)+\theta n\big )\big )\) in order to prove some statistical properties of integers n for which \(S_q(n)\) and \(S_q(n+1)\) belong to given arithmetic progressions. This extends the results obtained by Gelfond in 1968 and those obtained by Mauduit–Sárközy in 1996. 相似文献
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George E. Andrews Donny Passary James A. Sellers Ae Ja Yee 《The Ramanujan Journal》2017,42(2):347-361
We establish new connection formulae between Fibonacci polynomials and Chebyshev polynomials of the first and second kinds. These formulae are expressed in terms of certain values of hypergeometric functions of the type \(_2F_{1}\). Consequently, we obtain some new expressions for the celebrated Fibonacci numbers and their derivative sequences. Moreover, we evaluate some definite integrals involving products of Fibonacci and Chebyshev polynomials. 相似文献
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Fulgence Eyi-Obiang Youssef Ouknine Octave Moutsinga 《Journal of Theoretical Probability》2017,30(1):117-142
In papers by Yor, a remarkable class \(({\varSigma })\) of submartingales is introduced, which, up to technicalities, are submartingales \((X_{t})_{t\ge 0}\) whose increasing process is carried by the times t such that \(X_{t}=0\). These submartingales have several applications in stochastic analysis: for example, the resolution of Skorokhod embedding problem, the study of Brownian local times and the study of zeros of continuous martingales. The submartingales of class \(({\varSigma })\) have been extensively studied in a series of articles by Nikeghbali (part of them in collaboration with Najnudel, some others with Cheridito and Platen). On the other hand, stochastic calculus has been extended to signed measures by Ruiz de Chavez (Le théorème de Paul Lévy pour des mesures signées. Séminaire de probabilités (Strasbourg). Springer, Berlin, 1984) and Beghdadi-Sakrani (Calcul stochastique pour les mesures signées. Séminaire de probabilités (Strasbourg). Springer, Berlin, 2003). In Eyi Obiang et al. (Stochastics 86(1):70–86, 2014), the authors of the present paper have extended the notion of submartingales of class \(({\varSigma })\) to the setting of Ruiz de Chavez (1984) and Beghdadi-Sakrani (2003), giving two different classes of stochastic processes named classes \(\sum (H)\) and \(\sum _\mathrm{s}(H)\) where from tools of the theory of stochastic calculus for signed measures, the authors provide general frameworks and methods for dealing with processes of these classes. In this work, we first give some formulas of multiplicative decomposition for processes of these classes. Afterward, we shall establish some representation results allowing to recover any process of one of these classes from its final value and the last time it visited the origin. 相似文献
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Guram Donadze 《Proceedings Mathematical Sciences》2018,128(1):6
In [1], Anderson and Badawi conjectured that \(\mathrm{rad}(I)^n \subseteq I\) for every n-absorbing ideal I of a commutative ring. In this article, we prove their conjecture. We also prove related conjectures for radical ideals. 相似文献
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The Ramanujan Journal - Assuming the Riemann hypothesis, we prove that $$\begin{aligned} N_k(T) = \frac{T}{2\pi }\log \frac{T}{4\pi e} + O_k\bigg (\frac{\log T}{\log \log T}\bigg ), \end{aligned}$$... 相似文献
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The Ramanujan Journal - The two partition functions $$p_\omega (n)$$ and $$p_\nu (n)$$ were introduced by Andrews, Dixit and Yee, which are related to the third-order mock theta functions $$\omega... 相似文献
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Johannes Schoißengeier 《Monatshefte für Mathematik》1986,102(1):59-77
Estimates for
. Let and letB
1(x)={x}–1/2. In this paper we shall give best possible estimates for
. On the importance of this sum see the papers ofBehnke [2],Hardy andLittlewood [5],Hecke [6] andOstrowski [9].
Meinem verehrten Lehrer, Herrn Professor E. Hlawka zum siebzigsten Geburtstag gewidmet 相似文献
Meinem verehrten Lehrer, Herrn Professor E. Hlawka zum siebzigsten Geburtstag gewidmet 相似文献
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A combinatorial proof that the number of points of the space \(\overline {{M_{0,n}}} \left( {{F_q}} \right)\) satisfies the recurrent formula for the Poincaré polynomials of the space \(\overline {{M_{0,n}}} \left( \mathbb{C} \right)\) is obtained. 相似文献