相关论文: On the function sum(-k^2/s^2)
Let $\gamma$ denote imaginary parts of complex zeros of the Riemann zeta-function $\zeta(s)$. Certain sums over the $\gamma$'s are evaluated, by using the function $G(s) = \sum_{\gamma>0}\gamma^{-s}$ and other techniques. Some integrals…
Inspired by a question asked on the list {\tt mathfun}, we revisit {\em Kempner-like series}, i.e., harmonic sums $\sum' 1/n$ where the integers $n$ in the summation have ``restricted'' digits. First we give a short proof that $\lim_{k \to…
We give a Pieri-type formula for the sum of $K$-$k$-Schur functions $\sum_{\mu\le\lambda} g^{(k)}_{\mu}$ over a principal order ideal of the poset of $k$-bounded partitions under the strong Bruhat order, which sum we denote by…
Starting from some of Norman Levinson's results, we construct interesting examples of functions $f(s)$ such that for $s=\frac12+it$, we have $Z(t)=2\Re\{\pi^{-\frac{s}{2}}\Gamma(s/2)f(s)\}$. For example one such function is…
Estimates for $Z_2(s) = \int_1^|infty |\zeta(1/2+ix)|^4x^{-s}dx (\Re s > 1)$ are discussed, both pointwise and in mean square. It is shown how these estimates can be used to bound $E_2(T)$, the error term in the asymptotic formula for…
The paper deals with the problem of ideals of $H^\infty$: describe increasing functions $\phi\ge 0$ such that for all bounded analytic functions $f_1,f_2,...,f_n, \tau$ in the unit disc $D$ the condition $|\tau(z) | \le \phi(\sum_k…
We calculate some infinite sums containing the digamma function in closed-form. These sums are related either to the incomplete beta function or to the Bessel functions. The calculations yield interesting new results as by-products, such as…
It is argued that to arrive at a quantitative description of the surface tension of a liquid drop as a function of its inverse radius, it is necessary to include the bending rigidity k and Gaussian rigidity k_bar in its description. New…
In the present paper, using S.L. Sobolev's method, interpolation spline that minimizes the expression $\int_0^1(\varphi^{(m)}(x)+\omega^2\varphi^{(m-2)}(x))^2dx$ in the $K_2(P_m)$ space are constructed. Explicit formulas for the…
In this paper we prove some exponential inequalities involving the sinc function. We analyze and prove inequalities with constant exponents as well as inequalities with certain polynomial exponents. Also, we establish intervals in which…
We find two-sides estimates for the best uniform approximations of classes of convolutions of $2\pi$-periodic functions from unit ball of the space $L_p, 1 \le p <\infty,$ with fixed kernels, modules of Fourier coefficients of which satisfy…
We find the asymptotics of the series $\sum_{n=1}^\infty (-1)^n n^{-1} \exp(-t/n)$ as $t\to+\infty$. The answer is an oscillating function of $t$ dominated by $\exp(-(2\pi t)^{1/2})$. The intermediate step is to find the asymptotics of the…
We investigate {\bf explicit} universal estimate of finite Morse index solutions to polyharmonic equations. \,Differently to previous works \cite{BL2, DDF, fa, H1}, propose here a direct proof using a new interpolation inequality and a…
In this article we present certain formulas involving arithmetical functions. In the first part we study properties of sums and product formulas for general type of arithmetic functions. In the second part we apply these formulas to the…
In this paper certain classes of infinite sums involving special functions are evaluated analytically by application of basic quantum mechanical principles to simple models of half harmonic oscillator and a particle trapped inside an…
To prove by probabilistic methods that every $(n-1)$-dimensional section of the unit cube in $R^n$ has volume at most $\sqrt 2$, K. Ball made essential use of the inequality $$ \frac{1}{\pi}\int_{-\infty}^{\infty} \left(\frac{\sin^2…
We obtain a result concerning the stability under the interpolation with functional parameter method for the approximation spaces of Lorentz-Marcinkiewicz type and also for the approximation spaces generated by symmetric norming functions…
A representation for the Riemann zeta function valid for arbitrary complex $s=\sigma+it$ is $\zeta(s)=\sum_{n=0}^\infty A(n,s)$, where \[A(n,s)=\frac{2^{-n-1}}{1-2^{1-s}} \sum_{k=0}^n \left(\!\begin{array}{c}n\\k\end{array}\!\right)…
In this short note, we derive an upper-bound for the sum of two comparison functions, namely for the sum of a class K and an extended class K function. To the best of our knowledge, the relations derived in this note have not been…
We introduce a statistical quantity, known as the $K$ function, related to the integral of the two--point correlation function. It gives us straightforward information about the scale where clustering dominates and the scale at which…