相关论文: On error term estimates \`a la Walfisz for mean va…

Bernstein's theorem (also called Hausdorff--Bernstein--Widder theorem) enables the integral representation of a completely monotonic function. We introduce a finite completely monotonic function, which is a completely monotonic function…

Euler's totient function, $\varphi(n)$, which counts how many of $0,1,\dots,n-1$ are coprime to $n$, has an explicit asymptotic lower bound of $n/\log \log n$, modulo some constant. In this note, we generalise $\varphi$; given an…

We improve the Modified Winitzki's Approximation of the error function $erf(x)\cong \sqrt{1-e^{-x^2\frac{\frac{4}{\pi}+0.147x^2}{1+0.147x^2}}}$ which has error $|\varepsilon (x)| < 1.25 \cdot 10^{-4}$ $\forall x \ge 0$ till reaching 4…

We provide an abstract multivariate central limit theorem with the Lindeberg-type error bounded in terms of Lipschitz functions (Wasserstein 1-distance) or functions with bounded second or third derivatives. The result is proved by means of…

Asymptotic study on the partition function $p(n)$ began with the work of Hardy and Ramanujan. Later Rademacher obtained a convergent series for $p(n)$ and an error bound was given by Lehmer. Despite having this, a full asymptotic expansion…

We study the distribution of families of multiplicative functions among the coprime residue classes to moduli varying uniformly in a wide range, obtaining analogues of the Siegel--Walfisz Theorem for large classes of multiplicative…

For a real analytic periodic function $\phi:\mathbb{R}\to \mathbb{R}$, an integer $b\ge 2$ and $\lambda\in (1/b,1)$, we prove the following dichotomy for the Weierstrass-type function $W(x)=\sum\limits_{n\ge 0}{{\lambda}^n\phi(b^nx)}$:…

Let $x \mapsto x+ \alpha$ be a rotation on the circle and let $\varphi$ be a step function. We denote by $\varphi\_n (x)$ the corresponding ergodic sums $\sum\_{j=0}^{n-1} \varphi(x+j \alpha)$. Under an assumption on $\alpha$, for example…

With regard to a three-step estimation procedure, proposed without theoretical discussion by Li and You in Journal of Applied Statistics and Management, for a nonparametric regression model with time-varying regression function, local…

We suggest a new optimization technique for minimizing the sum $\sum_{i=1}^n f_i(x)$ of $n$ non-convex real functions that satisfy a property that we call piecewise log-Lipschitz. This is by forging links between techniques in computational…

Let $f\colon\mathbb{N}\rightarrow\mathbb{C}$ be an arithmetic function and consider the Beatty set $\mathcal{B}(\alpha) = \lbrace\, \lfloor n\alpha \rfloor : n\in\mathbb{N} \,\rbrace$ associated to a real number $\alpha$, where…

We show that the expected asymptotic for the sums $\sum_{X < n \leq 2X} \Lambda(n) \Lambda(n+h)$, $\sum_{X < n \leq 2X} d_k(n) d_l(n+h)$, and $\sum_{X < n \leq 2X} \Lambda(n) d_k(n+h)$ hold for almost all $h \in [-H,H]$, provided that…

Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem. Our main results are the asymptotic formulas $$ \int_1^X \Delta^3(x){\rm d}x = BX^{7/4} + O_\epsilon(X^{\beta+\epsilon}) \qquad(B > 0) $$ and $$ \int_1^X…

In this note we generalize and prove a recent conjecture of Varchenko concerning the number of critical points of a (multivalued) meromorphic function $\phi$ on an algebraic manifold. Under certain conditions, this number turns out to…

Let $\tau$ denote the divisor function, and $f$ be any multiplicative function that satisfies some mild hypotheses. We establish the asymptotic formula or non-trivial upper bound for the shifted convolution sum $\sum_{n \leq…

Let $f$ and $g$ be holomorphic or Maass cusp forms for $\rm SL_2(\mathbb{Z})$ with normalized Fourier coefficients $\lambda_f(n)$ and $\lambda_g(n)$, respectively. In this paper, we prove nontrivial estimates for the sum $$…

Let $p_n$ denote the $n$-th prime. In 2000, Panaitopol established the inequality $p_1 \cdots p_n > p_{n+1}^{n - \pi(n)}$ for all $n \geq 2$, where $\pi(x)$ is the prime counting function. In 2021, Yang and Liao refined this by introducing…

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 study minimizers of non-autonomous functionals \begin{align*} \inf_u \int_\Omega \varphi(x,|\nabla u|) \, dx \end{align*} when $\varphi$ has generalized Orlicz growth. We consider the case where the upper growth rate of $\varphi$ is…

Let F be an orthonormal basis of weight 2 cusp forms on Gamma_0(N). We show that various weighted averages of special values L(f \tensor chi, 1) over f in F are equal to 4 pi + O(N^{-1 + epsilon}). A previous result of Duke gives an error…