A new MnxOy/carbon nanorods derived from bimetallic Zn/Mn metal–organic framework as an efficient oxygen reduction reaction electrocatalyst for alkaline Zn-Air batteries

Journal of Solid State Electrochemistry - In this work, nanorods like bimetallic Zn/Mn metal–organic-frameworks (MOFs) are proposed as the precursor for preparing MnxOy/porous carbon...     

Shifted convolution sums of Fourier coefficients with squarefull kernel functions

Lithuanian Mathematical Journal - Let f(z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth normalized Fourier coefficient (Hecke...     

On a divisor problem related to the Epstein zeta-function, II

Recently by using the theory of modular forms and the Riemann zeta-function, Lü improved the estimates for the error term in a divisor problem related to the Epstein zeta-function established by Sankaranarayanan. In this short note, we are able to further sharpen some results of Sankaranarayanan and of Lü, and to establish corresponding Ω-estimates.     

Note on divisor function for quaternion algebras

Let a be an integral ideal in a quaternion algebra U over rational numbers Q which ramifies precisely at p and ∞, and d(a) be its divisor function. Recently, Kim and Zhang proved a quaternion analogue of the classical formula , and then established an asymptotic formula for the sum . In this note we improved the upper bound for the error term in the asymptotic formula, and then considered the quaternion analogue of another well-known formula .     

On averages of Fourier coefficients of Maass cusp forms

Let φ be a primitive Maass cusp form and t _φ (n) be its nth Fourier coefficient at the cusp infinity. In this short note, we are interested in the estimation of the sums ${\sum_{n \leq x}t_{\varphi}(n)}$ and ${\sum_{n \leq x}t_{\varphi}(n^2)}$ . We are able to improve the previous results by showing that for any ${\varepsilon > 0}$ $$\sum_{n \leq x}t_{\varphi}(n) \ll\, _{\varphi, \varepsilon} x^{\frac{1027}{2827} + \varepsilon} \quad {and}\quad\sum_{n \leq x}t_{\varphi}(n^2) \ll\,_{\varphi, \varepsilon} x^{\frac{489}{861} + \varepsilon}.$$      

Mean values connected with the Dedekind zeta-function of a non-normal cubic field

After Landau’s famous work, many authors contributed to some mean values connected with the Dedekind zetafunction. In this paper, we are interested in the integral power sums of the coefficients of the Dedekind zeta function of a non-normal cubic extension K ₃/?, i.e. $ S_{l,K_3 } (x) = \sum\nolimits_{m \leqslant x} {M^l (m)} $ , where M(m) denotes the number of integral ideals of the field K ₃ of norm m and l ∈ ?. We improve the previous results for $ S_{2,K_3 } (x) $ and $ S_{3,K_3 } (x) $ .     

On Fractional Power Moments of <Emphasis Type="Italic">L</Emphasis>-functions Associated with Certain Cusp Forms

This paper is devoted to upper and lower bounds of fractional moments of L-functions attached to certain cusp forms. The upper bound is proved under the analogue of the Riemann hypothesis.     

The sixth and eighth moments of Fourier coefficients of cusp forms

Let λ(n) be the nth normalized Fourier coefficient of a holomorphic Hecke eigencuspform f(z) of even integral weight k for the full modular group. In this paper we are able to prove the following results.

(i)

For any ε>0, we have

     

The generalized Bourgain-Sarnak-Ziegler criterion and its application to additively twisted sums on GLm

The Bourgain-Sarnak-Ziegler(BSZ) criterion was initially established for an arbitrary multiplicative function a(n) satisfying |a(n)| ≤1. Recently, Cafferata et al.(2020) showed that the condition |a(n)| ≤1 can be replaced by a(p) ? 1 to some extent. In this paper, we formulate and prove a further extension of the BSZ criterion, in which the restriction a(p) ? 1 is removed. As applications, we use it together with the analytic theory of automorphic L-functions to prove that there exist some cancellations in the sequence{λπ(n)e(n~kα)}n ≥1 on GL_m and the M?bius function is disjoint from this sequence.     

The ternary Goldbach–Vinogradov theorem with almost equal primes from the Beatty sequence

Let α ₁, α ₂, α ₃, β ₁, β ₂, β ₃ be real numbers with α ₁, α ₂, α ₃ >1. Suppose that each individual α _i is of a finite type and that at least one pair $\alpha_{i}^{-1}$ , $\alpha_{j}^{-1}$ is also of a finite type. In this paper we prove that every large odd integer n can be represented as $$n=p_{1}+p_{2}+p_{3}, $$ with p _i =n/3+O(n ^2/3(logn) ^c ) and $p_{i}\in\mathcal{B}_{i}$ , where c>0 is an absolute constant and $\mathcal{B}_{i}$ denotes the so-called Beatty sequence, i.e. $$\mathcal{B}_{i}=\bigl\{n\in\mathbb{N}: n=[\alpha_{i}m+ \beta_{i}] \mbox { for some } m\in\mathbb{Z}\bigr\}. $$