Power Moments of the Riesz Mean Error Term of Symmetric Square L-Function in Short Intervals

Let f(z) be a holomorphic Hecke eigenform of weight k with respect to SL(2,Z) and let L(s,sym2f)=∑n=1∞cnn−s,ℜs>1 denote the symmetric square L-function of f. In this paper, we consider the Riesz mean of the form Dρ(x;sym2f)=L(0,sym2f)Γ(ρ+1)xρ+Δρ(x;sym2f) and derive the asymptotic formulas for ∫T−HT+HΔρk(x;sym2f)dx, when k≥3.

BOCHNER–RIESZ MEANS ON BLOCK-SOBOLEV SPACES IN COMPACT LIE GROUP

On a compact Lie group $G$ of dimension $n$, we study the Bochner–Riesz mean $S_{R}^{\unicode[STIX]{x1D6FC}}(f)$ of the Fourier series for a function $f$. At the critical index $\unicode[STIX]{x1D6FC}=(n-1)/2$, we obtain the convergence rate for $S_{R}^{(n-1)/2}(f)$ when $f$ is a function in the block-Sobolev space. The main theorems extend some known results on the $m$-torus $\mathbb{T}^{m}$.

On the Riesz means of δ k (n)

International audience Let k ≥ 1 be an integer. Let δ k (n) denote the maximum divisor of n which is co-prime to k. We study the error term of the general m-th Riesz mean of the arithmetical function δ k (n) for any positive integer m ≥ 1, namely the error term E m,k (x) where 1 m! n≤x δ k (n) 1 − n x m = M m,k (x) + E m,k (x). We establish a non-trivial upper bound for E m,k (x) , for any integer m ≥ 1.