An arithmetical approach to the convergence problem of series of dilated functions and its connection with the Riemann Zeta function

详细信息    查看全文

• 作者：Michel J.G. Weber ^{michel.weber@math.unistra.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Systems of dilated functions ; Series ; Decomposition of squared sums ; FC sets ; GCD ; Arithmetical functions ; Dirichlet convolution ; Ω-theorem ; Riemann Zeta function ; Mean convergence ; Almost everywhere convergence
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：162
• 期：Complete
• 页码：137-179
• 全文大小：600 K

文摘

Given a periodic function f  , we study the convergence almost everywhere and in norm of the series ∑_kc_kf(kx)${\sum }_k{c}_{k}f(kx)$. Let 19a83a352b37e2">$f(x)={\sum }_{m=1}^{\infty }{a}_m\sin 2\pi mx$ where ${\sum }_{m=1}^{\infty }{a}_m^{2}d(m)<\infty$ and d(m)=∑_d|m1$d(m)={\sum }_{d|m}1$, and let f_n(x)=f(nx)$f_n(x)=f(nx)$. We show by using a new decomposition of squared sums that for any K⊂N$K\subset \mathbb{N}$ finite, ${\Vert {\sum }_{k\in K}{c}_k{f}_k\Vert }_2^2\le ({\sum }_{m=1}^{\infty }{a}_m^{2}d(m)){\sum }_{k\in K}{c}_k^{2}d(k^2)$. If $f^{(s)}(x)={\sum }_{j=1}^{\infty }\frac{\sin 2\pi jx}{j^s}$, s>1/2$s>1/2$, by only using elementary Dirichlet convolution calculus, we show that for 0<ε≤2s−1$0<\epsilon \le 2s-1$, $\zeta {(2s)}^{-1}{\Vert {\sum }_{k\in K}{c}_k{f}_k^{(s)}\Vert }_2^2\le \frac{1+\epsilon }{\epsilon }({\sum }_{k\in K}{|c_k|}^2{\sigma }_{1+\epsilon -2s}(k))$, where σ_h(n)=∑_d|nd^h${\sigma }_h(n)={\sum }_{d|n}{d}^h$. From this we deduce that if f∈BV(T)$f\in \mathrm{BV}(\mathbb{T})$, 〈f,1〉=0$〈f,1〉=0$ and then the series ∑_kc_kf_k${\sum }_k{c}_k{f}_k$ converges almost everywhere. This slightly improves a recent result, depending on a fine analysis on the polydisc [1, th. 3] (n_k=k$n_k=k$), where it was assumed that ${\sum }_k{c}_k^2{(\log \log k)}^{\gamma }$ converges for some γ>4$\gamma >4$. We further show that the same conclusion holds under the arithmetical condition for some 19a5a1a94" title="Click to view the MathML source">b>0$b>0$, or if a228d3e25412">${\sum }_k{c}_k^{2}d(k^2){(\log \log k)}^2<\infty$. We also derive from a recent result of Hilberdink an Ω-result for the Riemann Zeta function involving factor closed sets. As an application we find that simple conditions on T and ν   ensuring that for any σ>1/2$\sigma >1/2$, 0≤ε<σ$0\le \epsilon <\sigma$, we have We finally prove an important complementary result to Wintner's famous characterization of mean convergence of series ${\sum }_{k=0}^{\infty }{c}_k{f}_k$.