کد مقاله | کد نشریه | سال انتشار | مقاله انگلیسی | نسخه تمام متن |
---|---|---|---|---|
6415335 | 1630652 | 2016 | 43 صفحه PDF | دانلود رایگان |
Given a periodic function f, we study the convergence almost everywhere and in norm of the series âkckf(kx). Let f(x)=âm=1âamsinâ¡2Ïmx where âm=1âam2d(m)<â and d(m)=âd|m1, and let fn(x)=f(nx). We show by using a new decomposition of squared sums that for any KâN finite, ââkâKckfkâ22â¤(âm=1âam2d(m))âkâKck2d(k2). If f(s)(x)=âj=1âsinâ¡2Ïjxjs, s>1/2, by only using elementary Dirichlet convolution calculus, we show that for 0<εâ¤2sâ1, ζ(2s)â1ââkâKckfk(s)â22â¤1+εε(âkâK|ck|2Ï1+εâ2s(k)), where Ïh(n)=âd|ndh. From this we deduce that if fâBV(T), ãf,1ã=0 andâkck2(logâ¡logâ¡k)4(logâ¡logâ¡logâ¡k)2<â, then the series âkckfk converges almost everywhere. This slightly improves a recent result, depending on a fine analysis on the polydisc [1, th. 3] (nk=k), where it was assumed that âkck2(logâ¡logâ¡k)γ converges for some γ>4. We further show that the same conclusion holds under the arithmetical conditionâkck2(logâ¡logâ¡k)2+bÏâ1+1(logâ¡logâ¡k)b/3(k)<â, for some b>0, or if âkck2d(k2)(logâ¡logâ¡k)2<â. 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 Ï>1/2, 0â¤Îµ<Ï, we havemax1â¤tâ¤Tâ¡|ζ(Ï+it)|â¥C(Ï)(1Ïâ2ε(ν)ân|νÏâs+ε(n)2n2ε)1/2. We finally prove an important complementary result to Wintner's famous characterization of mean convergence of series âk=0âckfk.
Journal: Journal of Number Theory - Volume 162, May 2016, Pages 137-179