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

JOURNAL OF NUMBER THEORY

卷:162

Article

Weber, Michel J. G.¹ 

[1] IRMA, 10 Rue Gen Zimmer, F-67084 Strasbourg, France

关键词: Systems of dilated functions;    Series;    Decomposition of squared sums;    FC sets;    GCD;    Arithmetical functions;    Dirichlet convolution;    Omega-theorem;    Riemann Zeta function;    Mean convergence;    Almost everywhere convergence;   

DOI  :  10.1016/j.jnt.2015.10.002

来源: Elsevier

【 摘 要 】

Given a periodic function f , we study the convergence almost everywhere and in norm of the series Sigma(k) ck f(kx) . Let f(x)=Sigma(infinity)(m=1) a(m)(2) d(m) < infinity d(m)=Sigma d(/m) where Sigma(infinity)(m=1) a(m)(2) d(m)1/2 , by only using elementary Dirichlet convolution calculus, we show that for 0< epsilon <= 2s-1,zeta(2s)(-1) parallel to Sigma k is an element of k ck f(kx) <= 1+epsilon/epsilon(Sigma k is an element of k vertical bar xk vertical bar(2) sigma(1+epsilon)2s(k))(k)), where sigma h(n)=Sigma(dh)(d/n). From this we deduce that if f is an element of BV(T), < f,1 > =0 and Sigma(ck2)(k) (log log k)(4)/(log log log k)(2) 4. We further show that the same conclusion holds under the arithmetical condition for some b>0 , or if Sigma kc(k)(2)d(k(2))(loglogk)(2) < infinity . We also derive from a recent result of Hilberdink an Omega-result for the Riemann Zeta function involving factor closed sets. max(1 <= t <= T) vertical bar zeta(sigma+it)vertical bar >= C(sigma)(1/sigma-2 epsilon(v) Sigma(n/v) sigma-s+epsilon(n)(2/)n(2 epsilon))(1/2) . We finally prove an important complementary result to Wintner's famous characterization of mean convergence of series Sigma(infinity)(k=0) ckfk . (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

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