{"paper":{"title":"An arithmetical approach to the convergence problem of series of dilated functions and its connection with the Riemann Zeta function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michel Weber","submitted_at":"2014-07-10T08:14:46Z","abstract_excerpt":"Given a periodic function $f$, we study the convergence almost everywhere and in norm of the series $\\sum_{k} c_k f(kx)$. Let $f(x)= \\sum_{m=1}^\\infty a_m \\sin {2\\pi m x}$ where $\\sum_{m=1}^\\infty a_{m }^2d(m) <\\infty$ and $d(m)=\\sum_{d|m} 1$, and let $f_n(x) = f(nx)$. We show by using a new decomposition of squared sums that for any $K\\subset \\N$ finite, $ \\|\\sum_{k\\in K} c_k f_k \\|_2^2 \\le ( \\sum_{m=1}^\\infty a_{m }^2 d(m)\n  ) \\sum_{k\\in K } c_{k}^2d(k^2)$. If $f^s (x)= \\sum_{j=1}^\\infty \\frac{\\sin 2\\pi jx}{j^s}$,\n  $s>1/2$, by only using\n  elementary Dirichlet convolution calculus, we show "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.2722","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}