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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 ","authors_text":"Michel Weber","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-10T08:14:46Z","title":"An arithmetical approach to the convergence problem of series of dilated functions and its connection with the Riemann Zeta function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.2722","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:239bd003de138d1504d09080ebb41d74775987a3f8197c705e658507958424c9","target":"record","created_at":"2026-05-18T00:39:58Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"be738390d12592e12299fd4d29bd4779c8d82e4ec0eab6ca0077e4648f955df1","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-07-10T08:14:46Z","title_canon_sha256":"b88dfddc04b6cefffe083984724c4c997c5b9ec98373a6e61abbd404899be493"},"schema_version":"1.0","source":{"id":"1407.2722","kind":"arxiv","version":2}},"canonical_sha256":"f8ce1d8cecef7cc02eb160471228f707cdb0fc4cdf2054a97a7023920c3ea83b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f8ce1d8cecef7cc02eb160471228f707cdb0fc4cdf2054a97a7023920c3ea83b","first_computed_at":"2026-05-18T00:39:58.730850Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:39:58.730850Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"g2uEuGbOTd1WjqpWa6czKT2AHno+RLRCO4Ryxtx9RLyzzHDUZKR5Pw0CEiDe02pqACwGtKVqtZLXwNxXkEpXDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:39:58.731563Z","signed_message":"canonical_sha256_bytes"},"source_id":"1407.2722","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:239bd003de138d1504d09080ebb41d74775987a3f8197c705e658507958424c9","sha256:8fa467803fca8cac725187e9fbf391d06aec92ad49bbfc369af1892a53a558ba"],"state_sha256":"0978d2d6914f2d344febf2cc04f79ab42e9f0ef2df7e27afb8538332d0d76181"}