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Under reasonable conditions on the coefficient sequence $\\{c^j_n\\}_{n,j}$, we show that $$ \\lim_{T\\to \\infty}\\frac{1}{T} \\int_{0}^T \\Big| \\sum_{j\\in J_n} c^j_n\\,j^{it}\\Big| \\dd t\\sim \\big(\\frac{\\pi} {2}\\sum_{j\\in J_n} (c^j_n)^2\\big)^{1/2} $$ as $n\\to \\infty$. We also show by means of an elementary device that for all $0<\\a<2$, \\begin{eqnarray*}\n  \\lim_{T\\to \\infty} \\Big(\\frac{1}{T} \\int_{0}^T \\big| \\sum_{n=1}^N n^{-it}\\big|^\\a\\dd t\\Big)^{1/\\a} \\ge C_\\a\\, \\frac{ N^{\\frac{1}{2}}} {\\big( \\log N\\big)^{{\\frac{1}{\\"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1411.7291","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CV","submitted_at":"2014-11-26T16:33:01Z","cross_cats_sorted":[],"title_canon_sha256":"b4691b45718476913d0c3461ea8a0de5e5b59420c35276dea64c4fc017407463","abstract_canon_sha256":"d48bd54b2dfbe20bdfaae2955de683105d797cffcfa6ad396f5d0abc5e1663d9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:58.713935Z","signature_b64":"WMjeGT297CiNZ4cA1rFLMhtu1jWbrwUz8oXlTMXLy2fLvrw1UywQkC1/AWgjPCIwpbtppSN3NLubDun4m3cxBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"458c6811a6e57ae13507b20e9d366dd7389760752e363a4c578b91c51624d93d","last_reissued_at":"2026-05-18T00:39:58.713472Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:58.713472Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$\\boldsymbol L^{\\boldsymbol 1}$-Norm of Steinhaus chaose on the polydisc","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CV","authors_text":"Michel Weber","submitted_at":"2014-11-26T16:33:01Z","abstract_excerpt":"Let $J_n\\subset[1,n]$, $n=1,2,\\ldots$ be increasing sets of mutually coprime numbers. Under reasonable conditions on the coefficient sequence $\\{c^j_n\\}_{n,j}$, we show that $$ \\lim_{T\\to \\infty}\\frac{1}{T} \\int_{0}^T \\Big| \\sum_{j\\in J_n} c^j_n\\,j^{it}\\Big| \\dd t\\sim \\big(\\frac{\\pi} {2}\\sum_{j\\in J_n} (c^j_n)^2\\big)^{1/2} $$ as $n\\to \\infty$. We also show by means of an elementary device that for all $0<\\a<2$, \\begin{eqnarray*}\n  \\lim_{T\\to \\infty} \\Big(\\frac{1}{T} \\int_{0}^T \\big| \\sum_{n=1}^N n^{-it}\\big|^\\a\\dd t\\Big)^{1/\\a} \\ge C_\\a\\, \\frac{ N^{\\frac{1}{2}}} {\\big( \\log N\\big)^{{\\frac{1}{\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.7291","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1411.7291","created_at":"2026-05-18T00:39:58.713537+00:00"},{"alias_kind":"arxiv_version","alias_value":"1411.7291v1","created_at":"2026-05-18T00:39:58.713537+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1411.7291","created_at":"2026-05-18T00:39:58.713537+00:00"},{"alias_kind":"pith_short_12","alias_value":"IWGGQENG4V5O","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_16","alias_value":"IWGGQENG4V5OCNIH","created_at":"2026-05-18T12:28:33.132498+00:00"},{"alias_kind":"pith_short_8","alias_value":"IWGGQENG","created_at":"2026-05-18T12:28:33.132498+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/IWGGQENG4V5OCNIHWIHJ2NTN24","json":"https://pith.science/pith/IWGGQENG4V5OCNIHWIHJ2NTN24.json","graph_json":"https://pith.science/api/pith-number/IWGGQENG4V5OCNIHWIHJ2NTN24/graph.json","events_json":"https://pith.science/api/pith-number/IWGGQENG4V5OCNIHWIHJ2NTN24/events.json","paper":"https://pith.science/paper/IWGGQENG"},"agent_actions":{"view_html":"https://pith.science/pith/IWGGQENG4V5OCNIHWIHJ2NTN24","download_json":"https://pith.science/pith/IWGGQENG4V5OCNIHWIHJ2NTN24.json","view_paper":"https://pith.science/paper/IWGGQENG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1411.7291&json=true","fetch_graph":"https://pith.science/api/pith-number/IWGGQENG4V5OCNIHWIHJ2NTN24/graph.json","fetch_events":"https://pith.science/api/pith-number/IWGGQENG4V5OCNIHWIHJ2NTN24/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/IWGGQENG4V5OCNIHWIHJ2NTN24/action/timestamp_anchor","attest_storage":"https://pith.science/pith/IWGGQENG4V5OCNIHWIHJ2NTN24/action/storage_attestation","attest_author":"https://pith.science/pith/IWGGQENG4V5OCNIHWIHJ2NTN24/action/author_attestation","sign_citation":"https://pith.science/pith/IWGGQENG4V5OCNIHWIHJ2NTN24/action/citation_signature","submit_replication":"https://pith.science/pith/IWGGQENG4V5OCNIHWIHJ2NTN24/action/replication_record"}},"created_at":"2026-05-18T00:39:58.713537+00:00","updated_at":"2026-05-18T00:39:58.713537+00:00"}