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For $q$ prime, we compute the moments of $D(s,a/q)$ at the central point $s=1/2$, when averaging over $1\\leq a<q$.\n  As a consequence we deduce the asymptotic for the iterated moment of Dirichlet $L$-functions $\\sum_{\\chi_1,\\dots,\\chi_k\\mod q}|L(\\frac12,\\chi_1)|^2\\cdots |L(\\frac12,\\chi_k)|^2|L(\\frac12,\\chi_1\\cdots \\chi_k)|^2$, obtaining a power saving error term.\n  Also, we compute the moments of certain functions de"},"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":"1701.06601","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-01-23T19:40:35Z","cross_cats_sorted":[],"title_canon_sha256":"9f2a38433afc1dc3446dffb35afffef1173c1fed0b3c64f2a38fa964bddcf597","abstract_canon_sha256":"dfbb79ae910ff74baac55ff03bab0669e84c8d26ccf6fd9292a8e6a37329ac8e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:16.527376Z","signature_b64":"125Yz/RNgQeuYOTr32Tlrj+Hgn+c1BV2D1RNZQzSDWodUQROFLgn3wRv+xCm0jZxHfQX8XhrRklRWGc/ZqsuDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"da489e6865bf0398c3319e550a62bc46b70cd8317c70d1f5cf496f0a549c3c7d","last_reissued_at":"2026-05-17T23:50:16.526718Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:16.526718Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"High moments of the Estermann function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Sandro Bettin","submitted_at":"2017-01-23T19:40:35Z","abstract_excerpt":"For $a/q\\in\\mathbb{Q}$ the Estermann function is defined as $D(s,a/q):=\\sum_{n\\geq1}d(n)n^{-s}\\operatorname{e}(n\\frac aq)$ if $\\Re(s)>1$ and by meromorphic continuation otherwise. For $q$ prime, we compute the moments of $D(s,a/q)$ at the central point $s=1/2$, when averaging over $1\\leq a<q$.\n  As a consequence we deduce the asymptotic for the iterated moment of Dirichlet $L$-functions $\\sum_{\\chi_1,\\dots,\\chi_k\\mod q}|L(\\frac12,\\chi_1)|^2\\cdots |L(\\frac12,\\chi_k)|^2|L(\\frac12,\\chi_1\\cdots \\chi_k)|^2$, obtaining a power saving error term.\n  Also, we compute the moments of certain functions de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.06601","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":"1701.06601","created_at":"2026-05-17T23:50:16.526808+00:00"},{"alias_kind":"arxiv_version","alias_value":"1701.06601v1","created_at":"2026-05-17T23:50:16.526808+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.06601","created_at":"2026-05-17T23:50:16.526808+00:00"},{"alias_kind":"pith_short_12","alias_value":"3JEJ42DFX4BZ","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_16","alias_value":"3JEJ42DFX4BZRQZR","created_at":"2026-05-18T12:30:58.224056+00:00"},{"alias_kind":"pith_short_8","alias_value":"3JEJ42DF","created_at":"2026-05-18T12:30:58.224056+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/3JEJ42DFX4BZRQZRTZKQUYV4I2","json":"https://pith.science/pith/3JEJ42DFX4BZRQZRTZKQUYV4I2.json","graph_json":"https://pith.science/api/pith-number/3JEJ42DFX4BZRQZRTZKQUYV4I2/graph.json","events_json":"https://pith.science/api/pith-number/3JEJ42DFX4BZRQZRTZKQUYV4I2/events.json","paper":"https://pith.science/paper/3JEJ42DF"},"agent_actions":{"view_html":"https://pith.science/pith/3JEJ42DFX4BZRQZRTZKQUYV4I2","download_json":"https://pith.science/pith/3JEJ42DFX4BZRQZRTZKQUYV4I2.json","view_paper":"https://pith.science/paper/3JEJ42DF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1701.06601&json=true","fetch_graph":"https://pith.science/api/pith-number/3JEJ42DFX4BZRQZRTZKQUYV4I2/graph.json","fetch_events":"https://pith.science/api/pith-number/3JEJ42DFX4BZRQZRTZKQUYV4I2/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/3JEJ42DFX4BZRQZRTZKQUYV4I2/action/timestamp_anchor","attest_storage":"https://pith.science/pith/3JEJ42DFX4BZRQZRTZKQUYV4I2/action/storage_attestation","attest_author":"https://pith.science/pith/3JEJ42DFX4BZRQZRTZKQUYV4I2/action/author_attestation","sign_citation":"https://pith.science/pith/3JEJ42DFX4BZRQZRTZKQUYV4I2/action/citation_signature","submit_replication":"https://pith.science/pith/3JEJ42DFX4BZRQZRTZKQUYV4I2/action/replication_record"}},"created_at":"2026-05-17T23:50:16.526808+00:00","updated_at":"2026-05-17T23:50:16.526808+00:00"}