{"id":"397a9d17-c2d8-463b-9b96-2c14747d9892","arxiv_id":"1908.03658","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For every number field K, the Riemann hypothesis for ζ_K is equivalent to a specific o(q^{3/4-ε}) error term in the convergence of the discrete measures m_q to the measure κ/ζ_K(2) q dq.","lead":"The paper shows that the Extended Riemann Hypothesis for the Dedekind zeta function of any number field is equivalent to a precise rate of convergence of certain discrete arithmetic measures to a continuous measure. It generalizes Verjovsky's measure-theoretic formulation of the Riemann hypothesis from the rationals to arbitrary number fields.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Converse of Theorem 1.1(C) omits the test-function separation needed to turn holomorphy of M_f for every f into zero-freeness of ζ_K(2s); the gap is easily filled but is load-bearing.","rationale":"The reader's weakest assumption identifies exactly the locus where the central claim is least secure: the converse of Theorem 1.1(C). In that direction, the paper derives holomorphy of M_f(s) on Re(s)>1/4 for every test function f, and then immediately concludes that ζ_K(2s) has no zeros there. The missing step is a separation argument showing that for any possible zero ρ of ζ_K with Re(ρ)>1/2, some admissible f has nonvanishing Mellin transform at ρ. Without this, the conclusion is a non sequitur. The gap is real as written, but it is also very easily repaired: take f_ρ(q)=q^{-ρ}ψ(q) with any positive smooth bump ψ supported away from 0; its Mellin transform at ρ is positive. Thus the central mathematical claim is very likely correct, and the issue warrants a conditional revision rather than rejection. I agree with the reader's assessment. I would not downgrade the verdict; the paper's other flagged issue, the false identity in Lemma 4.3, is less central to the ERH equivalence and is likewise repairable or avoidable by replacing the lemma with the trivial consequence of the known Mertens bound. Overall, the reader's conditional verdict remains the appropriate one.","tokens_in":12686,"tokens_out":25077,"duration_ms":248522,"concrete_test":"","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central equivalence in Theorem 1.1(C) is proven in Section 4.4 (labelled 'Proof of Theorem D' in the manuscript). The assumption m_q(f)-m(f)=o(q^{3/4-ε}) for every f∈C_c^ℓ with n+1≤ℓ≤∞ gives, via the estimates in that section, that M_f(s)=∫_0^T m_q(f)q^{s-2}dq is holomorphic for Re(s)>1/4+ε, hence for Re(s)>1/4, apart from the possible simple pole at s=1. Equation (5) states M_f(s)=2 ζ_K(2s-1)/ζ_K(2s) ∫_0^∞ f(q)q^{2s-1}dq. If ρ is a zero of ζ_K with Re(ρ)>1/2, then s0=ρ/2 lies in Re(s)>1/4 and the denominator ζ_K(2s) vanishes at s0. To obtain a contradiction one must exhibit an allowed test function f with ∫_0^∞ f(q)q^{ρ-1}dq ≠ 0; otherwise the pole at s0 could be cancelled by a common zero of all Mellin transforms. The paper never supplies this separation argument, so the implication 'error bound for every f ⇒ ERH' does not follow as written. This is not a purely cosmetic omission: the 'only if' direction of the central claim literally depends on excluding such cancellation. The separation is true and easy to provide, as the concrete test shows, but the manuscript leaves it out.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper defines, for an algebraic number field K, discrete measures m_q on R+ by m_q(f)=∑_{a⊂o_K} q φ_K(a) f(q^{1/2} N(a)), and studies their weak convergence as q→0 to the absolutely continuous measure m(f)=κ/ζ_K(2)∫ f(q) q dq. The main result, Theorem 1.1(C), asserts that the Extended Riemann Hypothesis for ζ_K(s) is equivalent to the statement that for every f∈C_c^ℓ(R+) with n+1≤ℓ≤∞, one has m_q(f)=m(f)+o(q^{3/4-ε}) for all 0<ε<1/4, together with a one-parameter refinement involving zero-free half-planes. The proof is based on the Mellin transform identity M_f(s)=2ζ_K(2s-1)/ζ_K(2s) ∫ f(q) q^{2s-1} dq and on estimates for this transform combined with Mellin inversion. Further results give conditional rates of convergence under the Lindelöf hypothesis or under a generalized circle-problem hypothesis, and optimality statements for characteristic functions and certain continuous functions.","tokens_in":12953,"tokens_out":15089,"duration_ms":147345,"significance":"The explicit Mellin factorization M_f(s)=2φ_K(s) times the Mellin transform of f is the paper's main positive content; it cleanly exposes why smoothness of f translates into decay of M_f and hence into improved error exponents. If Theorem 1.1(C) were fully established, it would provide a natural generalization of Verjovsky's criterion for ζ(s) to Dedekind zeta functions and could serve as a useful reformulation of the ERH. The paper does not appear to assume the conclusion, and the factorization itself is a valuable contribution. However, as detailed below, two gaps in the proofs — the missing test-function separation in the implication from the error bound to RH, and an invalid identity in the proof of Lemma 4.3 — mean that the central claim is not yet rigorously established as written.","major_comments":[{"comment":"The proof that m_q(f)=m(f)+o(q^{3/4-ε}) for every allowed f implies the Riemann hypothesis is incomplete. Equation (5) shows M_f(s)=2ζ_K(2s-1)/ζ_K(2s)∫f(q)q^{2s-1}dq. If ρ is a zero of ζ_K with Re(ρ)>1/2, then s0=ρ/2 lies in Re(s)>1/4 and ζ_K(2s) vanishes at s0, so M_f(s) would have a pole at s0 unless the test-function factor ∫f(q)q^{ρ-1}dq also vanishes. The manuscript never proves that there exists an allowed f with this integral nonzero. Consequently, the conclusion that M_f(s) is holomorphic for Re(s)>1/4 for every f contradicts the existence of such a zero only under an additional separation argument. This is load-bearing: it is exactly the 'only if' direction of the central equivalence. The gap is fillable, for example by choosing f_t(q)=η(q)q^t with η∈C_c^∞ positive and t outside the zero set of the entire function t↦∫η(q)q^{ρ-1+t}dq, but the argument must be supplied in the paper.","section":"Section 4.4 (Proof of Theorem D; Theorem 1.1(C), implication 'error bound for every f ⇒ RH')"},{"comment":"The proof of Lemma 4.3 contains a false identity. The text states that if x+1 is restricted to values that are norms of prime integral ideals, then φ_K(x+1)=|{n⊂o | N(n)≤x}|=κx+O(x^{1-1/n}). This is incorrect: for a prime ideal p with N(p)=x+1, the definition gives φ_K(p)=N(p)-1=x, not the counting function of all integral ideals of norm at most x. Moreover the notation φ_K(⌊x+1⌋) is ambiguous because φ_K is defined on integral ideals, not on real numbers. As written, the contradiction argument in Lemma 4.3 does not go through. The lemma may be salvageable by a different argument, but this proof needs to be replaced.","section":"Lemma 4.3 (used in Theorem 1.1(D))"},{"comment":"The two implications in the 'furthermore' part of Theorem 1.1(C), concerning an arbitrary α∈(1/2,3/4), are stated but not proved. Section 4.4 only treats the case α=3/4. The converse direction ('no zeroes in Re(s)>2(1-α) implies equation (2)') requires estimates for M_f(s) on the line Re(s)=1-α+ε analogous to those in Lemma 3.4, and the forward direction requires the same test-function separation argument as above, adapted to the zero-free region. Since these statements are part of the theorem's claim, they need either a proof or an explicit restriction of the theorem to the case α=3/4.","section":"Theorem 1.1(C), 'furthermore' part (general α)"}],"minor_comments":[{"comment":"The proof of Theorem 1.1(A) is written only for f equal to a characteristic function of an interval; the text says the remaining cases 'can be proved similarly'. For a claim about all f∈C_c^0(R+), a standard approximation by step functions requires a quantitative statement about the error under the measure m_q, so at least a brief explanation of this approximation step should be added.","section":"Section 4.1 (Proof of Theorem A)"},{"comment":"The section titles do not match the theorem numbering: Section 4.2 proves Theorem 1.1(D), Section 4.3 proves Theorem 1.1(B), and Section 4.4 proves Theorem 1.1(C). This makes the structure difficult to follow and should be corrected.","section":"Section labels (Sections 4.2–4.4)"},{"comment":"The statement 'if 1/2 ≤ Re(s), ζ_K(2s)^{-1}=O(1)' is not literally correct on the line Re(s)=1/2, since 1/ζ_K(1+it) can grow slowly as t→∞; the standard zero-free region gives O(t^ε) for any ε>0. The final decay estimate in the lemma is unaffected, but the displayed assertion should be qualified.","section":"Lemma 3.3 proof"},{"comment":"The definitions of the functions F in Theorem 1.1(E)–(F) contain the typo 'for ≤1', which should read 'for t≤1'. The same typo appears in Section 4.5.","section":"Theorem 1.1(E)–(F) and Section 4.5"},{"comment":"The claimed bound MF(σ+it)=O(1/(1+|t|)^{1+1/4}) in Theorem (E) is stated without displaying the computation; from φ_K(σ+it)=O(t^{n/2+ε}) and the Beta-function factor of degree ⌊n/2⌋+2 one obtains the even/odd-parity exponents -2+ε and -3/2+ε, which imply the stated bound but should be shown explicitly.","section":"Section 4.5, estimates for MF_r(s)"}],"recommendation":"major_revision","confidential_remarks":"The core Mellin-transform identity is correct and useful, and the central equivalence is likely salvageable, but the proof as written has a genuine gap in the 'error bound for every f ⇒ RH' direction and an invalid argument in Lemma 4.3. The one-parameter 'furthermore' statements in Theorem 1.1(C) are asserted without proof; the authors should either prove them or restrict the theorem. The editor may also wish to ask the authors to clarify the novelty of the general-α statements relative to Verjovsky's scalar results [18,19]."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a real extension of Verjovsky's rational case, and the core Mellin-transform mechanism is sound. But the converse direction of the main equivalence has a load-bearing omitted step, and Lemma 4.3 has a false identity. Both are fixable, but they are not cosmetic.\n\nThe new content is the passage from K=Q to arbitrary number fields: the smoothness threshold n+1, the conditional error terms under Lindelöf and circle-problem hypotheses, and the omega results in (E) and (F). The Mellin relation M_f(s)=2 ζ_K(2s-1)/ζ_K(2s) ∫ f(q)q^{2s-1}dq is correctly derived, and the forward direction of Theorem 1.1(C) — from ERH to the o(q^{3/4-ε}) bound — is fine once you accept the standard estimates used in Lemma 3.4.\n\nThe first real problem is the converse in Section 4.4. From the assumed error bound for every f, you obtain that M_f(s) is holomorphic for Re(s)>1/4 apart from the pole at s=1. To conclude that ζ_K(2s) has no zeros there, you must rule out cancellation: if ζ_K(2s_0)=0, you need a test function f with ∫ f(q) q^{2s_0-1}dq ≠ 0. The manuscript simply says \"therefore RH is true.\" This is the same test-function separation that Verjovsky had to handle, and it is easy to supply — fix any s, choose f supported near a point where the factor q^{σ-1}e^{it log q} has positive real part — but the paper leaves it out. Without it, the \"if\" direction of the claimed equivalence is unproved as written.\n\nThe second issue is Lemma 4.3. In the proof, the identity\nΦ_K(x+1)/(x+1)^2 = Φ_K(x)/(x+1)^2 + φ_K(⌊x+1⌋)/(x+1)^2\nis wrong for number fields: several ideals can share the same norm, so the increment is a sum over all ideals with norm in (x,x+1], not a single totient value. The author then writes φ_K(x+1)=|{n: N(n)≤x}|, which mixes ideals and integers and is not the Euler totient of an ideal. The lemma's statement is likely true, and the proof could be repaired by grouping ideals by norm and using known counting estimates, but as written it does not work. This affects Theorem 1.1(D), which relies on the lemma; the omega result is probably correct but currently not established.\n\nThere are also smaller issues: Section 4.4 is titled \"Proof of Theorem D\" when it proves (C), and the notation φ_K with an integer argument conflates an integral ideal with its norm. These are easy to clean up.\n\nOverall: the paper is worth a serious referee. The central equivalence is almost certainly correct and genuinely extends Verjovsky's work, but the converse gap in Theorem 1.1(C) and the broken Lemma 4.3 need to be fixed before publication. I would recommend conditional acceptance after major revision, with careful checking of the omega results.","headline":"A genuine but rough extension of Verjovsky's discrete-measure criterion to all number fields; the main equivalence is almost certainly right, but the converse direction has a repairable omitted separation argument and Lemma 4.3 has a false identity.","tokens_in":13559,"tokens_out":5222,"would_cite":true,"duration_ms":47205,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11M26","11N37","11R42"],"pacs":[],"model":"deepseek-v4-flash","headline":"For an algebraic number field K, the extended Riemann hypothesis for its Dedekind zeta function is equivalent to a single convergence-rate statement for discrete totient-weighted measures.","keywords":["Dedekind zeta function","Extended Riemann Hypothesis","discrete measures","Mellin transform","Euler totient function","algebraic number field","rate of convergence","zero-free regions"],"falsifier":"To settle the converse, check the missing separation property at a hypothetical zero: fix a field $K$ and a point $s_0$ with $\\mathrm{Re}(s_0)>1/4$ and $\\zeta_K(2s_0)=0$, and compute $\\int_0^\\infty f(q)q^{2s_0-1}dq$ for a spanning family of $C^{n+1}_c(\\mathbb{R}_+)$. If every such integral vanished, the 'every $f$' error bound would hold despite ERH failing; if any is nonzero, the pole argument contradicts the assumed error bound. The paper's own family $f(t)=(1-t)^{n+2}$ on $(0,1)$ and $0$ elsewhere gives the integral $(n+2)!/(2s_0(2s_0+1)\\cdots(2s_0+n+2))$, which is nonzero at every such $s_0$.","tokens_in":12413,"feed_emoji":"🔢","tokens_out":18187,"duration_ms":170271,"temperature":0.7,"pith_summary":"The paper claims that the extended Riemann hypothesis for the Dedekind zeta function $\\zeta_K(s)$ of an algebraic number field $K$ is exactly a rate-of-convergence statement. For a field of degree $n=[K:\\mathbb{Q}]$, define discrete measures $m_q(f)=\\sum_{\\mathfrak{a}} q\\,\\varphi_K(\\mathfrak{a})\\,f(q^{1/2}N(\\mathfrak{a}))$ over integral ideals $\\mathfrak{a}$; the claim is that $\\zeta_K$ satisfies the Riemann hypothesis if and only if $m_q(f)$ approaches $m(f)=\\frac{\\kappa}{\\zeta_K(2)}\\int f(q)\\,q\\,dq$ with error $o(q^{3/4-\\varepsilon})$ as $q\\to 0$ for every compactly supported test function with at least $n+1$ derivatives and every $\\varepsilon<1/4$. The paper also proves an unconditional $o(q^{1/2})$ error term at lower regularity, a quantitative version tying error exponents to zero-free half-planes, and optimality examples showing rougher test functions cannot reach the $1/2$ exponent. A sympathetic reader should care because the result turns an analytic conjecture about zeros into a concrete distributional statement about arithmetic objects.","feed_headline":"One convergence rate equals the extended Riemann hypothesis","feed_subtitle":"For number fields, the zeta-function's zeros reduce to how fast arithmetic sums approach a continuous measure.","key_machinery":"The central object is the family of discrete measures $m_q$ on $\\mathbb{R}_+$, defined arithmetically by $m_q(f)=\\sum_{\\mathfrak{a}} q\\,\\varphi_K(\\mathfrak{a})\\,f(q^{1/2}N(\\mathfrak{a}))$, together with its Mellin transform $M_f(s)$. The load-bearing identity is $M_f(s)=2\\,\\frac{\\zeta_K(2s-1)}{\\zeta_K(2s)}\\int_0^\\infty f(q)q^{2s-1}dq$: it identifies the analytic continuation of the transform with a ratio of Dedekind zeta functions, so a zero of $\\zeta_K(2s)$ at $s_0$ creates a pole of $M_f$ unless the test function's Mellin integral vanishes there. The proof then uses two gears: integration by parts, which makes $M_f$ decay on vertical lines when $f$ has enough derivatives (the threshold $\\ell\\ge n+1$), and the Riemann–Lebesgue lemma, which converts that decay into the claimed error term after Mellin inversion.","core_discovery":"On the paper's own terms, the discovery is Theorem 1.1(C): the Riemann hypothesis for the Dedekind zeta function holds if and only if for every $f\\in C^{\\ell}_c(\\mathbb{R}_+)$ with $n+1\\le \\ell\\le\\infty$, one has $m_q(f)=m(f)+o(q^{3/4-\\varepsilon})$ as $q\\to 0$, for all $0<\\varepsilon<1/4$. The same theorem gives an interpolating version: for $\\alpha\\in(1/2,3/4)$, the error $o(q^{\\alpha-\\varepsilon})$ holds for all such $f$ if and only if $\\zeta_K(s)$ has no zeros in the half-plane $\\mathrm{Re}(s)>2(1-\\alpha)$. The mechanism is the Mellin transform identity $M_f(s)=2\\,\\frac{\\zeta_K(2s-1)}{\\zeta_K(2s)}\\int_0^\\infty f(q)q^{2s-1}dq$, which converts zeros of $\\zeta_K$ into poles of $M_f$; Mellin inversion and contour shifting then convert holomorphy past $\\mathrm{Re}(s)=1/4$ into the asserted error term. Before the equivalence, the paper establishes an unconditional error term $o(q^{1/2})$ for test functions with at least $\\lfloor n/2\\rfloor+2$ derivatives, and shows that characteristic functions of intervals have error with $\\limsup_{q\\to0} q^{-\\alpha}|m_q(f)-m(f)|=\\infty$ for every $\\alpha>1/2$.","pith_inferences":["Beyond the paper: the missing separation step in the converse can be supplied by the paper's own Beta-function family; taking $f(t)=(1-t)^{n+2}$ on $(0,1)$ and $0$ elsewhere gives Mellin transform $(n+2)!/(2s(2s+1)\\cdots(2s+n+2))$, which does not vanish in $\\mathrm{Re}(s)>1/4$, so the 'error bound for every $f$ implies ERH' direction is repairable if the contour estimates are valid.","Beyond the paper: because the argument uses only the Euler product, the functional equation, and Phragmén–Lindelöf bounds for $\\zeta_K$, the same measure reformulation should extend to other L-functions with these features, such as Hecke L-functions, yielding analogous 'L-function hypothesis as convergence rate' equivalences.","Beyond the paper: the quantitative form suggests an empirical test for a fixed number field: compute $m_q(f)-m(f)$ for the Beta-family test functions at small $q$ and compare the observed decay rate with the predicted zero-free half-plane $\\mathrm{Re}(s)>2(1-\\alpha)$; the relation is checkable at finite precision."],"forward_implications":["If the extended Riemann hypothesis holds for $K$, then every compactly supported test function with at least $n+1$ derivatives satisfies $m_q(f)=m(f)+o(q^{3/4-\\varepsilon})$ as $q\\to0$, for all $\\varepsilon<1/4$.","Conversely, if that error bound holds for every such $f$, then $\\zeta_K$ has no zeros off the critical line, so the extended Riemann hypothesis follows.","The quantitative version couples the error exponent to zero-free half-planes: an error $o(q^{\\alpha-\\varepsilon})$ for all such $f$ is equivalent to $\\zeta_K(s)$ having no zeros with $\\mathrm{Re}(s)>2(1-\\alpha)$, for $\\alpha\\in(1/2,3/4)$.","Without any unproved hypothesis, test functions with at least $\\lfloor n/2\\rfloor+2$ derivatives already give $m_q(f)=m(f)+o(q^{1/2})$, and this exponent cannot be improved by replacing $f$ with the characteristic function of an interval, whose error satisfies $\\limsup_{q\\to0} q^{-\\alpha}|m_q(f)-m(f)|=\\infty$ for every $\\alpha>1/2$."],"supporting_citations":[{"why":"Supplies the discrete-measure setup and the Mellin-transform criterion whose number-field generalization is proved here.","marker":"[19]"},{"why":"The original study of these measures and of the optimality of the 1/2 exponent for interval indicators.","marker":"[18]"},{"why":"Connects arithmetic counting sums of this shape to zeta zeros through periodic-orbit asymptotics, motivating the reformulation.","marker":"[14]"},{"why":"The classical asymptotic for totient sums that Proposition 4.2 generalizes to number fields.","marker":"[8]"},{"why":"A classical consequence of the Riemann hypothesis used in Lemma 3.4 to control the reciprocal of $\\zeta_K(2s)$ on shifted contours.","marker":"[7]"},{"why":"Gives the count of integral ideals with norm at most $x$, the input used in Lemma 4.1 to derive the totient sum.","marker":"[9]"},{"why":"Supplies the standard analytic properties of $\\zeta_K$, including the functional equation and the growth bounds used in the contour estimates.","marker":"[6]"}],"fun_headline_variants":["Extended RH: a measure convergence rate","Measure error speed decides extended RH","Dedekind zeta zeros from a measure rate","The extended RH is just a convergence rate"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the admissible smooth test functions are rich enough to detect every possible zero of $\\zeta_K(2s)$ in the half-plane $\\mathrm{Re}(s)>1/4$: if such a zero existed, at least one function in $C^{n+1}_c(\\mathbb{R}_+)$ would have a nonzero weighted integral at that point, and the paper does not supply this separation step.","fun_headline_variants_meta":{"raw":{"variants":["Extended RH: a measure convergence rate","Measure error speed decides extended RH","Dedekind zeta zeros from a measure rate","The extended RH is just a convergence rate"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000376,"raw_usage":{"total_tokens":2008,"prompt_tokens":952,"completion_tokens":1056,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":568,"completion_tokens_details":{"reasoning_tokens":1001}},"tokens_in":568,"tokens_out":1056,"duration_ms":12076,"temperature":1.0,"reasoning_tokens":1001,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:08:01.921232+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"To settle the converse, check the missing separation property at a hypothetical zero: fix a field $K$ and a point $s_0$ with $\\mathrm{Re}(s_0)>1/4$ and $\\zeta_K(2s_0)=0$, and compute $\\int_0^\\infty f(q)q^{2s_0-1}dq$ for a spanning family of $C^{n+1}_c(\\mathbb{R}_+)$. If every such integral vanished, the 'every $f$' error bound would hold despite ERH failing; if any is nonzero, the pole argument contradicts the assumed error bound. The paper's own family $f(t)=(1-t)^{n+2}$ on $(0,1)$ and $0$ elsewhere gives the integral $(n+2)!/(2s_0(2s_0+1)\\cdots(2s_0+n+2))$, which is nonzero at every such $s_0$.","supporting_citations":[{"cited_title":"Verjovsky","cited_arxiv_id":null,"evidence_quote":"Supplies the discrete-measure setup and the Mellin-transform criterion whose number-field generalization is proved here."},{"cited_title":"Verjovsky","cited_arxiv_id":null,"evidence_quote":"The original study of these measures and of the optimality of the 1/2 exponent for interval indicators."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Connects arithmetic counting sums of this shape to zeta zeros through periodic-orbit asymptotics, motivating the reformulation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The classical asymptotic for totient sums that Proposition 4.2 generalizes to number fields."},{"cited_title":"Littlewood","cited_arxiv_id":null,"evidence_quote":"A classical consequence of the Riemann hypothesis used in Lemma 3.4 to control the reciprocal of $\\zeta_K(2s)$ on shifted contours."},{"cited_title":"Ram Murty and Jeanine Van Order","cited_arxiv_id":null,"evidence_quote":"Gives the count of integral ideals with norm at most $x$, the input used in Lemma 4.1 to derive the totient sum."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the standard analytic properties of $\\zeta_K$, including the functional equation and the growth bounds used in the contour estimates."}],"review_version":1}