# References — Chapter ᚑ(O): The prime mileposts *Teclam Áirme, the Mathematical Compendium — a copy for AI readers of https://www.teclamairme.com/#ref-ch-18, as of 2026-09-30. Links are relative to https://www.teclamairme.com/fortheaireaders/.* [Guide for AI readers](index.md) · [The chapter](ch18.md) *Each entry: one plain sentence saying exactly what it is cited for, its status, then every source with a note saying what that source actually carries. "Own computation" / "Own reading" entries are the author's own work, not outside sources.* ### 18.1 {#r-18.1} For Re(s)>1, the Riemann zeta function is the series sum of n^(-s); elsewhere it is defined by analytic continuation, is meromorphic, and has its only singularity at s=1, a simple pole with residue 1. An analytic continuation is unique: two analytic functions equal on a set with a limit point are equal throughout. *Status: confirmed* - [NIST Digital Library of Mathematical Functions, §25.2 Definition and Expansions](https://dlmf.nist.gov/25.2) — DLMF states the defining series for Re(s)>1 and says the continuation is meromorphic with only a simple pole at s=1 and residue 1. - [NIST DLMF §1.10, Functions of a complex variable](https://dlmf.nist.gov/1.10) — “If f₂(z), analytic in D₂, equals f₁(z) on an arc in D = D₁ ∩ D₂, or on just an infinite number of points with a limit point in D, then they are equal throughout D and f₂(z) is called an analytic continuation of f₁(z).” ### 18.2 {#r-18.2} The special values are ζ(0)=-1/2, ζ(2)=π²/6, ζ(4)=π⁴/90, and ζ(6)=π⁶/945; more generally the positive even values are rational multiples of powers of π involving Bernoulli numbers. ζ(0) = −1/2 follows from the functional equation and the residue at s = 1. Euler found the values ζ(2n) in E41 (written 1735, published 1740) and gave the Bernoulli-number form in his 1755 book. For odd positive integers no such simple expression is known. *Status: confirmed* - [NIST Digital Library of Mathematical Functions, §25.6 Integer Arguments](https://dlmf.nist.gov/25.6) — equations 25.6.1 and 25.6.2 list these values and the general even-value formula. - [MIT 18.785 Number Theory I (Fall 2015), Lecture 16](https://math.mit.edu/classes/18.785/2015fa/LectureNotes16.pdf) — read from the PDF: “We can determine the value of ζ(0) via the functional equation. We know that ζ(s) has a pole of residue 1 at s = 1 … thus ζ(0) = −1/2.” - [Jeffrey C. Lagarias, Euler’s constant: Euler’s work and modern developments, arXiv:1303.1856](https://arxiv.org/abs/1303.1856) — read from the PDF: “In his 1755 book on differential calculus Euler [E212, Part II, Chap. 6, Arts. 141–153] gave closed forms for ζ(2n) explicitly expressed in terms of the Bernoulli numbers.” - [Euler Archive, E41, De summis serierum reciprocarum](https://scholarlycommons.pacific.edu/euler-works/41/) — “Euler finds the values of ζ(2n), where ζ is now named as the Riemann zeta function”; Written Date 1735, Published Date 1740. - [Bernoulli number, Wikipedia](https://en.wikipedia.org/wiki/Bernoulli_number) — its table gives B₆ = 1/42. - [Riemann zeta function, Wikipedia](https://en.wikipedia.org/wiki/Riemann_zeta_function) — read as raw wikitext: “For odd positive integers, no such simple expression is known.” ### 18.3 {#r-18.3} Euler solved the Basel sum as π²/6 and later identified the zeta Euler product in 1737; the problem is named for Basel, the home city of Euler and of the Bernoullis, who had failed at it, and solving it made Euler’s reputation; the historical date of the Basel result is convention-sensitive because the work was written/presented in 1734–1735 and published in 1740. *Status: judgement* - [Euler Archive / Pacific University, De summis serierum reciprocarum](https://scholarlycommons.pacific.edu/euler-works/41/) — the source identifies Euler’s paper and gives its dates: Written Date 1735, Published Date 1740. - [Nature, The Zeta-Function of Riemann](https://www.nature.com/articles/126600c0) — the historical note dates Euler’s prime-product observation to 1737. - [Jeffrey C. Lagarias, Euler’s constant: Euler’s work and modern developments, arXiv:1303.1856 (Bull. AMS 50 (2013) 527–628)](https://arxiv.org/abs/1303.1856) — the same survey, free on arXiv: Euler solved the Basel problem by 1735, and his three proofs that ζ(2) = π²/6 (E41, read 5 December 1735) were published in 1740. - [Wikipedia, Basel problem](https://en.wikipedia.org/wiki/Basel_problem) — the problem was posed by Pietro Mengoli in 1650; it is named for Basel as the home town of Euler and of the Bernoulli family, who had attempted it without success; and Euler’s 1734 solution brought him immediate fame at twenty-eight. Editorial note: Sources distinguish discovery, presentation, and publication dates. We keep the snapshot’s “found by Euler in 1734” as a discovery date, but the review should not silently imply that the paper was published in 1734. ### 18.4 {#r-18.4} Euler’s product is ζ(s)=∏p(1-p^(-s))^(-1) for Re(s)>1, because expanding the geometric factors and using unique factorization lists each positive integer once. The proof uses only the geometric series and the fundamental theorem of arithmetic; at s = 1 the divergence of the harmonic series gives infinitely many primes. *Status: confirmed* - [NIST Digital Library of Mathematical Functions, §25.2](https://dlmf.nist.gov/25.2) — the zeta-function section includes the Euler product representation and its domain. - [Riemann zeta function, Wikipedia](https://en.wikipedia.org/wiki/Riemann_zeta_function) — read as raw wikitext: “The proof of Euler’s identity uses only the formula for the geometric series and the fundamental theorem of arithmetic. Since the harmonic series, obtained when s = 1, diverges, Euler’s formula, which becomes ∏_p p/(p−1), implies that there are infinitely many primes.” ### 18.5 {#r-18.5} The functional equation relates ζ(s) to ζ(1-s) through the sine and gamma factors; its sine factor gives the trivial zeros ζ(-2n)=0 for n≥1, and the completed xi function is symmetric under s↔1-s. ξ is entire and its zeros are exactly the nontrivial zeros of ζ; Riemann established the functional equation in 1859. *Status: confirmed* - [NIST Digital Library of Mathematical Functions, §25.4 Reflection Formulas](https://dlmf.nist.gov/25.4) — equations 25.4.2–25.4.4 give the reflection formula and xi-function definition/symmetry. - [NIST Digital Library of Mathematical Functions, §25.10 Zeros](https://dlmf.nist.gov/25.10) — DLMF explicitly states that the functional equation implies ζ(-2n)=0 and calls these the trivial zeros. - [Riemann zeta function, Wikipedia](https://en.wikipedia.org/wiki/Riemann_zeta_function) — read as raw wikitext: “The xi function is an entire function whose zeros are exactly the non-trivial zeros of the Riemann zeta function”; “The functional equation was established by Riemann in his 1859 paper “On the Number of Primes Less Than a Given Magnitude.”” ### 18.6 {#r-18.6} At s=1 the harmonic series diverges and ζ has a simple pole with residue 1, its only singularity; consequently the reciprocal function 1/ζ(s), after analytic continuation as a reciprocal, has a removable singularity and takes the value 0 at s=1. The divergence of the harmonic series was first proved by Nicole Oresme in 1350. *Status: confirmed* - [NIST Digital Library of Mathematical Functions, §25.2](https://dlmf.nist.gov/25.2) — DLMF states the simple pole and residue 1: “It is a meromorphic function whose only singularity in ℂ is a simple pole at s = 1, with residue 1.” - [NIST Digital Library of Mathematical Functions, §5.2 Gamma Function](https://dlmf.nist.gov/5.2) — the analogous reciprocal-at-a-pole convention is stated for 1/Γ; the value here follows directly from the Laurent expansion of ζ at its simple pole. - [Harmonic series (mathematics), Wikipedia](https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)) — read as raw wikitext: “The divergence of the harmonic series was first proven in 1350 by Nicole Oresme.” Editorial note: Part 2 item on 1/ζ(1): we retain the existing wording. This is the value of the analytically continued reciprocal function at the removable singularity, not a claim that the divergent series has a finite value. ### 18.7 {#r-18.7} The first nontrivial zero has height about 14.134725, the second about 21.022040, and the third about 25.010858; Gram published the first list in 1903. Gram’s 1903 computation showed the first 10 zeros, those of height at most 50, on the critical line. Montgomery’s 1973 pair-correlation conjecture uses the normalization 2π/log T, the average spacing near height T; Dyson pointed out that it matches random Hermitian matrices. *Status: confirmed* - [A. M. Odlyzko, Tables of zeros of the Riemann zeta function](https://www-users.cse.umn.edu/~odlyzko/zeta_tables/index.html) — (file zeros1, first three lines): the first three nontrivial heights are 14.134725142, 21.022039639 and 25.010857580. - [Riemann hypothesis, Wikipedia](https://en.wikipedia.org/wiki/Riemann_hypothesis) — read as raw wikitext: “J. P. Gram (1903) used the Euler–Maclaurin formula and discovered Gram’s law. He showed that all 10 zeros with imaginary part at most 50 range lie on the critical line”; 2004, 10¹³: “Xavier Gourdon (2004) and Patrick Demichel used the Odlyzko–Schönhage algorithm”; “The most extensive computer search by Platt and Trudgian for counterexamples of the Riemann hypothesis has verified it for |t| ≤ 3.0001753328×10¹²,” the page’s footnote citing Platt and Trudgian (2021) (the arXiv preprint is of April 2020, the journal paper of 2021); its table of zero computations opens with 1859?, 3 zeros: “B. Riemann used the Riemann–Siegel formula (unpublished, but reported in Siegel 1932).” - [Montgomery’s pair correlation conjecture, Wikipedia](https://en.wikipedia.org/wiki/Montgomery%27s_pair_correlation_conjecture) — read as raw wikitext: a conjecture “made by Hugh Montgomery (1973)”; “(The factor 2π/log(T) is a normalization factor that can be thought of informally as the average spacing between zeros with imaginary part about T.)”; “which, as Freeman Dyson pointed out to him, is the same as the pair correlation function of random Hermitian matrices.” - [C. L. Siegel, On Riemann’s Nachlass for Analytic Number Theory (1932), tr. Barkan and Sklar, arXiv:1810.05198](https://arxiv.org/pdf/1810.05198) — “For the smallest positive zero he finds the value of a1 = 14.1386; Gram has calculated the value 14.1347, different by less than 3 parts per thousand.” ### 18.8 {#r-18.8} Rigorous computation by Platt and Trudgian verified the Riemann hypothesis up to height 3,000,175,332,800, which is the lowest 12,363,153,437,138 nontrivial zeros. This claim line used to open with “Hardy proved in 1914 that infinitely many zeta zeros lie on the critical line,” which neither source below carries: DLMF §25.10 says only that ζ(½ + it) has infinitely many zeros with t real, naming nobody, and the full text of Platt and Trudgian never mentions Hardy. 18.22 was written to supply the attribution and the year, and the claim line was narrowed to match on 17 September 2026. Their paper is Bulletin of the London Mathematical Society 53 (2021), pp. 792–797, published online 22 January 2021, doi:10.1112/blms.12460, from the preprint arXiv:2004.09765 of 21 April 2020. Gram checked the first 10 zeros in 1903; Gourdon reached 10¹³ in 2004; Platt and Trudgian’s verification (preprint 2020) appeared in 2021. *Status: confirmed* - [NIST Digital Library of Mathematical Functions, §25.10 Zeros](https://dlmf.nist.gov/25.10) — DLMF states that there are infinitely many zeros on the critical line (because Z(t) changes sign infinitely often, ζ(½ + it) has infinitely many zeros with t real) and describes their computation via the function Z(t) and the Riemann–Siegel formula; it does not name Hardy. - [Platt and Trudgian, The Riemann hypothesis is true up to 3·10^12 (arXiv:2004.09765)](https://arxiv.org/abs/2004.09765) — Theorem 1 reads “The Riemann hypothesis is true up to height 3 000 175 332 800. That is, the lowest 12 363 153 437 138 non-trivial zeroes ρ have ℜρ = 1/2.” - [Riemann hypothesis, Wikipedia](https://en.wikipedia.org/wiki/Riemann_hypothesis) — read as raw wikitext: “J. P. Gram (1903) used the Euler–Maclaurin formula and discovered Gram’s law. He showed that all 10 zeros with imaginary part at most 50 range lie on the critical line”; 2004, 10¹³: “Xavier Gourdon (2004) and Patrick Demichel used the Odlyzko–Schönhage algorithm”; “The most extensive computer search by Platt and Trudgian for counterexamples of the Riemann hypothesis has verified it for |t| ≤ 3.0001753328×10¹²,” the page’s footnote citing Platt and Trudgian (2021) (the arXiv preprint is of April 2020, the journal paper of 2021). ### 18.9 {#r-18.9} Apéry proved ζ(3) irrational in 1978; whether ζ(3) is transcendental remains open. *Status: confirmed* - [Waldschmidt, Zeta Values](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/ZetaValuesRIMS2009.pdf) — Theorem 4 identifies Apéry’s 1978 irrationality theorem and gives ζ(3)’s value. - [OEIS A002117](https://oeis.org/A002117) — the entry records Apéry’s 1978 irrationality result and that the rational-multiple-of-π³ question is unresolved. ### 18.10 {#r-18.10} The value ζ(1/2) is approximately -1.4603545088; no source cited here settles whether it is irrational. *Status: unconfirmed* - [NIST Digital Library of Mathematical Functions, §25.10](https://dlmf.nist.gov/25.10) — DLMF confirms the critical-line framework but does not establish an arithmetic classification for ζ(1/2). - [Wolfram MathWorld, Riemann Zeta Function](https://mathworld.wolfram.com/RiemannZetaFunction.html) — gives ζ(1/2) = −1.46035450880…, a negative number. Editorial note: The numerical value and its sign are confirmed; what remains unconfirmed is the arithmetic status. No source settles whether ζ(1/2) is irrational, so the status-open wording is retained. The formula carries |ζ(½)|, and the tip, the chapter key and the card now say so explicitly, because ζ(1/2) itself is negative. ### 18.11 {#r-18.11} The prime-counting function satisfies π(10)=4, π(100)=25, and π(10^6)=78498; the notation π(x) here counts primes, not the circle constant. *Status: confirmed* - [Prime-counting function, Encyclopaedia/standard reference summary](https://en.wikipedia.org/wiki/Prime-counting_function) — the definition and the listed values are given explicitly; its table runs to π(10²⁹), and “The values for 10²⁷, 10²⁸, and 10²⁹ were announced by David Baugh and Kim Walisch in 2015, 2020, and 2022, respectively.” (exponents as typeset on the page). ### 18.12 {#r-18.12} The prime number theorem π(x)~x/log x was conjectured by Gauss and Legendre and proved independently by Hadamard and de la Vallée Poussin in 1896 using the zero-free line Re(s)=1. Wiener showed that the prime number theorem is equivalent to ζ having no zeros on Re(s) = 1. *Status: confirmed* - [MacTutor, Jacques Hadamard biography](https://mathshistory.st-andrews.ac.uk/Biographies/Hadamard/) — MacTutor dates the proof to 1896 and credits Hadamard and de la Vallée Poussin using complex analysis. - [NIST Digital Library of Mathematical Functions, §25.10](https://dlmf.nist.gov/25.10) — DLMF connects the absence of zeros on Re(s)=1 with the prime number theorem and cites both 1896 proofs. - [Riemann Hypothesis, Wolfram MathWorld](https://mathworld.wolfram.com/RiemannHypothesis.html) — “Wiener showed that the prime number theorem is literally equivalent to the assertion that the Riemann zeta function ζ(s) has no zeros on σ = 1.” ### 18.13 {#r-18.13} For the Mertens function M(x)=Σn≤x μ(n), the Riemann hypothesis is equivalent to M(x)=O(x^(1/2+ε)) for every ε>0, while the stronger Mertens conjecture |M(x)|<√x was disproved by Odlyzko and te Riele in 1985. *Status: confirmed* - [Wolfram MathWorld, Mertens Conjecture](https://mathworld.wolfram.com/MertensConjecture.html) — the page states the RH equivalence and records the disproof of the stronger conjecture: “The Mertens conjecture was proved false by Odlyzko and te Riele (1985). Their proof is indirect and does not produce a specific counterexample.” - [Wikipedia, Mertens conjecture](https://en.wikipedia.org/wiki/Mertens_conjecture) — the history and Odlyzko–te Riele 1985 disproof are stated. ### 18.14 {#r-18.14} Dirichlet introduced L-functions in 1837 to prove that every arithmetic progression with coprime first term and difference contains infinitely many primes; the nonvanishing L(1, χ) step is central. *Status: confirmed* - [Encyclopedia of Mathematics, Dirichlet L-function](https://encyclopediaofmath.org/wiki/Dirichlet-L-function) — the entry dates Dirichlet’s introduction of L-functions to 1837 and states their role in primes in arithmetic progressions. - [MIT 18.785 Lecture Notes, Dirichlet L-functions](https://math.mit.edu/classes/18.785/2016fa/LectureNotes18.pdf) — the notes state Dirichlet’s 1837 theorem and the coprime progression condition. ### 18.15 {#r-18.15} Gram points are defined by θ(t)=nπ for the Riemann–Siegel theta function; g₀, the first above the lowest zero, is approximately 17.8456, and Gram’s law is only a rule of thumb rather than a theorem: in the long run it fails for about a quarter of all Gram intervals. *Status: confirmed* - [NIST Digital Library of Mathematical Functions, §25.10 Zeros](https://dlmf.nist.gov/25.10) — DLMF defines the real Hardy Z-function and the Riemann–Siegel theta function used to locate critical-line zeros. - [Wikipedia, Riemann–Siegel theta function](https://en.wikipedia.org/wiki/Riemann%E2%80%93Siegel_theta_function) — the page gives the Gram-point convention and the first point near 17.8456; “Today we know that, in the long run, Gram’s law fails for about 1/4 of all Gram-intervals to contain exactly 1 zero of the Riemann zeta function.” - [Gram Point, Wolfram MathWorld](https://mathworld.wolfram.com/GramPoint.html) — a Gram point gₙ satisfies θ(gₙ) = nπ, numbered from n = 0; its table gives g₀ = 17.8455995404. ### 18.16 {#r-18.16} At s=2, ζ(2) times the reciprocal Euler product is 1, and the p=2 Euler factor is (1-2^-2)^-1=4/3; multiplying all prime factors reconstructs π²/6. The full product equals ζ(2) = π²/6. *Status: confirmed* - [NIST Digital Library of Mathematical Functions, §25.2 and §25.6](https://dlmf.nist.gov/25.2) — the Euler product and ζ(2)=π²/6 are stated in the DLMF sections. - **Own computation** (the author's own work) — At p=2, (1-2^-2)^-1=(1-1/4)^-1=4/3; multiplying ζ(2) by ∏p(1-p^-2) cancels the product exactly. - [NIST DLMF §25.6, Zeta function: integer arguments](https://dlmf.nist.gov/25.6) — 25.6.1: ζ(2) = π²/6. ### 18.17 {#r-18.17} Riemann’s memoir Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse appeared in the Monatsberichte der Berliner Akademie in November 1859; in it he argues that ξ(t) can vanish only for the imaginary part of t between ½i and −½i, that is for Re(s) between 0 and 1 inclusive, and records the hypothesis as “very probable” while setting the proof aside. Its length is given as six manuscript pages or as nine printed ones. *Status: confirmed* - [Clay Mathematics Institute, Riemann’s 1859 Manuscript](https://www.claymath.org/collections/riemanns-1859-manuscript/) — “Just six manuscript pages in length,” first published in the Monatsberichte der Berliner Akademie, November 1859. - [On the Number of Prime Numbers less than a Given Quantity, translated by David R. Wilkins (Clay Mathematics Institute)](https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-translation.pdf) — carries the epigraph verbatim — “it is very probable that all roots are real. Certainly one would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts, as it appears unnecessary for the next objective of my investigation.” — and the argument confining the roots to the closed strip. - [Wikipedia, On the Number of Primes Less Than a Given Magnitude](https://en.wikipedia.org/wiki/On_the_Number_of_Primes_Less_Than_a_Given_Magnitude) — calls it “a seminal 9-page paper” published in the November 1859 Monatsberichte. Editorial note: The two live sources count different objects: Clay counts Riemann’s handwritten manuscript at six pages, Wikipedia the printed article at nine. Neither supports the “eight pages” the text used to give, so the text now says only that the paper runs to fewer than ten pages, which both sources support. ### 18.18 {#r-18.18} The regularized value ζ(−1) = −1/12 is the one used to compute the Casimir force for a scalar field in one dimension; the three-dimensional parallel-plate calculation regularizes with ζ(−3) = 1/120 and gives an energy per unit area of −ħcπ²/720a³. *Status: confirmed* - [Wikipedia, 1 + 2 + 3 + 4 + ⋯](https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF) — “The regularization of 1 + 2 + 3 + 4 + ⋯ is also involved in computing the Casimir force for a scalar field in one dimension,” and the same value is what fixes the twenty-six dimensions of bosonic string theory. - [Wikipedia, Casimir effect](https://en.wikipedia.org/wiki/Casimir_effect) — the parallel-plate derivation runs through “ζ(−3) = 1/120” and yields ⟨E⟩/A = −ħcπ²/720a³. Editorial note: The page had attributed the two-metal-plate calculation to ζ(−1) = −1/12. That is the one-dimensional scalar case; the plates need ζ(−3). Both are now named. ### 18.19 {#r-18.19} The Riemann Hypothesis is one of the seven Clay Millennium Prize Problems, each carrying a prize of $1,000,000, and a proof would sharpen the known estimates for how the primes are distributed. *Status: confirmed* - [Clay Mathematics Institute, Millennium Problems](https://www.claymath.org/millennium-problems/) — “the Board of Directors of CMI designated a $7 million prize fund for the solutions to these problems, with $1 million allocated to the solution of each problem,” and the Riemann Hypothesis is one of the seven listed. - [Platt and Trudgian, The Riemann hypothesis is true up to 3·10^12 (arXiv:2004.09765), §3.1](https://arxiv.org/abs/2004.09765) — “Of course, on the Riemann hypothesis much more is known: Schoenfeld proved that |ψ(x) − x| ≤ (1/8π)√x log²x, (x > 59),” a sharper error term than the unconditional bounds the same section collects. ### 18.20 {#r-18.20} The de Bruijn–Newman constant Λ is a real number for which the Riemann hypothesis is equivalent to Λ ≤ 0; Rodgers and Tao proved Λ ≥ 0, so the hypothesis is true if and only if Λ = 0. *Status: confirmed* - [Rodgers and Tao, The De Bruijn–Newman constant is non-negative (Forum of Mathematics Pi 8, 2020)](https://arxiv.org/abs/1801.05914) — the abstract defines Λ by the heat-flow deformation, states that the Riemann hypothesis is equivalent to Λ ≤ 0, and proves Newman’s conjecture Λ ≥ 0. - [Platt and Trudgian, The Riemann hypothesis is true up to 3·10^12 (arXiv:2004.09765), §3.4](https://arxiv.org/abs/2004.09765) — “The Riemann hypothesis is equivalent to the statement that Λ ≤ 0. Rodgers and Tao proved that Λ ≥ 0.,” with Corollary 2 giving Λ ≤ 0.2. ### 18.21 {#r-18.21} Analytic continuation is unique: if f1 is analytic in a domain D1 and f2 is analytic in D2, and the two agree on an arc in D = D1 ∩ D2, or on just an infinite number of points with a limit point in D, then they agree throughout D, and f2 is called an analytic continuation of f1. *Status: confirmed* - [NIST Digital Library of Mathematical Functions, §1.10(ii) Analytic Continuation](https://dlmf.nist.gov/1.10) — states the uniqueness in so many words — “Let f1(z) be analytic in a domain D1. If f2(z), analytic in D2, equals f1(z) on an arc in D = D1 ∩ D2, or on just an infinite number of points with a limit point in D, then they are equal throughout D and f2(z) is called an analytic continuation of f1(z).” Editorial note: Added in the V14 prose pass for the new card `analcont`. Chapter ᚑ(O)’s sentence “That is analytic continuation: a well-behaved function of a complex variable is so rigid that knowing it on a small patch fixes it everywhere” carried no footnote of its own, and 18.1 (DLMF §25.2) supplies only the fact that ζ is continued, not the uniqueness theorem that makes the continuation the function rather than a function. This entry carries the theorem. ### 18.22 {#r-18.22} Hardy proved in 1914 that infinitely many nontrivial zeros of the Riemann zeta function lie on the critical line; whether all of them do is still unknown. *Status: confirmed* - [Wolfram MathWorld, Riemann Hypothesis](https://mathworld.wolfram.com/RiemannHypothesis.html) — states “In 1914, Hardy proved that an infinite number of values for t can be found for which ζ(1/2 + it) = 0 and t is real (Havil 2003, p. 213),” and immediately adds “However, it is not known if all nontrivial roots satisfy Re(s) = 1/2.” Editorial note: Added in the V14 prose pass. The chapter’s sentence “Hardy proved in 1914 that infinitely many zeros are on the line” was footnoted 18.8 alone, and the V12 sweep had already recorded that neither of 18.8’s sources carries the attribution or the year — DLMF §25.10 says only that ζ(½ + it) has infinitely many zeros with t real, naming nobody, and the full text of Platt and Trudgian never mentions Hardy. This entry supplies both. 18.8 continues to carry the verification height, which is what it does support. ### 18.23 {#r-18.23} Euler’s product expansion first appeared in Variae observationes circa series infinitas (Eneström E72), written in 1737 and published in 1744 in Commentarii academiae scientiarum Petropolitanae 9, 160–188. *Status: confirmed* - [Euler Archive / University of the Pacific, E72 Variae observationes circa series infinitas](https://scholarlycommons.pacific.edu/euler-works/72/) — the record gives “Written Date 1737,” “Published Date 1744,” the citation “Commentarii academiae scientiarum Petropolitanae, Volume 9, pp. 160-188,” and the content summary “This is the paper in which Euler product expansions first appeared.” Editorial note: Added in the V14 prose pass to carry the 1737 date the chapter uses three times. The V12 sweep recorded that the date rested on two sources a reader cannot open: originalsources.com returns a sign-in page under HTTP 200, and the 1930 Nature note gives no detail. The Euler Archive record is open, names the paper, and dates it directly. This is the companion of 18.3’s E41 record for the 1734 Basel result. ### 18.24 {#r-18.24} Ramanujan wrote the identity 1 + 2 + 3 + 4 + ... = -1/12 in his second letter to G. H. Hardy, of 27 February 1913. *Status: confirmed* - [1 + 2 + 3 + 4 + ⋯, Wikipedia](https://en.wikipedia.org/wiki/1_%2B_2_%2B_3_%2B_4_%2B_%E2%8B%AF) — read as raw HTML, § Ramanujan summation: “Ramanujan wrote in his second letter to G. H. Hardy, dated 27 February 1913: ‘Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich’s Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + ⋯ = −1/12 under my theory.” So the date, the addressee and the identity are all on that page. Editorial note: Added 17 September 2026 for the B6 pass. The #zetam1 card said “Ramanujan states it in his letter to Hardy of 27 February 1913” and footnoted 18.18, which is the Casimir entry: its claim line is about the regularized value used for the Casimir force and neither of its two sources mentions Ramanujan, Hardy or 1913. A reader following that marker landed on a physics calculation. Grepping references.html for “Ramanujan” before this entry existed returned r-5.9, r-6.11, r-6.12, r-6.31 and r-17.5, none of them about the letter. ### 18.25 {#r-18.25} The chapter epigraph is Riemann’s, from Ueber die Anzahl der Primzahlen unter einer gegebenen Groesse of 1859, in David Wilkins’s English translation; it is the sentence in which the Riemann hypothesis is first stated and immediately set aside. *Status: confirmed* - [Bernhard Riemann, On the Number of Prime Numbers less than a Given Quantity, translated by David R. Wilkins, at the Clay Mathematics Institute](https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-translation.pdf) — fetched, read as extracted PDF text. The epigraph appears word for word: “One now finds indeed approximately this number of real roots within these limits, and it is very probable that all roots are real. Certainly one would wish for a stricter proof here; I have meanwhile temporarily put aside the search for this after some fleeting futile attempts, as it appears unnecessary for the next objective of my investigation.” The Clay page hosting it credits the transcription and translation to David Wilkins by permission and the manuscript images to the Niedersächsische Staats- und Universitätsbibliothek Göttingen. Editorial note: Added 17 September 2026 to source the chapter epigraph, which carried no marker. This is the strongest of the fourteen: the quotation was checked against the translation the footer names, on the site that publishes it, and matches exactly, including the footer’s gloss that the roots are those of Riemann’s transformed function. ### 18.26 {#r-18.26} The fundamental theorem of arithmetic says every integer greater than 1 is a product of primes in exactly one way up to order; it can be derived from Euclid’s Elements, Book VII propositions 30, 31 and 32 and Book IX proposition 14. It is the reason 1 is not counted as a prime, and it fails for general rings of algebraic integers, which is why unique factorization domains are singled out by name. *Status: confirmed* - [Fundamental theorem of arithmetic, Wikipedia](https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic) — read as raw HTML: “The fundamental theorem can be derived from Book VII, propositions 30, 31 and 32, and Book IX, proposition 14 of Euclid’s Elements”; “1 is not considered a prime number: if 1 were prime, then factorization into primes would not be unique”; and “the theorem does not hold for algebraic integers,” which is what unique factorization domains name. - [Fundamental Theorem of Arithmetic, MathWorld](https://mathworld.wolfram.com/FundamentalTheoremofArithmetic.html) — the statement: “every positive integer (except the number 1) can be represented in exactly one way apart from rearrangement as a product of one or more primes.” ### 18.27 {#r-18.27} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Euler had guessed the real-variable form of the functional equation in 1749; Riemann proved the functional equation in his 1859 paper; Gram observed in 1903 that zeros tend to fall between consecutive Gram points (Gram’s law); the Mertens conjecture was made by Stieltjes in a letter to Hermite of 1885 and in print by Mertens in 1897; Mertens conjectured |M(x)| < sqrt(x) in 1897; Kotnik and te Riele improved the upper bound for a counterexample to the Mertens conjecture in 2006, to about e^(1.59×10^40), and a later preprint lowers it to about e^(1.017×10^29), and Kim and Nguyen lowered it in 2024 to about e^(1.96×10^19); Gauss conjectured the prime number theorem around 1792; Legendre published a version of the conjecture in 1798; Chebyshev proved in 1850 that pi(x) is within constant factors of x/ln x. *Status: confirmed* - [E352 Remarques sur un beau rapport entre les séries des puissances tant directes que réciproques, Euler Archive (University of the Pacific)](https://scholarlycommons.pacific.edu/euler-works/352/) — read raw: “Euler evaluates several values of the now-named Riemann zeta function, and finds some functional relations for it. ... Published Date 1768 ... Written Date 1749.” - [Riemann zeta function, Wikipedia](https://en.wikipedia.org/wiki/Riemann_zeta_function) — read raw: “The functional equation was established by Riemann in his 1859 paper “On the Number of Primes Less Than a Given Magnitude” and used to construct the analytic continuation in the first place..” - [Riemann hypothesis, Wikipedia](https://en.wikipedia.org/wiki/Riemann_hypothesis) — read raw: “J. P. Gram (1903) used the Euler–Maclaurin formula and discovered Gram’s law. He showed that all 10 zeros with imaginary part at most 50 range lie on the critical line with real part 1/2”; “Trudgian (2011) showed that both Gram’s law and Rosser’s rule fail in a positive proportion of cases.”; Turing’s check “that Z has the correct sign at several consecutive Gram points” “is still the usual method used.” - [Mertens conjecture, Wikipedia](https://en.wikipedia.org/wiki/Mertens_conjecture) — read raw: “It was conjectured by Thomas Joannes Stieltjes, in an 1885 letter to Charles Hermite (reprinted in Stieltjes (1905)), and again in print by Mertens (1897)”; “In 2006, Kotnik and te Riele improved the upper bound and showed that there are infinitely many values of n for which m(n) › 1.2184, but without giving any specific value for such an n..” - [Prime number theorem, Wikipedia](https://en.wikipedia.org/wiki/Prime_number_theorem) — read raw: “Carl Friedrich Gauss considered the same question at age 15 or 16 “in the year 1792 or 1793,” according to his own recollection in 1849.”; “Adrien-Marie Legendre conjectured in 1797 or 1798 that π(a) is approximated by the function a / (A log a + B), where A and B are unspecified constants. In the second edition of his book on number theory (1808) he then made a more precise conjecture”; “In two papers from 1848 and 1850, the Russian mathematician Pafnuty Chebyshev attempted to prove the asymptotic law of distribution of prime numbers..” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim. ### 18.28 {#r-18.28} Dates and attributions printed on this chapter’s glossary cards and in its field notes, each checked against the source listed: Erdős and Selberg each published an elementary proof of the prime number theorem in 1949; Riemann’s 1859 explicit formula writes pi(x) as a sum over the zeros of zeta; de Bruijn’s 1950 deformation of the xi-related function H; Newman defined the constant Lambda in 1976; Riemann’s only paper on number theory, 1859, made s complex and continued zeta to the plane; Euler worked formally with divergent sums in the 1740s. Riemann’s memoir shows ξ(t) can vanish only for Im t between ½i and −½i, that is 0 ≤ Re(s) ≤ 1. The explicit formula’s dominant term, li(x), comes from the pole at s = 1, the nontrivial zeros control the oscillations of the primes around it, and von Koch proved that the Riemann hypothesis implies the best possible bound for the error of the prime number theorem. *Status: confirmed* - [Prime number theorem, Wikipedia](https://en.wikipedia.org/wiki/Prime_number_theorem) — read raw: “Several different proofs of it were found, including the “elementary” proofs of Atle Selberg (1949) and Paul Erdős (1949).”; “Carl Friedrich Gauss considered the same question at age 15 or 16 “in the year 1792 or 1793,” according to his own recollection in 1849.” - [Riemann hypothesis, Wikipedia](https://en.wikipedia.org/wiki/Riemann_hypothesis) — read raw: “their occurrence in his explicit formula for the number of primes π(x) less than or equal to a given number x, which he published in his 1859 paper “On the Number of Primes Less Than a Given Magnitude..””; “the dominant term li(x) comes from the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivial zeros”; “This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their “expected” positions.”; “Helge von Koch proved that the Riemann hypothesis implies the “best possible” bound for the error of the prime number theorem.” (read 29 September 2026 from the cached copy of 24 September). - [De Bruijn–Newman constant, Wikipedia](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Newman_constant) — read raw: “De Bruijn showed in 1950 that H has only real zeros if λ≥1/2, and moreover, that if H has only real zeros for some λ, H also has only real zeros if λ is replaced by any larger value.”; “Newman proved in 1976 the existence of a constant Λ for which the “if and only if” claim holds; and this then implies that Λ is unique.”; “Brad Rodgers and Terence Tao proved that Λ ≥ 0, so the Riemann hypothesis is equivalent to Λ = 0..” - [On the Number of Primes Less Than a Given Magnitude, Wikipedia](https://en.wikipedia.org/wiki/On_the_Number_of_Primes_Less_Than_a_Given_Magnitude) — read raw: “is a seminal 9-page paper by Bernhard Riemann published in the November 1859 edition of the Monatsberichte der Königlich Preußischen Akademie der Wissenschaften zu Berlin..” - [1 − 2 + 3 − 4 + ⋯, Wikipedia](https://en.wikipedia.org/wiki/1_%E2%88%92_2_%2B_3_%E2%88%92_4_%2B_%E2%8B%AF) — read raw: “In a 1749 report, Leonhard Euler acknowledges that the series diverges but prepares to sum it anyway.” - [Riemann, On the Number of Prime Numbers less than a Given Quantity, tr. D. R. Wilkins (Clay Mathematics Institute)](https://www.claymath.org/wp-content/uploads/2023/04/Wilkins-translation.pdf) — read from the PDF: “it follows that the function ξ(t) can only vanish if the imaginary part of t lies between ½i and −½i.” Editorial note: Added 18 September 2026 by the V42 citation sweep. citecheck.py found these years printed on a card or in a paragraph whose cited entries did not carry them; each was then verified against a source fetched raw that day, and the quotation in each source note is verbatim.