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11: 27.12 Asymptotic Formulas: Primes

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27.12.1

\lim_{n\to\infty}\frac{p_{n}}{n\ln n}=1,

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Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : positive integer and $p,p_{1},\ldots$ : prime numbers
Permalink:: http://dlmf.nist.gov/27.12.E1
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.12 and Ch.27

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27.12.2

p_{n}>n\ln n,

n=1,2,\dots

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Symbols:: $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : positive integer and $p,p_{1},\ldots$ : prime numbers
Permalink:: http://dlmf.nist.gov/27.12.E2
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.12 and Ch.27

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27.12.4

\pi\left(x\right)\sim\sum_{k=1}^{\infty}\frac{(k-1)!\,x}{(\ln x)^{k}}.

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Symbols:: $\sim$ : Poincaré asymptotic expansion, $!$ : factorial (as in $n!$ ), $\ln\NVar{z}$ : principal branch of logarithm function, $\pi\left(\NVar{x}\right)$ : number of primes not exceeding a number, $k$ : positive integer and $x$ : real number
Permalink:: http://dlmf.nist.gov/27.12.E4
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.12 and Ch.27

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\pi\left(x\right)-\operatorname{li}\left(x\right)

changes sign infinitely often as

x\to\infty

; see Littlewood (1914), Bays and Hudson (2000). … ►

27.12.7

\left|\pi\left(x\right)-\operatorname{li}\left(x\right)\right|<\frac{1}{8\pi}% \sqrt{x}\,\ln x.

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Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{li}\left(\NVar{x}\right)$ : logarithmic integral, $\ln\NVar{z}$ : principal branch of logarithm function, $\pi\left(\NVar{x}\right)$ : number of primes not exceeding a number and $x$ : real number
Referenced by:: §27.12
Permalink:: http://dlmf.nist.gov/27.12.E7
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.12, §27.12 and Ch.27

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12: 27.11 Asymptotic Formulas: Partial Sums

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27.11.2

\sum_{n\leq x}d\left(n\right)=x\ln x+(2\gamma-1)x+O\left(\sqrt{x}\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\gamma$ : Euler’s constant, $d_{\NVar{k}}\left(\NVar{n}\right)$ : divisor function, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : positive integer and $x$ : real number
A&S Ref:: 24.3.3 III
Referenced by:: §27.11, §27.11, Erratum (V1.1.8) for Section 27.11
Permalink:: http://dlmf.nist.gov/27.11.E2
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

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27.11.3

\sum_{n\leq x}\frac{d\left(n\right)}{n}=\frac{1}{2}(\ln x)^{2}+2\gamma\ln x+O% \left(1\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\gamma$ : Euler’s constant, $d_{\NVar{k}}\left(\NVar{n}\right)$ : divisor function, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : positive integer and $x$ : real number
Referenced by:: §27.11
Permalink:: http://dlmf.nist.gov/27.11.E3
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

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27.11.6

\sum_{n\leq x}\phi\left(n\right)=\frac{3}{{\pi}^{2}}x^{2}+O\left(x\ln x\right).

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : positive integer and $x$ : real number
A&S Ref:: 24.3.2 III (in slightly different form)
Permalink:: http://dlmf.nist.gov/27.11.E6
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

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27.11.7

\sum_{n\leq x}\frac{\phi\left(n\right)}{n}=\frac{6}{{\pi}^{2}}x+O\left(\ln x% \right).

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\phi\left(\NVar{n}\right)$ : Euler’s totient, $\pi$ : the ratio of the circumference of a circle to its diameter, $\ln\NVar{z}$ : principal branch of logarithm function, $n$ : positive integer and $x$ : real number
Permalink:: http://dlmf.nist.gov/27.11.E7
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

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27.11.8

\sum_{p\leq x}\frac{1}{p}=\ln\ln x+A+O\left(\frac{1}{\ln x}\right),

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Symbols:: $O\left(\NVar{x}\right)$ : order not exceeding, $\ln\NVar{z}$ : principal branch of logarithm function, $p,p_{1},\ldots$ : prime numbers, $x$ : real number and $A$ : constant
Permalink:: http://dlmf.nist.gov/27.11.E8
Encodings:: pMML, png, TeX
Change of Notation (effective with 1.0.10):: The notation for logarithm has been changed to $\ln$ from $\mathrm{log}$ .
See also:: Annotations for §27.11 and Ch.27

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13: 1.13 Differential Equations

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§1.13(iv) Change of Variables

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Transformation of the Point at Infinity

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Elimination of First Derivative by Change of Dependent Variable

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Elimination of First Derivative by Change of Independent Variable

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Liouville Transformation

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14: 4.33 Maclaurin Series and Laurent Series

… ►For expansions that correspond to (4.19.4)–(4.19.9), change

z

i z

and use (4.28.8)–(4.28.13).

15: 25.10 Zeros

… ►Because

|Z(t)|=|\zeta\left(\frac{1}{2}+it\right)|

Z(t)

vanishes at the zeros of

\zeta\left(\frac{1}{2}+it\right)

, which can be separated by observing sign changes of

Z(t)

. Because

Z(t)

changes sign infinitely often,

\zeta\left(\frac{1}{2}+it\right)

has infinitely many zeros with

t

real. … ►By comparing

N(T)

with the number of sign changes of

Z(t)

we can decide whether

\zeta\left(s\right)

has any zeros off the line in this region. Sign changes of

Z(t)

are determined by multiplying (25.9.3) by

\exp\left(i\vartheta(t)\right)

to obtain the Riemann–Siegel formula: …

16: 22.9 Cyclic Identities

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22.9.7

\kappa=\operatorname{dn}\left(2K\left(k\right)/3,k\right),

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Defines:: $\kappa$ : change of variable (locally)
Symbols:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind and $k$ : modulus
Referenced by:: §22.9(iii), §22.9(iv)
Permalink:: http://dlmf.nist.gov/22.9.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

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22.9.8

s_{1,3}^{(4)}s_{2,3}^{(4)}+s_{2,3}^{(4)}s_{3,3}^{(4)}+s_{3,3}^{(4)}s_{1,3}^{(4% )}=\frac{\kappa^{2}-1}{k^{2}},

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Symbols:: $k$ : modulus, $s_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable
Referenced by:: Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10), Erratum (V1.0.25) for Equations (22.9.8), (22.9.9) and (22.9.10)
Permalink:: http://dlmf.nist.gov/22.9.E8
Encodings:: pMML, png, TeX
Correction (effective with 1.0.25):: Originally this equation was written incorrectly for all $s_{m,p}^{(4)}$ with $p=2$ . It has been corrected by writing $p=3$ throughout.
Suggested 2019-11-07 by Juan Miguel Nieto
See also:: Annotations for §22.9(ii), §22.9(ii), §22.9 and Ch.22

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22.9.13

s_{1,3}^{(4)}s_{2,3}^{(4)}s_{3,3}^{(4)}=-\frac{1}{1-\kappa^{2}}\left(s_{1,3}^{% (4)}+s_{2,3}^{(4)}+s_{3,3}^{(4)}\right),

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Symbols:: $s_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable
Permalink:: http://dlmf.nist.gov/22.9.E13
Encodings:: pMML, png, TeX
See also:: Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

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22.9.14

c_{1,3}^{(4)}c_{2,3}^{(4)}c_{3,3}^{(4)}=\frac{\kappa^{2}}{1-\kappa^{2}}\left(c% _{1,3}^{(4)}+c_{2,3}^{(4)}+c_{3,3}^{(4)}\right),

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Symbols:: $c_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable
Permalink:: http://dlmf.nist.gov/22.9.E14
Encodings:: pMML, png, TeX
See also:: Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

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22.9.15

d_{1,3}^{(2)}d_{2,3}^{(2)}d_{3,3}^{(2)}=\frac{\kappa^{2}+k^{2}-1}{1-\kappa^{2}% }\left(d_{1,3}^{(2)}+d_{2,3}^{(2)}+d_{3,3}^{(2)}\right),

ⓘ

Symbols:: $k$ : modulus, $d_{m,p}^{(r)}$ : cyclic quantity and $\kappa$ : change of variable
Permalink:: http://dlmf.nist.gov/22.9.E15
Encodings:: pMML, png, TeX
See also:: Annotations for §22.9(iii), §22.9(iii), §22.9 and Ch.22

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17: 9.7 Asymptotic Expansions

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9.7.1

\zeta=\tfrac{2}{3}z^{\ifrac{3}{2}}.

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Defines:: $\zeta$ : change of variable (locally)
Symbols:: $z$ : complex variable
Permalink:: http://dlmf.nist.gov/9.7.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §9.7(i), §9.7 and Ch.9

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9.7.5

\operatorname{Ai}\left(z\right)\sim\frac{e^{-\zeta}}{2\sqrt{\pi}z^{1/4}}\sum_{% k=0}^{\infty}(-1)^{k}\frac{u_{k}}{\zeta^{k}},

|\operatorname{ph}z|\leq\pi-\delta

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta$ : change of variable and $u_{s}$ : expansion coefficient
Source:: Olver (1997b, (1.07), pp. 392 and 413)
Referenced by:: (9.10.4), (9.10.6), §9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)
Permalink:: http://dlmf.nist.gov/9.7.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §9.7(ii), §9.7 and Ch.9

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9.7.6

\operatorname{Ai}'\left(z\right)\sim-\frac{z^{1/4}e^{-\zeta}}{2\sqrt{\pi}}\sum% _{k=0}^{\infty}(-1)^{k}\frac{v_{k}}{\zeta^{k}},

|\operatorname{ph}z|\leq\pi-\delta

ⓘ

Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $\operatorname{ph}$ : phase, $k$ : nonnegative integer, $z$ : complex variable, $\delta$ : small positive constant, $\zeta$ : change of variable and $v_{s}$ : expansion coefficient
Source:: Olver (1997b, (1.07), pp. 392 and 413)
Referenced by:: §9.7(iii), §9.7(iv), §9.7(iv), §9.7(v), Erratum (V1.0.17) for Subsection 9.7(iv)
Permalink:: http://dlmf.nist.gov/9.7.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §9.7(ii), §9.7 and Ch.9

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9.7.20

R_{n}(z)=(-1)^{n}\sum_{k=0}^{m-1}(-1)^{k}u_{k}\frac{G_{n-k}\left(2\zeta\right)% }{\zeta^{k}}+R_{m,n}(z),

ⓘ

Defines:: $R_{n}$ : remainder function (locally)
Symbols:: $G_{\NVar{p}}\left(\NVar{z}\right)$ : rescaled terminant function, $k$ : nonnegative integer, $z$ : complex variable, $\zeta$ : change of variable, $m$ : index, $n$ : index and $u_{s}$ : expansion coefficient
Proof sketch:: Use the methods of Olver (1991b, 1993a).
Permalink:: http://dlmf.nist.gov/9.7.E20
Encodings:: pMML, png, TeX
See also:: Annotations for §9.7(v), §9.7 and Ch.9

►

9.7.21

S_{n}(z)=(-1)^{n-1}\sum_{k=0}^{m-1}(-1)^{k}v_{k}\frac{G_{n-k}\left(2\zeta% \right)}{\zeta^{k}}+S_{m,n}(z),

ⓘ

Defines:: $S_{n}$ : remainder function (locally)
Symbols:: $G_{\NVar{p}}\left(\NVar{z}\right)$ : rescaled terminant function, $k$ : nonnegative integer, $z$ : complex variable, $\zeta$ : change of variable, $m$ : index, $n$ : index and $v_{s}$ : expansion coefficient
Proof sketch:: Use the methods of Olver (1991b, 1993a).
Permalink:: http://dlmf.nist.gov/9.7.E21
Encodings:: pMML, png, TeX
See also:: Annotations for §9.7(v), §9.7 and Ch.9

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18: 22.2 Definitions

… ►With ►

22.2.3

\zeta=\frac{\pi z}{2K\left(k\right)},

ⓘ

Symbols:: $\pi$ : the ratio of the circumference of a circle to its diameter, $K\left(\NVar{k}\right)$ : Legendre’s complete elliptic integral of the first kind, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/22.2.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

►

22.2.4

\operatorname{sn}\left(z,k\right)=\frac{\theta_{3}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{1}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{ns}\left(z,k\right)},

ⓘ

Defines:: $\operatorname{ns}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{sn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Referenced by:: §23.6(ii)
Permalink:: http://dlmf.nist.gov/22.2.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

►

22.2.5

\operatorname{cn}\left(z,k\right)=\frac{\theta_{4}\left(0,q\right)}{\theta_{2}% \left(0,q\right)}\frac{\theta_{2}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{nc}\left(z,k\right)},

ⓘ

Defines:: $\operatorname{cn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{nc}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Permalink:: http://dlmf.nist.gov/22.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

►

22.2.6

\operatorname{dn}\left(z,k\right)=\frac{\theta_{4}\left(0,q\right)}{\theta_{3}% \left(0,q\right)}\frac{\theta_{3}\left(\zeta,q\right)}{\theta_{4}\left(\zeta,q% \right)}=\frac{1}{\operatorname{nd}\left(z,k\right)},

ⓘ

Defines:: $\operatorname{dn}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function and $\operatorname{nd}\left(\NVar{z},\NVar{k}\right)$ : Jacobian elliptic function
Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $q$ : nome, $z$ : complex, $k$ : modulus and $\zeta$ : change of variable
Referenced by:: §22.20(ii)
Permalink:: http://dlmf.nist.gov/22.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §22.2 and Ch.22

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19: 34.1 Special Notation

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34.1.1

\left(j_{1}\;m_{1}\;j_{2}\;m_{2}|j_{1}\;j_{2}\;j_{3}\,\,m_{3}\right)=(-1)^{j_{% 1}-j_{2}+m_{3}}(2j_{3}+1)^{\frac{1}{2}}\begin{pmatrix}j_{1}&j_{2}&j_{3}\\ m_{1}&m_{2}&-m_{3}\end{pmatrix};

ⓘ

Symbols:: $\begin{pmatrix}\NVar{j_{1}}&\NVar{j_{2}}&\NVar{j_{3}}\\ \NVar{m_{1}}&\NVar{m_{2}}&\NVar{m_{3}}\end{pmatrix}$ : $\mathit{3j}$ symbol and $j,j_{r}$ : non-negative integers or non-negative integers plus one half.
Sources:: Edmonds (1974, p. 46, Eq. (3.7.3)); Rotenberg et al. (1959, p. 1, Eq. (1.1a))
Permalink:: http://dlmf.nist.gov/34.1.E1
Encodings:: pMML, png, TeX
Clarification (effective with 1.0.15):: A wording change reflects that the Clebsch–Gordan coefficients are an alternative formulation of angular momentum problems, rather than alternative notation for the $\mathit{3j}$ . The sign of $m_{3}$ was changed for clarity.
See also:: Annotations for §34.1 and Ch.34

…