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11: 27.11 Asymptotic Formulas: Partial Sums

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27.11.3 $$\sum _{n\le x}\frac{d\left(n\right)}{n}=\frac{1}{2}{(\ln x)}^2+2\gamma \ln x+O\left(1\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\gamma$: Euler’s constant, $d_k\left(n\right)$: divisor function, $\ln z$: 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 $\log$.

See also:

Annotations for §27.11 and Ch.27

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27.11.6 $$\sum _{n\le x}\varphi \left(n\right)=\frac{3}{{\pi }^2}{x}^2+O\left(x\ln x\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: 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 $\log$.

See also:

Annotations for §27.11 and Ch.27

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27.11.7 $$\sum _{n\le x}\frac{\varphi \left(n\right)}{n}=\frac{6}{{\pi }^2}x+O\left(\ln x\right).$$

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\pi$: the ratio of the circumference of a circle to its diameter, $\ln z$: 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 $\log$.

See also:

Annotations for §27.11 and Ch.27

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27.11.8 $$\sum _{p\le x}\frac{1}{p}=\ln \ln x+A+O\left(\frac{1}{\ln x}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\ln z$: principal branch of logarithm function, $p,p_1,\mathrm{\dots }$: 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 $\log$.

See also:

Annotations for §27.11 and Ch.27

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27.11.10 $$\sum _{p\le x}\frac{\ln p}{p}=\ln x+O\left(1\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\ln z$: principal branch of logarithm function, $p,p_1,\mathrm{\dots }$: prime numbers and $x$: real number

Permalink:

http://dlmf.nist.gov/27.11.E10

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.11 and Ch.27

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12: 10.15 Derivatives with Respect to Order

§10.15 Derivatives with Respect to Order

… ►For the notations $\mathrm{Ci}$ and $\mathrm{Si}$ see §6.2(ii). … ►

13: 19.1 Special Notation

… ►In Abramowitz and Stegun (1964, Chapter 17) the functions (19.1.1) and (19.1.2) are denoted, in order, by $K(\alpha )$, $E(\alpha )$, $\mathrm{\Pi }(n\backslash \alpha )$, $F(\varphi \backslash \alpha )$, $E(\varphi \backslash \alpha )$, and $\mathrm{\Pi }(n;\varphi \backslash \alpha )$, where $\alpha =\arcsin k$ and $n$ is the ${\alpha }^2$ (not related to $k$) in (19.1.1) and (19.1.2). …

14: 27.2 Functions

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27.2.3 $$\pi \left(x\right)\sim \frac{x}{\ln x}.$$

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Symbols:

$\sim$: asymptotic equality, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

§25.16(i), §27.11

Permalink:

http://dlmf.nist.gov/27.2.E3

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►An equivalent form states that the $n$th prime $p_n$ (when the primes are listed in increasing order) is asymptotic to $n\ln n$ as $n\to \mathrm{\infty }$: ►

27.2.4 $$p_n\sim n\ln n.$$

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Symbols:

$\sim$: asymptotic equality, $\ln z$: principal branch of logarithm function, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.2.E4

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►It is the special case $k=2$ of the function $d_k\left(n\right)$ that counts the number of ways of expressing $n$ as the product of $k$ factors, with the order of factors taken into account. … ►

27.2.14 $$\mathrm{\Lambda }\left(n\right)=\ln p,$$ $n=p^a$,

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Defines:

$\mathrm{\Lambda }\left(n\right)$: Mangoldt’s function

Symbols:

$\ln z$: principal branch of logarithm function, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a$: primitive root

Permalink:

http://dlmf.nist.gov/27.2.E14

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

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15: 2.5 Mellin Transform Methods

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2.5.4 $$ℳI\left(z\right)=ℳf\left(1-z\right)ℳh\left(z\right).$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $f(x)$: locally integrable function, $I(x)$: convolution integral and $h(x)$: function

Keywords:

Mellin transform

Permalink:

http://dlmf.nist.gov/2.5.E4

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(i), §2.5 and Ch.2

… ►In the half-plane $\mathrm{\Re }z>\max (0,-2\nu )$, the product $ℳf\left(1-z\right)ℳh\left(z\right)$ has a pole of order two at each positive integer, and … ►

2.5.26 $$ℳf\left(z\right)=ℳf_1\left(z\right)+ℳf_2\left(z\right)$$

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Symbols:

$ℳ\left(f\right)\left(s\right)$: Mellin transform, $f(x)$: locally integrable function and $f_j(t)$: truncated functions

Keywords:

Mellin transform

Referenced by:

§2.5(ii)

Permalink:

http://dlmf.nist.gov/2.5.E26

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Mellin transform was changed to $ℳf\left(z\right)$ from $ℳ(f;z)$.

See also:

Annotations for §2.5(ii), §2.5 and Ch.2

… ►(The last order estimate follows from the Riemann–Lebesgue lemma, §1.8(i).) … ►In the notation of (2.5.18) and (2.5.20), $\kappa =0$, ${\beta }_s=s+1$, and $c=0$. …

16: 27.12 Asymptotic Formulas: Primes

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27.12.5 $$\left|\pi \left(x\right)-\mathrm{li}\left(x\right)\right|=O\left(x\exp \left(-c{(\ln x)}^{1/2}\right)\right),$$ $x\to \mathrm{\infty }$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

Erratum (V1.0.18) for Equation (27.12.5)

Permalink:

http://dlmf.nist.gov/27.12.E5

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.18):

Replaced $\sqrt{\ln x}$ with ${(\ln x)}^{1/2}$ to be consistent with other equations in this section.

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

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27.12.6 $$\left|\pi \left(x\right)-\mathrm{li}\left(x\right)\right|=O\left(x\exp \left(-d{(\ln x)}^{3/5}{(\ln \ln x)}^{-1/5}\right)\right).$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number, $d$: positive integer and $x$: real number

Permalink:

http://dlmf.nist.gov/27.12.E6

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

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27.12.8 $$\frac{\mathrm{li}\left(x\right)}{\varphi \left(m\right)}+O\left(x\exp \left(-\lambda (\alpha ){(\ln x)}^{1/2}\right)\right),$$ $m\le {(\ln x)}^{\alpha }$, $\alpha >0$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\varphi \left(n\right)$: Euler’s totient, $\exp z$: exponential function, $\mathrm{li}\left(x\right)$: logarithmic integral, $\ln z$: principal branch of logarithm function, $m$: positive integer and $x$: real number

Referenced by:

Erratum (V1.0.17) for Equation (27.12.8)

Permalink:

http://dlmf.nist.gov/27.12.E8

Encodings:

pMML, png, TeX

Errata (effective with 1.0.17):

Originally the first term was given incorrectly by $\frac{x}{\varphi \left(m\right)}$.

Reported 2017-12-04 by Gergő Nemes

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.12, §27.12 and Ch.27

…

17: 1.17 Integral and Series Representations of the Dirac Delta

… ►Formal interchange of the order of integration in the Fourier integral formula ((1.14.1) and (1.14.4)): … ►Formal interchange of the order of summation and integration in the Fourier summation formula ((1.8.3) and (1.8.4)): … ►

1.17.25 $$\delta \left(\cos {\theta }_1-\cos {\theta }_2\right)\delta \left({\varphi }_1-{\varphi }_2\right)=\sum _{\mathrm{\ell }=0}^{\mathrm{\infty }}\sum _{m=-\mathrm{\ell }}^{\mathrm{\ell }}{Y}_{\mathrm{\ell },m}({\theta }_1,{\varphi }_1)\overline{Y_{\mathrm{\ell },m}({\theta }_2,{\varphi }_2)}.$$

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Symbols:

$\delta \left(x-a\right)$: Dirac delta (or Dirac delta function), $\overline{z}$: complex conjugate, $\cos z$: cosine function, $Y_{l,m}(\theta ,\varphi )$: spherical harmonic and $m$: nonnegative integer

Referenced by:

§1.17(iii)

Permalink:

http://dlmf.nist.gov/1.17.E25

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.19):

As a notational clarification, the second spherical harmonic in the sum over $\mathrm{\ell }$ has been explicity marked up as a complex conjugate.

See also:

Annotations for §1.17(iii), §1.17(iii), §1.17 and Ch.1

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18: 3.4 Differentiation

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First-Order

►For partial derivatives we use the notation $u_{t,s}=u(x_0+th,y_0+sh)$. … ►

Second-Order

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3.4.28 $${\nabla }^2{u}_{0,0}=\frac{1}{h^2}(u_{1,0}+u_{0,1}+u_{-1,0}+u_{0,-1}-4u_{0,0})+O\left(h^2\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding and $u$: function

A&S Ref:

25.3.30

Permalink:

http://dlmf.nist.gov/3.4.E28

Encodings:

pMML, png, TeX

See also:

Annotations for §3.4(iii), §3.4(iii), §3.4 and Ch.3

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Fourth-Order

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19: 18.39 Applications in the Physical Sciences

… ►The nature of, and notations and common vocabulary for, the eigenvalues and eigenfunctions of self-adjoint second order differential operators is overviewed in §1.18. …

20: 30.1 Special Notation

§30.1 Special Notation

►(For other notation see Notation for the Special Functions.) … ►These notations are similar to those used in Arscott (1964b) and Erdélyi et al. (1955). … ►

Other Notations

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