Mobius%20inversion

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11: 27.11 Asymptotic Formulas: Partial Sums

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27.11.12 $$\sum _{n\le x}\mu \left(n\right)=O\left(x{\mathrm{e}}^{-C\sqrt{\ln x}}\right),$$ $x\to \mathrm{\infty }$,

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Symbols:

$O\left(x\right)$: order not exceeding, $\mu \left(n\right)$: Möbius function, $\mathrm{e}$: base of natural logarithm, $\ln z$: principal branch of logarithm function, $n$: positive integer, $x$: real number and $C$: constant

A&S Ref:

24.3.1 III

Referenced by:

§27.11

Permalink:

http://dlmf.nist.gov/27.11.E12

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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27.11.13 $$\underset{x\to \mathrm{\infty }}{lim}\frac{1}{x}\sum _{n\le x}\mu \left(n\right)=0,$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $n$: positive integer and $x$: real number

Referenced by:

§27.11

Permalink:

http://dlmf.nist.gov/27.11.E13

Encodings:

pMML, png, TeX

See also:

Annotations for §27.11 and Ch.27

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27.11.14 $$\underset{x\to \mathrm{\infty }}{lim}\sum _{n\le x}\frac{\mu \left(n\right)}{n}=0,$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.1 III

Referenced by:

§27.11

Permalink:

http://dlmf.nist.gov/27.11.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §27.11 and Ch.27

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27.11.15 $$\underset{x\to \mathrm{\infty }}{lim}\sum _{n\le x}\frac{\mu \left(n\right)\ln n}{n}=-1.$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $\ln z$: principal branch of logarithm function, $n$: positive integer and $x$: real number

A&S Ref:

24.3.1 III

Referenced by:

§27.11, §27.11

Permalink:

http://dlmf.nist.gov/27.11.E15

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: 20 Theta Functions

Chapter 20 Theta Functions

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13: 4.27 Sums

§4.27 Sums

►For sums of trigonometric and inverse trigonometric functions see Gradshteyn and Ryzhik (2000, Chapter 1), Hansen (1975, §§14–42), Oberhettinger (1973), and Prudnikov et al. (1986a, Chapter 5).

14: 27.10 Periodic Number-Theoretic Functions

… ►It can also be expressed in terms of the Möbius function as a divisor sum: ►

27.10.5 $$c_k\left(n\right)=\sum _{d|\left(n,k\right)}d\mu \left(\frac{k}{d}\right).$$

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Symbols:

$\mu \left(n\right)$: Möbius function, $c_k\left(n\right)$: Ramanujan’s sum, $\left(m,n\right)$: greatest common divisor (gcd), $d$: positive integer, $k$: positive integer and $n$: positive integer

Permalink:

http://dlmf.nist.gov/27.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §27.10 and Ch.27

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15: 4.47 Approximations

§4.47 Approximations

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§4.47(i) Chebyshev-Series Expansions

►Clenshaw (1962) and Luke (1975, Chapter 3) give 20D coefficients for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arctan$, $\mathrm{arcsinh}$. … ►Hart et al. (1968) give $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\cot$, $\arcsin$, $\arccos$, $\arctan$, $\sinh$, $\cosh$, $\tanh$, $\mathrm{arcsinh}$, $\mathrm{arccosh}$. … ►Luke (1975, Chapter 3) supplies real and complex approximations for $\ln$, $\exp$, $\sin$, $\cos$, $\tan$, $\arctan$, $\mathrm{arcsinh}$. …

16: 4.29 Graphics

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§4.29(i) Real Arguments

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Figure 4.29.2: Principal values of $\mathrm{arcsinh}x$ and $\mathrm{arccosh}x$. … Magnify

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Figure 4.29.4: Principal values of $\mathrm{arctanh}x$ and $\mathrm{arccoth}x$. … Magnify

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§4.29(ii) Complex Arguments

… ►The surfaces for the complex hyperbolic and inverse hyperbolic functions are similar to the surfaces depicted in §4.15(iii) for the trigonometric and inverse trigonometric functions. …

17: 18.40 Methods of Computation

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§18.40(ii) The Classical Moment Problem

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Stieltjes Inversion via (approximate) Analytic Continuation

… ►Results of low ($2$ to $3$ decimal digits) precision for $w(x)$ are easily obtained for $N\sim 10$ to $20$. … ►

Histogram Approach

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Derivative Rule Approach

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18: 4.1 Special Notation

… ►The main purpose of the present chapter is to extend these definitions and properties to complex arguments $z$. ►The main functions treated in this chapter are the logarithm $\ln z$, $\mathrm{Ln}z$; the exponential $\exp z$, ${\mathrm{e}}^z$; the circular trigonometric (or just trigonometric) functions $\sin z$, $\cos z$, $\tan z$, $\csc z$, $\sec z$, $\cot z$; the inverse trigonometric functions $\arcsin z$, $\mathrm{Arcsin}z$, etc. ; the hyperbolic trigonometric (or just hyperbolic) functions $\sinh z$, $\cosh z$, $\tanh z$, $\mathrm{csch}z$, $\mathrm{sech}z$, $\coth z$; the inverse hyperbolic functions $\mathrm{arcsinh}z$, $\mathrm{Arcsinh}z$, etc. ►Sometimes in the literature the meanings of $\ln$ and $\mathrm{Ln}$ are interchanged; similarly for $\arcsin z$ and $\mathrm{Arcsin}z$, etc. …${\sin }^{-1}z$ for $\arcsin z$ and ${\mathrm{Sin}}^{-1}z$ for $\mathrm{Arcsin}z$.

19: 1.9 Calculus of a Complex Variable

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1.9.6 $$\omega =\arctan \left(\left|y/x\right|\right)\in [0,\frac{1}{2}\pi ].$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $[a,b]$: closed interval, $\in$: element of, $\arctan z$: arctangent function and $\left|x\right|$: absolute value of $x$

A&S Ref:

3.7.4

Permalink:

http://dlmf.nist.gov/1.9.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §1.9(i), §1.9(i), §1.9 and Ch.1

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Bilinear Transformation

… ►Other names for the bilinear transformation are fractional linear transformation, homographic transformation, and Möbius transformation. … ►

§1.9(vii) Inversion of Limits

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20: 26.13 Permutations: Cycle Notation

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26.13.6 $$\left(j,k\right)=\left(k-1,k\right)\left(k-2,k-1\right)\mathrm{\cdots }\left(j+1,j+2\right)\left(j,j+1\right)\left(j+1,j+2\right)\mathrm{\cdots }\left(k-1,k\right).$$

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Symbols:

$\left(S\right)$: cycle, $j$: nonnegative integer and $k$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.13.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.13 and Ch.26

… ►Given a permutation $\sigma \in {𝔖}_n$, the inversion number of $\sigma$, denoted $inv(\sigma )$, is the least number of adjacent transpositions required to represent $\sigma$. …