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11: 25.10 Zeros

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25.10.1 $$Z(t)\equiv \exp \left(\mathrm{i}\vartheta (t)\right)\zeta \left(\frac{1}{2}+\mathrm{i}t\right),$$

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Symbols:

$\zeta \left(s\right)$: Riemann zeta function, $\equiv$: equals by definition, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $Z(t)$: zeros function and $\vartheta (t)$: function

Keywords:

definition

Source:

Titchmarsh (1986b, (4.17.2), p. 89)

Permalink:

http://dlmf.nist.gov/25.10.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.10(i), §25.10 and Ch.25

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25.10.2 $$\vartheta (t)\equiv \mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)-\frac{1}{2}t\ln \pi$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\overline{z}$: complex conjugate, $\equiv$: equals by definition, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $\mathrm{ph}$: phase, $z$: complex variable, $\vartheta (t)$: function and $\chi (s)$: function

Keywords:

definition

Proof sketch:

Derivable from Titchmarsh (1986b, third equation on p. 89), namely $\vartheta (t)\equiv -\frac{1}{2}\mathrm{ph}\chi (\frac{1}{2}+\mathrm{i}t)$, by using (25.9.2), (1.9.18), (1.9.20), $\mathrm{\Gamma }\left(\overline{z}\right)=\overline{\mathrm{\Gamma }\left(z\right)}$, and (4.8.10).

Permalink:

http://dlmf.nist.gov/25.10.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §25.10(i), §25.10 and Ch.25

►is chosen to make $Z(t)$ real, and $\mathrm{ph}\mathrm{\Gamma }\left(\frac{1}{4}+\frac{1}{2}\mathrm{i}t\right)$ assumes its principal value. … ►The error term $R(t)$ can be expressed as an asymptotic series that begins … ►Riemann also developed a technique for determining further terms. …

12: Guide to Searching the DLMF

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Terms, Phrases and Expressions

►Search queries are made up of terms, textual phrases, and math expressions, combined with Boolean operators: ►

a textual word, a number, or a math symbol.

… ►If you do not want a term or a sequence of terms in your query to undergo math processing, you should quote them as a phrase. … ►

Single-letter terms

…

13: 27.12 Asymptotic Formulas: Primes

… ►There exists a positive constant $c$ such that …The best available asymptotic error estimate (2009) appears in Korobov (1958) and Vinogradov (1958): there exists a positive constant $d$ such that … ►

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

… ►For example, if $2^n\cancel{\equiv }2(modn)$, then $n$ is composite. … ►A Carmichael number is a composite number $n$ for which $b^n\equiv b(modn)$ for all $b\in \mathbb{N}$. …

14: 27.11 Asymptotic Formulas: Partial Sums

… ►where $\gamma$ is Euler’s constant (§5.2(ii)). … ►The error terms given here are not necessarily the best known. …where $\gamma$ again is Euler’s constant. …where $A$ is a constant. … ►for some positive constant $C$, …

15: Errata

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31.11.6 $$K_j=\frac{(j+\alpha -\mu -1)(j+\beta -\mu -1)(j+\gamma -\mu -1)(j-\mu )}{(2j+\lambda -\mu -1)(2j+\lambda -\mu -2)}$$

The sign has been corrected and the final term in the numerator $(j+\lambda -1)$ has been corrected to be $(j-\mu )$.

Suggested by Hans Volkmer on 2022-06-02

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31.11.8 $$M_j=\frac{(j-\alpha +\lambda +1)(j-\beta +\lambda +1)(j-\gamma +\lambda +1)(j+\lambda )}{(2j+\lambda -\mu +1)(2j+\lambda -\mu +2)}$$

The sign has been corrected and the final term in the numerator $(j-\mu +1)$ has been corrected to be $(j+\lambda )$.

Suggested by Hans Volkmer on 2022-06-02

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19.20.11 $$R_J(0,y,z,p)=\frac{3}{2p\sqrt{z}}\ln \left(\frac{16z}{y}\right)-\frac{3}{p}{R}_C(z,p)+O\left(y\ln y\right),$$

as $y\to 0+$, $p$ ($\ne 0$) real, we have added the constant term $\frac{-3}{p}{R}_C(z,p)$ and the order term $O\left(y\ln y\right)$, and hence $\sim$ was replaced by $=$.

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Four of the terms were rewritten for improved clarity.

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31.12.3 $$\frac{d^{2}w}{{dz}^2}-\left(\frac{\gamma }{z}+\delta +z\right)\frac{dw}{dz}+\frac{\alpha z-q}{z}w=0$$

Originally the sign in front of the second term in this equation was $+$. The correct sign is $-$.

Reported 2013-10-31 by Henryk Witek.

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16: 31.17 Physical Applications

… ►The problem of adding three quantum spins $𝐬$, $𝐭$, and $𝐮$ can be solved by the method of separation of variables, and the solution is given in terms of a product of two Heun functions. … ►

$${𝐉}^2\mathrm{\Psi }(𝐱)\equiv {(𝐬+𝐭+𝐮)}^2\mathrm{\Psi }(𝐱)=j(j+1)\mathrm{\Psi }(𝐱),$$

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$${𝐻}_s\mathrm{\Psi }(𝐱)\equiv (-2𝐬\cdot 𝐭-(2/a)𝐬\cdot 𝐮)\mathrm{\Psi }(𝐱)=h_s\mathrm{\Psi }(𝐱),$$

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$\gamma =-2s,$

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17: 2.6 Distributional Methods

… ►Motivated by Watson’s lemma (§2.3(ii)), we substitute (2.6.2) in (2.6.1), and integrate term by term. …Inserting (2.6.2) into (2.6.1) and integrating formally term-by-term, we obtain … ►In terms of the convolution product …The replacement of $f(t)$ by its asymptotic expansion (2.6.9), followed by term-by-term integration leads to convolution integrals of the form … ►On inserting this identity into (2.6.54), we immediately encounter divergent integrals of the form …

18: 13.9 Zeros

… ►Inequalities for ${\varphi }_r$ are given in Gatteschi (1990), and identities involving infinite series of all of the complex zeros of $M(a,b,x)$ are given in Ahmed and Muldoon (1980). … ►

13.9.9 $$z=\pm (2n+a)\pi \mathrm{i}+\ln \left(-\frac{\mathrm{\Gamma }\left(a\right)}{\mathrm{\Gamma }\left(b-a\right)}{\left(\pm 2n\pi \mathrm{i}\right)}^{b-2a}\right)+O\left(n^{-1}\ln n\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{i}$: imaginary unit, $\ln z$: principal branch of logarithm function, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.9(i)

Permalink:

http://dlmf.nist.gov/13.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §13.9(i), §13.9 and Ch.13

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13.9.11 $$T(a,b)=\lfloor -a\rfloor +1,$$ , $\mathrm{\Gamma }\left(a\right)\mathrm{\Gamma }\left(a-b+1\right)>0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\lfloor x\rfloor$: floor of $x$ and $T(a,b)$: number of zeros

Permalink:

http://dlmf.nist.gov/13.9.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §13.9(ii), §13.9 and Ch.13

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13.9.12 $$T(a,b)=\lfloor -a\rfloor ,$$ , ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\lfloor x\rfloor$: floor of $x$ and $T(a,b)$: number of zeros

Permalink:

http://dlmf.nist.gov/13.9.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §13.9(ii), §13.9 and Ch.13

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13.9.16 $$a=-n-\frac{2}{\pi }\sqrt{zn}-\frac{2z}{{\pi }^2}+\frac{1}{2}b+\frac{1}{4}+\frac{z^2\left(\frac{1}{3}-4{\pi }^{-2}\right)+z-{(b-1)}^2+\frac{1}{4}}{4\pi \sqrt{zn}}+O\left(\frac{1}{n}\right),$$

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Symbols:

$O\left(x\right)$: order not exceeding, $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $n$: nonnegative integer and $z$: complex variable

Referenced by:

§13.9(ii), DLMF Update; Version 1.0.13, Erratum (V1.0.12) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16), Erratum (V1.0.13) for Equation (13.9.16)

Permalink:

http://dlmf.nist.gov/13.9.E16

Encodings:

pMML, png, TeX

Errata (effective with 1.0.13):

In applying changes in Version 1.0.12 to this equation, an editing error was made; it has been corrected.

Reported 2016-09-12 by Adri Olde Daalhuis

Errata (effective with 1.0.12):

Originally this equation was expressed in terms of the asymptotic symbol $\sim$. As a consequence of the use of the $O$ order symbol on the right hand side, $\sim$ was replaced by $=$.

Reported 2016-07-11 by Rudi Weikard

See also:

Annotations for §13.9(ii), §13.9 and Ch.13

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19: 25.12 Polylogarithms

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25.12.1 $${\mathrm{Li}}_2\left(z\right)\equiv \sum _{n=1}^{\mathrm{\infty }}\frac{z^n}{n^2},$$ $|z|\le 1$.

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Defines:

${\mathrm{Li}}_2\left(z\right)$: dilogarithm

Symbols:

$\equiv$: equals by definition, $n$: nonnegative integer, $x$: real variable and $z$: complex variable

Keywords:

definition

Source:

Maximon (2003, (3.1), p. 2808)

A&S Ref:

27.7.2 (with $z=1-x$)

Referenced by:

§25.12(i), §25.12(i)

Permalink:

http://dlmf.nist.gov/25.12.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(i), §25.12 and Ch.25

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25.12.11 $${\mathrm{Li}}_s\left(z\right)\equiv \frac{z}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}}{{\mathrm{e}}^x-z}dx,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\int$: integral, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $x$: real variable, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Lewin (1981, (7.188), p. 236)

Referenced by:

(25.12.16), §25.12(ii)

Permalink:

http://dlmf.nist.gov/25.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

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25.12.14 $$F_s(x)=\frac{1}{\mathrm{\Gamma }\left(s+1\right)}{\int }_0^{\mathrm{\infty }}\frac{t^s}{{\mathrm{e}}^{t-x}+1}dt,$$ $s>-1$,

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Defines:

$F_s(x)$: Fermi–Dirac integral (locally)

Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $x$: real variable and $s$: complex variable

Keywords:

definition, Fermi–Dirac integral, improper integral, integral representation

Source:

Dingle (1957b, (1), p. 226)

Referenced by:

(25.12.16), 3rd item, 5th item

Permalink:

http://dlmf.nist.gov/25.12.E14

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(iii), §25.12 and Ch.25

… ►Sometimes the factor $1/\mathrm{\Gamma }\left(s+1\right)$ is omitted. … ►In terms of polylogarithms …

20: 18.35 Pollaczek Polynomials

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18.35.5 $${\int }_{-1}^1{P}_n^{(\lambda )}(x;a,b)P_m^{(\lambda )}(x;a,b)w^{(\lambda )}(x;a,b)dx=\frac{\mathrm{\Gamma }\left(2\lambda +n\right)}{n!(\lambda +a+n)}{\delta }_{n,m},$$ $a\ge b\ge -a$, $\lambda >0$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, $P_n^{(\lambda )}(x;a,b)$: Pollaczek polynomial, $dx$: differential of $x$ , $!$: factorial (as in $n!$), $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Ismail (2009, Theorem 5.4.2)(proved)

Referenced by:

(18.35.6), §18.35(ii), Erratum (V1.2.0) for Equation (18.35.5)

Permalink:

http://dlmf.nist.gov/18.35.E5

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

This equation which was previously only valid for $n\ne m$, was updated to give the full normalization and the constraints which were originally given in (18.35.6) were moved to this equation. The constraint $\lambda >-\frac{1}{2}$ has been updated to be $\lambda >0$.

See also:

Annotations for §18.35(ii), §18.35 and Ch.18

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18.35.6 $$w^{(\lambda )}(\cos \theta ;a,b)={\pi }^{-1}{\mathrm{e}}^{(2\theta -\pi ){\tau }_{a,b}(\theta )}{\left(2\sin \theta \right)}^{2\lambda -1}{\left|\mathrm{\Gamma }\left(\lambda +\mathrm{i}{\tau }_{a,b}(\theta )\right)\right|}^2,$$ .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\sin z$: sine function, $w(x)$: weight function and ${\tau }_{a,b}(\theta )$

Source:

Ismail (2009, (5.4.13))

Referenced by:

(18.35.5), §18.35(ii), §18.39(iv), §18.39(iv), Erratum (V1.2.0) for Equation (18.35.5)

Permalink:

http://dlmf.nist.gov/18.35.E6

Encodings:

pMML, png, TeX

Modification (effective with 1.2.0):

The constraints in this equation have been moved to (18.35.5) except for .

See also:

Annotations for §18.35(ii), §18.35 and Ch.18

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$w_{x_k}^{(\lambda )}(a,b)={\displaystyle \frac{{\rho }^{2k-1}{\left(1-{\rho }^2\right)}^{2\lambda +1}\mathrm{\Gamma }\left(2\lambda +k\right)}{2\mathrm{\Delta }k!}},$

… ►

18.35.6_5 $${\int }_{-1}^1{P}_n^{(\lambda )}(x;a,b,c)P_m^{(\lambda )}(x;a,b,c)w^{(\lambda )}(x;a,b,c)dx=\frac{\mathrm{\Gamma }\left(c+1\right)\mathrm{\Gamma }\left(2\lambda +c+n\right)}{{\left(c+1\right)}_n(\lambda +a+c+n)}{\delta }_{n,m},$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\delta }_{j,k}$: Kronecker delta, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $Q_n^{(\lambda )}(x;a,b,c)$: Pollaczek polynomial, $dx$: differential of $x$ , $\int$: integral, $w(x)$: weight function, $m$: nonnegative integer, $n$: nonnegative integer and $x$: real variable

Proved:

Pollaczek (1950, (3))(proved)

Permalink:

http://dlmf.nist.gov/18.35.E6_5

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.35(ii), §18.35 and Ch.18

… ►This expansion is in terms of the Airy function $\mathrm{Ai}\left(x\right)$ and its derivative (§9.2), and is uniform in any compact $\theta$-interval in $(0,\mathrm{\infty })$. …