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31: 24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. …

32: Bibliography V

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Representation of an odd number as a sum of three primes (Russian). Dokl. Akad. Nauk SSSR 15, pp. 169–172 (Russian).

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External Links:

ZentralBlatt

Referenced by:

§27.13(ii).

Encodings:

BibTeX

…

33: Bibliography W

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Prime Divisors of the Bernoulli and Euler Numbers. In Number Theory for the Millennium, III (Urbana, IL, 2000), pp. 357–374.

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External Links:

FullText, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.20.

Encodings:

BibTeX

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34: 24.15 Related Sequences of Numbers

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24.15.9 $$p\frac{B_n}{n}\equiv S(p-1+n,p-1)(mod{p}^2),$$ $1\le n\le p-2$,

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Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

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24.15.10 $$\frac{2n-1}{4n}{p}^2{B}_{2n}\equiv S(p+2n,p-1)(mod{p}^3),$$ $2\le 2n\le p-3$.

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Symbols:

$B_n$: Bernoulli numbers, $S(n,k)$: Stirling number of the second kind, $\equiv$: modular equivalence, $n$: integer and $p$: prime

Referenced by:

§24.15(iii)

Permalink:

http://dlmf.nist.gov/24.15.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §24.15(iii), §24.15 and Ch.24

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35: 27.20 Methods of Computation: Other Number-Theoretic Functions

§27.20 Methods of Computation: Other Number-Theoretic Functions

►To calculate a multiplicative function it suffices to determine its values at the prime powers and then use (27.3.2). For a completely multiplicative function we use the values at the primes together with (27.3.10). …

36: 9.9 Zeros

… ►On the real line, $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ each have an infinite number of zeros, all of which are negative. … ►However, $\mathrm{Bi}\left(z\right)$ and ${\mathrm{Bi}}^{\prime }\left(z\right)$ each have an infinite number of complex zeros. …

37: 26.10 Integer Partitions: Other Restrictions

… ► $p({𝒟}^{\prime }3,n)$ denotes the number of partitions of $n$ into parts with difference at least 3, except that multiples of 3 must differ by at least 6. … ►

26.10.15 $$p({𝒟}^{\prime }3,n)=p(\in A_{1,6},n).$$

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Symbols:

$\in$: element of, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $n$: nonnegative integer and $A_{j,k}$: set

Permalink:

http://dlmf.nist.gov/26.10.E15

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(iv), §26.10 and Ch.26

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38: 26.18 Counting Techniques

… ►The number of positive integers $\le N$ that are not divisible by any of the primes $p_1,p_2,\mathrm{\dots },p_n$ (§27.2(i)) is …

39: 25.11 Hurwitz Zeta Function

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25.11.22 $${\zeta }^{\prime }(1-2n,\frac{1}{2})=-\frac{B_{2n}\ln 2}{n\cdot 4^n}-\frac{(2^{2n-1}-1){\zeta }^{\prime }\left(1-2n\right)}{2^{2n-1}},$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Keywords:

derivative

Source:

Miller and Adamchik (1998, (17), p. 205)

Permalink:

http://dlmf.nist.gov/25.11.E22

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

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25.11.23 $${\zeta }^{\prime }(1-2n,\frac{1}{3})=-\frac{\pi (9^n-1)B_{2n}}{8n\sqrt{3}(3^{2n-1}-1)}-\frac{B_{2n}\ln 3}{4n\cdot 3^{2n-1}}-\frac{{(-1)}^n{\psi }^{(2n-1)}\left(\frac{1}{3}\right)}{2\sqrt{3}{(6\pi )}^{2n-1}}-\frac{\left(3^{2n-1}-1\right){\zeta }^{\prime }\left(1-2n\right)}{2\cdot 3^{2n-1}},$$ $n=1,2,3,\mathrm{\dots }$.

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function and $n$: nonnegative integer

Keywords:

derivative

Source:

Miller and Adamchik (1998, (18), p. 205)

Permalink:

http://dlmf.nist.gov/25.11.E23

Encodings:

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See also:

Annotations for §25.11(vi), §25.11(vi), §25.11 and Ch.25

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25.11.32 $${\int }_0^a{x}^n\psi \left(x\right)dx={(-1)}^{n-1}{\zeta }^{\prime }\left(-n\right)+{(-1)}^n{H}_n\frac{B_{n+1}}{n+1}-\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right)H_k\frac{B_{k+1}(a)}{k+1}{a}^{n-k}+\sum _{k=0}^n{(-1)}^k\left(\genfrac{}{}{0.0pt}{}{n}{k}\right){\zeta }^{\prime }(-k,a)a^{n-k},$$ $n=1,2,\mathrm{\dots }$, $\mathrm{\Re }a>0$,

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $\left(\genfrac{}{}{0.0pt}{}{m}{n}\right)$: binomial coefficient, $dx$: differential of $x$ , $\psi \left(z\right)$: psi (or digamma) function, $H_n$: harmonic number, $\int$: integral, $\mathrm{\Re }$: real part, $k$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $a$: real or complex parameter and $h(n)$: harmonic number

Keywords:

definite integral, integral representation

Source:

Adamchik (1998, (17), p. 197)

Referenced by:

Erratum (V1.1.4) for Notation

Permalink:

http://dlmf.nist.gov/25.11.E32

Encodings:

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Notation (effective with 1.1.4):

The notation previously used for the harmonic number $h(n)$ (resp. $h(k)$) has been replaced to be $H_n$ (resp. $H_k$).

Suggested 2021-08-23 by Gergő Nemes

See also:

Annotations for §25.11(viii), §25.11 and Ch.25

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25.11.34 $$n{\int }_0^a{\zeta }^{\prime }(1-n,x)dx={\zeta }^{\prime }(-n,a)-{\zeta }^{\prime }\left(-n\right)+\frac{B_{n+1}-B_{n+1}\left(a\right)}{n(n+1)},$$ $n=1,2,\mathrm{\dots }$, $\mathrm{\Re }a>0$.

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Symbols:

$B_n$: Bernoulli numbers, $B_n\left(x\right)$: Bernoulli polynomials, $\zeta (s,a)$: Hurwitz zeta function, $\zeta \left(s\right)$: Riemann zeta function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part, $n$: nonnegative integer, $x$: real variable and $a$: real or complex parameter

Keywords:

definite integral, integral representation

Source:

Adamchik (1998, (15), p. 196)

Permalink:

http://dlmf.nist.gov/25.11.E34

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(viii), §25.11 and Ch.25

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25.11.44 $${\zeta }^{\prime }(-1,a)-\frac{1}{12}+\frac{1}{4}{a}^2-\left(\frac{1}{12}-\frac{1}{2}a+\frac{1}{2}{a}^2\right)\ln a\sim -\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k+2}}{(2k+2)(2k+1)2k}{a}^{-2k},$$

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Symbols:

$B_n$: Bernoulli numbers, $\zeta (s,a)$: Hurwitz zeta function, $\sim$: Poincaré asymptotic expansion, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $a$: real or complex parameter

Keywords:

asymptotic approximation

Source:

Elizalde (1986, (18), p. 349)

Permalink:

http://dlmf.nist.gov/25.11.E44

Encodings:

pMML, png, TeX

See also:

Annotations for §25.11(xii), §25.11 and Ch.25

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40: 20.14 Methods of Computation

… ►Then ${\tau }^{\prime }=-1/\tau =-\mathrm{i}\pi /\ln \left(0.9\right)$ and $q^{\prime }={\mathrm{e}}^{\mathrm{i}\pi {\tau }^{\prime }}=\exp \left({\pi }^2/\ln \left(0.9\right)\right)=(2.07\mathrm{\dots })\times {10}^{-41}$. Hence the first term of the series (20.2.3) for ${\theta }_3\left(z{\tau }^{\prime }\right|{\tau }^{\prime })$ suffices for most purposes. In theory, starting from any value of $\tau$, a finite number of applications of the transformations $\tau \to \tau +1$ and $\tau \to -1/\tau$ will result in a value of $\tau$ with $\mathrm{\Im }\tau \ge \sqrt{3}/2$; see §23.18. …