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11: Peter L. Walker

… ►Walker’s published work has been mainly in real and complex analysis, with excursions into analytic number theory and geometry, the latter in collaboration with Professor Mowaffaq Hajja of the University of Jordan. … ►

20 Theta Functions

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12: 20 Theta Functions

Chapter 20 Theta Functions

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13: 24.2 Definitions and Generating Functions

§24.2 Definitions and Generating Functions

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§24.2(i) Bernoulli Numbers and Polynomials

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§24.2(ii) Euler Numbers and Polynomials

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Table 24.2.1: Bernoulli and Euler numbers.

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$n$	$B_n$	$E_n$
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14: Tom M. Apostol

… ►Apostol was born on August 20, 1923. … ►He was internationally known for his textbooks on calculus, analysis, and analytic number theory, which have been translated into five languages, and for creating Project MATHEMATICS!, a series of video programs that bring mathematics to life with computer animation, live action, music, and special effects. … ►In 1998, the Mathematical Association of America (MAA) awarded him the annual Trevor Evans Award, presented to authors of an exceptional article that is accessible to undergraduates, for his piece entitled “What Is the Most Surprising Result in Mathematics?” (Answer: the prime number theorem). … ►

27 Functions of Number Theory

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15: 6.16 Mathematical Applications

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§6.16(ii) Number-Theoretic Significance of $\mathrm{li}\left(x\right)$

►If we assume Riemann’s hypothesis that all nonreal zeros of $\zeta \left(s\right)$ have real part of $\frac{1}{2}$ (§25.10(i)), then ►

6.16.5 $$\mathrm{li}\left(x\right)-\pi (x)=O\left(\sqrt{x}\ln x\right),$$ $x\to \mathrm{\infty }$,

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Symbols:

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

A&S Ref:

5.1.50

Permalink:

http://dlmf.nist.gov/6.16.E5

Encodings:

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See also:

Annotations for §6.16(ii), §6.16 and Ch.6

►where $\pi (x)$ is the number of primes less than or equal to $x$. … ►

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Figure 6.16.2: The logarithmic integral $\mathrm{li}\left(x\right)$, together with vertical bars indicating the value of $\pi (x)$ for $x=10,20,\mathrm{\dots },1000$. Magnify

16: 26.9 Integer Partitions: Restricted Number and Part Size

§26.9 Integer Partitions: Restricted Number and Part Size

… ► $p_k\left(n\right)$ denotes the number of partitions of $n$ into at most $k$ parts. See Table 26.9.1. … ►It follows that $p_k\left(n\right)$ also equals the number of partitions of $n$ into parts that are less than or equal to $k$. …

17: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ► $\left(\genfrac{}{}{0.0pt}{}{n}{n_1,n_2,\mathrm{\dots },n_k}\right)$ is the number of ways of placing $n=n_1+n_2+\mathrm{\cdots }+n_k$ distinct objects into $k$ labeled boxes so that there are $n_j$ objects in the $j$th box. It is also the number of $k$-dimensional lattice paths from $(0,0,\mathrm{\dots },0)$ to $(n_1,n_2,\mathrm{\dots },n_k)$. … ► $M_2$ is the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with $a_1$ cycles of length 1, $a_2$ cycles of length 2, $\mathrm{\dots }$, and $a_n$ cycles of length $n$: ►

26.4.7 $$M_2=\frac{n!}{1^{a_1}(a_1!){\mathrm{\hspace{0.17em}2}}^{a_2}(a_2!)\mathrm{\cdots }{n}^{a_n}(a_n!)}.$$

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Symbols:

$!$: factorial (as in $n!$), $n$: nonnegative integer, $a_n$: nonnegative integers and $M_2$: number of permutations

Referenced by:

§26.13

Permalink:

http://dlmf.nist.gov/26.4.E7

Encodings:

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Annotations for §26.4(i), §26.4 and Ch.26

… ► $M_3$ is the number of set partitions of $\{1,2,\mathrm{\dots },n\}$ with $a_1$ subsets of size 1, $a_2$ subsets of size 2, $\mathrm{\dots }$, and $a_n$ subsets of size $n$: …

18: 26.10 Integer Partitions: Other Restrictions

… ► $p(𝒟,n)$ denotes the number of partitions of $n$ into distinct parts. $p_m(𝒟,n)$ denotes the number of partitions of $n$ into at most $m$ distinct parts. $p(𝒟k,n)$ denotes the number of partitions of $n$ into parts with difference at least $k$. …$p(𝒪,n)$ denotes the number of partitions of $n$ into odd parts. $p(\in S,n)$ denotes the number of partitions of $n$ into parts taken from the set $S$. …

19: 26.2 Basic Definitions

… ►The total number of partitions of $n$ is denoted by $p\left(n\right)$. … ►

Table 26.2.1: Partitions $p\left(n\right)$.

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$n$	$p\left(n\right)$	$n$	$p\left(n\right)$	$n$	$p\left(n\right)$
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3	3	20	627	37	21637
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20: 26.11 Integer Partitions: Compositions

… ► $c\left(n\right)$ denotes the number of compositions of $n$, and $c_m\left(n\right)$ is the number of compositions into exactly $m$ parts. $c(\in T,n)$ is the number of compositions of $n$ with no 1’s, where again $T=\{2,3,4,\mathrm{\dots }\}$. … ►

26.11.1 $$c\left(0\right)=c(\in T,0)=1.$$

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Symbols:

$\in$: element of, $c\left(n\right)$: number of compositions of $n$ , $c(\mathrm{condition},n)$: restricted number of compositions of $n$ and $T$: set

Permalink:

http://dlmf.nist.gov/26.11.E1

Encodings:

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See also:

Annotations for §26.11 and Ch.26

… ►The Fibonacci numbers are determined recursively by … ►Additional information on Fibonacci numbers can be found in Rosen et al. (2000, pp. 140–145).