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1: 27.18 Methods of Computation: Primes

§27.18 Methods of Computation: Primes

►An overview of methods for precise counting of the number of primes not exceeding an arbitrary integer $x$ is given in Crandall and Pomerance (2005, §3.7). …

2: 26.18 Counting Techniques

§26.18 Counting Techniques

… ►Note that this is also one of the counting problems for which a formula is given in Table 26.17.1. …

3: 26.20 Physical Applications

… ►Applications of combinatorics, especially integer and plane partitions, to counting lattice structures and other problems of statistical mechanics, of which the Ising model is the principal example, can be found in Montroll (1964), Godsil et al. (1995), Baxter (1982), and Korepin et al. (1993). …

4: 26.22 Software

… ►For algorithms for counting and analyzing combinatorial structures see Knuth (1993), Nijenhuis and Wilf (1975), and Stanton and White (1986).

5: 25.18 Methods of Computation

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§25.18(i) Function Values and Derivatives

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6: 25.10 Zeros

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§25.10(ii) Riemann–Siegel Formula

►Riemann developed a method for counting the total number $N(T)$ of zeros of $\zeta \left(s\right)$ in that portion of the critical strip with . …

7: 10.58 Zeros

… ►For $n\ge 0$ the $m$th positive zeros of ${𝗃}_n\left(x\right)$, ${𝗃}_n^{\prime }\left(x\right)$, ${𝗒}_n\left(x\right)$, and ${𝗒}_n^{\prime }\left(x\right)$ are denoted by $a_{n,m}$, $a_{n,m}^{\prime }$, $b_{n,m}$, and $b_{n,m}^{\prime }$, respectively, except that for $n=0$ we count $x=0$ as the first zero of ${𝗃}_0^{\prime }\left(x\right)$. …

8: 26.5 Lattice Paths: Catalan Numbers

… ►It counts the number of lattice paths from $(0,0)$ to $(n,n)$ that stay on or above the line $y=x$. …

9: 26.13 Permutations: Cycle Notation

… ►The Stirling cycle numbers of the first kind, denoted by $\left[\genfrac{}{}{0.0pt}{}{n}{k}\right]$, count the number of permutations of $\{1,2,\mathrm{\dots },n\}$ with exactly $k$ cycles. …

10: 27.2 Functions

… ►There is great interest in the function $\pi \left(x\right)$ that counts the number of primes not exceeding $x$. … ►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. …