# number-theoretic functions

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## 11—15 of 15 matching pages

##### 11: 27.2 Functions

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►Other examples of number-theoretic functions treated in this chapter are as follows.
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27.2.9
$$d\left(n\right)=\sum _{d|n}1$$

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##### 12: 27.11 Asymptotic Formulas: Partial Sums

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►The behavior of a number-theoretic function
$f(n)$ for large $n$ is often difficult to determine because the function values can fluctuate considerably as $n$ increases.
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27.11.2
$$\sum _{n\le x}d\left(n\right)=x\mathrm{ln}x+(2\gamma -1)x+O\left(\sqrt{x}\right),$$

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##### 13: 6.16 Mathematical Applications

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►It occurs with Fourier-series expansions of all piecewise continuous functions.
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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 …where $\pi (x)$ is the number of primes less than or equal to $x$. …##### 14: 24.19 Methods of Computation

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###### §24.19(i) Bernoulli and Euler Numbers and Polynomials

… ►A similar method can be used for the Euler numbers based on (4.19.5). … ►###### §24.19(ii) Values of ${B}_{n}$ Modulo $p$

►For number-theoretic applications it is important to compute ${B}_{2n}\phantom{\rule{veryverythickmathspace}{0ex}}(modp)$ for $2n\le p-3$; in particular to find the*irregular pairs*$(2n,p)$ for which ${B}_{2n}\equiv 0\phantom{\rule{veryverythickmathspace}{0ex}}(modp)$. … ►Buhler et al. (1992) uses the expansion

24.19.3
$$\frac{{t}^{2}}{\mathrm{cosh}t-1}=-2\sum _{n=0}^{\mathrm{\infty}}(2n-1){B}_{2n}\frac{{t}^{2n}}{(2n)!},$$

and computes inverses modulo $p$ of the left-hand side. Multisectioning techniques are applied in implementations. See also Crandall (1996, pp. 116–120).

##### 15: Bibliography M

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Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.
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Methods of Theoretical Physics.
Vol. 1, McGraw-Hill Book Co., New York.
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Methods of Theoretical Physics.
Vol. 2, McGraw-Hill Book Co., New York.
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Asymptotic expansions of oblate spheroidal wave functions and their characteristic numbers.
J. Reine Angew. Math. 211, pp. 33–47.
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Asymptotic expansions of prolate spheroidal wave functions and their characteristic numbers.
J. Reine Angew. Math. 212, pp. 26–48.
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