# number-theoretic functions

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## 11—15 of 15 matching pages

##### 11: 27.2 Functions
Other examples of number-theoretic functions treated in this chapter are as follows. …
27.2.9 $d\left(n\right)=\sum_{d\mathbin{|}n}1$
##### 12: 27.11 Asymptotic Formulas: Partial Sums
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. …
27.11.2 $\sum_{n\leq x}d\left(n\right)=x\ln x+(2\gamma-1)x+O\left(\sqrt{x}\right),$
##### 13: 6.16 Mathematical Applications
It occurs with Fourier-series expansions of all piecewise continuous functions. … …
###### §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 $\tfrac{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
###### §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 $B_{2n}\pmod{p}$ for $2n\leq p-3$; in particular to find the irregular pairs $(2n,p)$ for which $B_{2n}\equiv 0\pmod{p}$. …
• Buhler et al. (1992) uses the expansion

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
• Magma (website) Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.
• P. M. Morse and H. Feshbach (1953a) Methods of Theoretical Physics. Vol. 1, McGraw-Hill Book Co., New York.
• P. M. Morse and H. Feshbach (1953b) Methods of Theoretical Physics. Vol. 2, McGraw-Hill Book Co., New York.
• H. J. W. Müller (1962) Asymptotic expansions of oblate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 211, pp. 33–47.
• H. J. W. Müller (1963) Asymptotic expansions of prolate spheroidal wave functions and their characteristic numbers. J. Reine Angew. Math. 212, pp. 26–48.