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1: 27.18 Methods of Computation: Primes

… ►An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … ► …

2: 27.1 Special Notation

§27.1 Special Notation

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$d,k,m,n$	positive integers (unless otherwise indicated).
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3: 27 Functions of Number Theory

Chapter 27 Functions of Number Theory

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4: 27.17 Other Applications

… ► ►There are also applications of number theory in many diverse areas, including physics, biology, chemistry, communications, and art. …

5: 27.9 Quadratic Characters

§27.9 Quadratic Characters

►For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. … ►If $p,q$ are distinct odd primes, then the quadratic reciprocity law states that … ►If an odd integer $P$ has prime factorization $P={\prod }_{r=1}^{\nu \left(n\right)}{p}_r^{a_r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod }_{r=1}^{\nu \left(n\right)}{(n|p_r)}^{a_r}$, with $(n|1)=1$. …

6: 20.12 Mathematical Applications

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§20.12(i) Number Theory

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7: Wadim Zudilin

… ►His research interests are primarily focused on applications of special functions in different parts of number theory. …He is a member of several editorial boards including the series Monographs in Number Theory published by World Scientific. …

8: David M. Bressoud

… ►Bressoud has published numerous papers in number theory, combinatorics, and special functions. … 227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special Relativity, published by Springer-Verlag in 1992, A Radical Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S. …

9: 27.13 Functions

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§27.13(i) Introduction

►Whereas multiplicative number theory is concerned with functions arising from prime factorization, additive number theory treats functions related to addition of integers. …The subsections that follow describe problems from additive number theory. … ►

§27.13(ii) Goldbach Conjecture

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§27.13(iii) Waring’s Problem

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10: 27.8 Dirichlet Characters

§27.8 Dirichlet Characters

… ►An example is the principal character (mod $k$): … ►If $\left(n,k\right)=1$, then the characters satisfy the orthogonality relation … ►A divisor $d$ of $k$ is called an induced modulus for $\chi$ if … ►If $k$ is odd, then the real characters (mod $k$) are the principal character and the quadratic characters described in the next section.