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Appendix C Notation

This is a quick guide to possibly unfamiliar notation. Page numbers or references usually refer to the first appearance of a notation with that meaning, occasionally to a definition.
Symbol Description Location
\(\mathbb{Z}\) (ring of) integers Definition 1.0.1
\(\mathbb{N}\) counting numbers (starting at zero) Definition 1.0.1
\(a\mid b\) \(a\) is a divisor of \(b\) Definition 1.2.5
\(\gcd(a,b)\) greatest common divisor of \(a\) and \(b\) Definition 2.2.1
\(\lfloor x\rfloor\) greatest integer (floor) function Definition 3.3.3
\(a \equiv b \text{ (mod }n)\) \(a\) is congruent to \(b\) modulo \(n\) Definition 4.1.1
\([a]\) the equivalence class of \(a\) modulo some fixed \(n\) Definition 4.4.1
\(a^{-1}\) multiplicative inverse of a number modulo some fixed \(n\) Definition 5.3.4
\(\prod_{i=1}^n p_i\) product of unspecified, possible identical, primes Theorem 6.3.2
\(\prod p\) short form for product of primes Example 6.3.3
\(\prod q\) alternate short form for product of primes Example 6.3.3
\(\prod_{i=1}^n p_i^{e_i}\) product of unspecified distinct prime power Example 6.3.4
\(\prod p^e\) short form for product of prime powers Example 6.3.4
\(p^k\parallel n\) for \(p\) prime, \(p^k\mid n\) but \(p^{k+1}\) does not divide \(n\) Definition 6.4.5
\(n!\) \(n\) factorial Definition 6.4.6
\(\mathbb{Z}_n\) (ring of) integers modulo \(n\) Definition 8.1.1
\(A\setminus \{a\}\) the set of all elements in \(A\) except \(a\in A\) Example 8.3.4
\(|G|\) order of a group \(G\) Definition 8.3.8
\(|x|\) order of a group element \(x\in G\) Definition 8.3.10
\(U_n\) group of units modulo \(n\) Definition 9.1.2
\(\phi(n)\) order of the group of units of \(n\) (Euler function) Definition 9.2.1
\(\varphi(n)\) alternate notation for Euler \(\phi\) function Definition 9.2.1
\(F_n\) Fermat number \(2^{2^n}+1\) Definition 12.1.1
\(M_n\) Mersenne number \(2^n-1\) Definition 12.1.6
\(r_2(n)\) number of different ways to write \(n\) as a sum of two squares Exercise 13.7.7
\(\mathbb{Z}[i]\) Gaussian integers \(\{a+bi\mid a,b\in\mathbb{Z}\}\) Definition 14.1.2
\(\mathbb{C}\) complex numbers Definition 14.1.2
\(r_k(n)\) number of different ways to write \(n\) as a sum of \(k\) perfect squares Example 14.2.3
\(QR\) abbreviation for ‘quadratic residue’ Definition 16.3.1
\(Q_p\) group of quadratic residues of \(p\) Definition 16.4.2
\(\left(\frac{a}{p}\right)\) Legendre symbol, for \(p\) an odd prime Definition 16.6.1
\(aE\) multiples of positive even numbers less than \(p\) by \(a\) Definition 17.2.2
\(\overline{aE}\) set of nonnegative remainders of elements of \(aE\) modulo \(p\) Definition 17.2.2
\(r_{a,e}\) remainder modulo \(p\) of the element \(ae\) of \(aE\) Definition 17.2.2
\(\left(\frac{a}{n}\right)\) Jacobi symbol, \(n\) odd Definition 17.4.9
\(R\) sum \(\sum_{\text{even }e,\; 0<e<p}\left\lfloor\frac{qe}{p}\right\rfloor\) in proof of quadratic reciprocity Paragraph
\(\mu\) sum \(\sum_{f=1}^{(p-1)/2}\left\lfloor\frac{qf}{p}\right\rfloor\) in proof of quadratic reciprocity Paragraph
\(r(n)\) alternate notation for \(r_2(n)\) Definition 18.2.1
\(\sigma_k(n)\) sum of \(k\)th powers of divisors of \(n\) Definition 19.1.1
\(\tau(n)\) number of (positive) divisors of \(n\) Remark 19.1.2
\(\sigma(n)\) sum of (positive) divisors of \(n\) Remark 19.1.2
\(u(n)\) unit function Definition 19.2.9
\(N(n)\) identity function Definition 19.2.9
\(\sigma^{-1}(n)\) abundancy index of \(n\) Fact 19.4.9
\(O(g(x))\) ‘Big Oh’ notation that a function is less in absolute value than \(Cg(x)\text{,}\) for some constant \(C\) Definition 20.1.2
\(\log(n)\) natural (base \(e\)) logarithm Definition 20.3.3
\(\gamma\) Euler-Mascheroni gamma constant, limit of difference between the harmonic series and natural logarithm Definition 20.3.10
\(\Gamma\) Gamma function factorial extension Remark 20.3.11
\(\pi(x)\) prime counting function Definition 21.0.1
\(\phi(n,a)\) number of integers coprime to first \(a\) primes Definition 21.1.7
\(p_a\) the \(a\)th prime Definition 21.1.7
\(Li(x)\) logarithmic integral \(\int_2^x \frac{dt}{\log(t)}\) Definition 21.2.2
\(\Theta(x)\) Chebyshev theta function Definition 21.4.3
\(a(n)\) prime number indicator function Definition 21.4.7
\(p\#\) primorial (product of primes up to \(p\)) Definition 22.2.7
\(C_2\) twin prime constant Remark 22.3.6
\(\mu(n)\) Moebius function of \(n\) Definition 23.1.1
\(f \star g\) Dirichlet product of \(f\) and \(g\) as arithmetic functions Definition 23.2.3
\(I(n)\) Dirichlet product identity function Definition 23.3.1
\(\omega(n)\) number of unique prime divisors of \(n\) Definition 23.3.3
\(\nu(n)\) alternate notation for \(\omega(n)\) Definition 23.3.3
\(\lambda(n)\) Liouville’s function Definition 23.3.4
\(\zeta(s)\) Riemann zeta function Definition 24.2.1
\(J(x)\) auxiliary function in Riemann explicit formula Definition 25.4.2