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 |
