We list in the following table the various symbols that are used in our posts about number theory. We assume all variables to represent integer numbers.

Symbol | Meaning | Implicit restrictions | Reference post |
---|---|---|---|

p, q | Prime numbers (by convention) | The definition of prime number | |

p_i | i-th prime number: p_1 = 2, p_2 = 3, p_3 = 5, etcetera | i \geq 1 | The definition of prime number |

q_i | Number taken from a list of prime numbers not necessarily sorted or complete. For example if the list is 5, 17, 3, then q_1 := 5, q_2 := 17 and q_3 := 3. | i \geq 1 | The definition of prime number |

\binom{n}{k} | Binomial “n on k“ | n > 0, 0 \leq k \leq n | Binomial coefficient (Wikipedia) |

b \mid a, b \nmid a | “b divides a“, “b doesn’t divide a“ | b \neq 0 | The definition of prime number |

\left \lfloor \frac{a}{b} \right \rfloor | Integer part of \frac{a}{b}, rounded down | b \neq 0 | |

\left \lceil \frac{a}{b} \right \rceil | Integer part of \frac{a}{b}, rounded up | b \neq 0 | |

a = o(b) | a is a “small oh” of b | a and b are two functions defined over integer or real numbers, with real values | Elements of asymptotic analysis: Definition A.1, Definition A.3, Definition A.4 |

a = O(b) | a is a “big Oh” of b | a and b are two functions defined over integer or real numbers, with real values | Elements of asymptotic analysis: Definition A.1, Definition A.3, Definition A.4 |

a \asymp b | a and b have the same order of magnitude | a and b are two functions defined over integer or real numbers, with real values | Elements of asymptotic analysis: Definition A.1, Definition A.3, Definition A.4 |

a \sim b | a and b are asymptotically equivalent | a and b are two functions defined over integer or real numbers, with real values | Elements of asymptotic analysis: Definition A.2, Definition A.3, Definition A.4 |

\theta^{\star}(x) | \prod_{p \leq x} p | x > 0 | The product of the first prime numbers: an overestimation, Definition N.4 |

\psi^{\star}(x) | \mathrm{LCM}(1, \dots, x) | x > 0 | The least common multiple of the first positive integers, Definition N.5 |

\pi(x) | Number of primes less than or equal to x | x > 0 | The least common multiple of the first positive integers, Definition N.6 |

\theta(x) | \log \theta^{\star}(x) | x > 0 | Chebyshev’s Theorem, Definition N.7 |

\psi(x) | \log \psi^{\star}(x) | x > 0 | Chebyshev’s Theorem, Definition N.7 |

\overline{f}(x) | Simple extension of f | f: I \rightarrow \mathbb{R} is a function defined on the set I \subseteq \mathbb{Z}. As a consequence, the function \overline{f} is defined on the set \overline{I} := \bigcup_{n \in I} [n, n+1). | From integer numbers to real numbers, Definition N.8 |

\widetilde{f}(x) | Extension of f | f: I \rightarrow \mathbb{R} is a function defined on the set I \subseteq \mathbb{Z}. The function \widetilde{f} is defined on \overline{I} := \bigcup_{n \in I} [n, n+1). | From integer numbers to real numbers, Definition N.9 |

We list in the following table the definitions explicitly introduced in our posts about number theory.

Definition | Statement |
---|---|

N.1: Prime number, first definition | A prime number is an integer number greater than 1, which is divisible obly by itself and 1.. |

N.2: Divisibility | An integer a is divisible by an integer b, with b \neq 0, if a = b c for some integer c. If a is divisible by b we write b \mid a (“b divides a“), otherwise we write b \nmid a (“b does not divide a“). |

N.3: Prime number, second definition | A prime number is an integer number p > 1 such that, if p \mid bc, then p \mid b or p \mid c, for any integers b and c. In other terms, p cannot divide a product of integers bc without dividing at least one of the two factors. |

N.4: The product of the first prime numbers | The following function is defined: \theta^{\star}(x) := \prod_{p \leq x} p where x is a positive integer number. |

N.5: Least common multiples of the positive integers up to x | We define the function \psi^{\star}(x) := \mathrm{LCM}(1, \dots, x), where x > 0 is an integer. |

N.6: Number of primes less than or equal to x | We define the function \pi(x) := |\{\textrm{prime numbers} \leq x\}|, where x is a positive integer. |

N.7: Logarithmic functions \theta(x) and \psi(x) | We define the functions \theta(x) := \log \theta^{\star} and \psi(x) := \log \psi^{\star}(x), where x is a positive integer. |

N.8: Simple extension of a function defined on integer numbers, to real numbers | Let f: I \rightarrow \mathbb{R} a function defined on a set I \subseteq \mathbb{Z}. On the set \overline{I} := \bigcup_{n \in I} [n, n+1), the function \overline{f}: \overline{I} \rightarrow \mathbb{R} is defined as: \forall n \in I\ \forall x \in [n, n+1): \overline{f}(x) := f(n). We’ll call the function \overline{f} “simple extension of f to real numbers”, or simply “simple extension of f“. |

N.9: Extension of a function defined on integer numbers, to real numbers | Let f: I \rightarrow \mathbb{R} be a function defined on the set I \subseteq \mathbb{Z}. Let \overline{I} := \bigcup_{n \in I} [n, n+1) and \widetilde{f}: \overline{I} \rightarrow \mathbb{R} a function such that \widetilde{f}_{\mid I} = f. We’ll say this function \widetilde{f} is an “extension of f to real numbers”, or simply an “extension of f“. |