We’ll 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, unless otherwise specified.
| Symbol | Meaning | Implicit restrictions | Reference post |
|---|---|---|---|
| p, q | Prime numbers (by convention) | The definition of prime number | |
| p_i | Usually it indicates i-th prime number: p_1 = 2, p_2 = 3, p_3 = 5, etcetera; otherwise, the meaning is the same as q_i (see next row). | 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 |
| n, m, x, d | Integer numbers (by convention) | ||
| t, u | Real numbers (by convention) | ||
| \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 |
| \Lambda^{\star}(x) | \begin{cases} p & \begin{aligned}\text{if $x = p^m$ for some prime $p$} \\ \text{and some positive integer $m$}\end{aligned} \\ 1 & \text{otherwise}\end{cases} | x > 0 | 13. The factorial function and the \Lambda^{\star} function, Definition N.10 |
| \Lambda(x) | \begin{cases} \log p & \begin{aligned}\text{if $x = p^m$ for some prime $p$} \\ \text{and some positive integer $m$}\end{aligned} \\ 0 & \text{otherwise}\end{cases} | x > 0 | 15. Chebyshev’s Theorem: strong version, Definition N.11 |
| \mathrm{Li}(x) | \int_2^n \frac{1}{\log x} dx | x > 0 | 16. The prime number Theorem, Definition N.12 |
| W(t) | \frac{\overline{\psi}(t) - t}{t} | t \geq 1 | 17. The functions W and V, Definition N.13 |
| V(u) | \frac{\overline{\psi}(e^u) - e^u}{e^u} | u \geq 0 | 17. Functions W and V, Definition N.13 |
| \alpha | \limsup_{x \to +\infty} |V(\log x)| | 18. The integral mean value and the absolute error function R, Definition N.14 | |
| \beta | \limsup_{x \to +\infty} \frac{1}{\log x} \int_0^{\log x} |V(u)| du | 18. The integral mean value and the absolute error function R, Definition N.14 | |
| R(t) | \overline{\psi}(t) - t | t \geq 1 | 18. The integral mean value and the absolute error function R, Definition N.15 |
| P_n | Set of the square-free divisors of n which are the product of an even number of prime factors | n \gt 0 | The Möbius function and its connection with the function Λ, Definition N.17 |
| D_n | Set of the square-free divisors of n which are the product of an odd number of prime factors | n \gt 0 | The Möbius function and its connection with the function Λ, Definition N.17 |
| Q_n | Set of the divisors of n which are not square-free | n \gt 0 | The Möbius function and its connection with the function Λ, Definition N.17 |
| \mu(d) | \begin{cases} 1 & \begin{aligned} & \text{if $d$ is square-free} \\ & \text{and is the product of an even} \\ & \text{number of prime factors} \end{aligned} \\ \\ -1 & \begin{aligned} & \text{if $d$ is square-free} \\ & \text{and is the product of an odd} \\ & \text{number of prime factors} \end{aligned} \\ \\ 0 & \text{if $d$ is not square-free} \end{cases} | d \gt 0 | The Möbius function and its connection with the function Λ, Definition N.18 |
| \sum_{d \mid n} | \sum_{d \in \{\text{divisors of }n\}} | n \gt 0 | Some important summations, Definition N.19 |
| \prod_{d \mid n} | \prod_{d \in \{\text{divisors of }n\}} | n \gt 0 | Some important summations, Definition N.19 |
| \sum_{ab = n} | \sum_{(a,b) \in \{ (a,b) \in \mathbb{N}^{\star} \times \mathbb{N}^{\star} \mid ab = n\}} | n \gt 0 | Some important summations, Definition N.20 |
| \prod_{ab = n} | \prod_{(a,b) \in \{ (a,b) \in \mathbb{N}^{\star} \times \mathbb{N}^{\star} \mid ab = n\}} | n \gt 0 | Some important summations, Definition N.20 |
| \sum_{ab \mid n} | \sum_{(a,b) \in \{(a,b) \in \mathbb{N}^{\star} \times \mathbb{N}^{\star} \ \mid\ ab \mid n\} } | n \gt 0 | Some important summations, Definition N.21 |
| \prod_{ab \mid n} | \prod_{((a,b) \in \{(a,b) \in \mathbb{N}^{\star} \times \mathbb{N}^{\star} \ \mid\ ab \mid n\} } | n \gt 0 | Some important summations, Definition N.21 |
| \sum_{n \leq x} | \sum_{n = 1}^x | n \gt 0 | Some important summations, Definition N.22 |
| \prod_{n \leq x} | \prod_{n = 1}^x | n \gt 0 | Some important summations, Definizione N.22 |
| \sum_{ab \leq n} | \sum_{(a,b) \in \{ (a,b) \in \mathbb{N}^{\star} \times \mathbb{N}^{\star} \mid ab \leq n\}} | n \gt 0 | Some important summations, Definition N.23 |
| d(n) | number of positive divisors of n | n \gt 0 | Arithmetic functions: definitions and some properties, Definition N.24 |
| \omega(n) | number of distinct prime factors of n | n \gt 0 | Arithmetic functions: definitions and some properties, Definition N.25 |
| \phi(n) | number of positive integers coprime with n and less than it | n \gt 0 | Arithmetic functions: definitions and some properties, Definition N.26 |
We’ll list in the following table the definitions explicitly introduced in our posts about number theory.
| Definition | Statement |
|---|---|
| N.1: Prime number | 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.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“. |
| N.10: Function \Lambda^{\star} | \Lambda^{\star}(x) := \begin{cases} p & \text{if $x = p^m$ for some prime $p$ and some positive integer $m$} \\ 1 & \text{otherwise}\end{cases} |
| N.11: Function \Lambda | \Lambda(x) := \log \Lambda^{\star}(x) = \begin{cases} \log p & \text{if $x = p^m$ for some prime $p$ and some positive integer $m$} \\ 0 & \text{otherwise}\end{cases} |
| N.12: Function \mathrm{Li} (logarithmic integral) |
We define the function \mathrm{Li}: \mathbb{N}^{\star} \rightarrow \mathbb{R}, called logarithmic integral, such that for all x \in \mathbb{N}^{\star}: \mathrm{Li}(x) := \int_2^n \frac{1}{\log x} dx
|
| N.13: Functions W and V |
We define the following functions W: [1, +\infty) \rightarrow \mathbb{R} and V: \mathbb{R}_{+} \rightarrow \mathbb{R}: \begin{aligned} W(t) := & \frac{\overline{\psi}(t) - t}{t} \\ V := & W \circ \mathrm{exp} \end{aligned}
|
| N.14: Constants \alpha and \beta |
Given the integer variable x \gt 0, we define the following constants: \alpha := \limsup_{x \to +\infty} |V(\log x)|
\beta := \limsup_{x \to +\infty} \frac{1}{\log x} \int_0^{\log x} |V(u)| du
|
| N.15: Function R |
We define the following function R: [1, +\infty) \rightarrow \mathbb{R}: R(t) := \overline{\psi}(t) - t
for all t \in [1, +\infty). |
| N.16: Square-free integer | An integer number is called square-free if it’s not divisible by any square number greater than 1. |
| N.17: Sets of divisors of a positive integer number |
Let n \in \mathbb{N}^{\star}. The following sets are defined:
|
| N.18: Möbius function |
We define the function \mu: \mathbb{N}^{\star} \rightarrow \{-1, 1, 0\} such that: \mu(d) := \begin{cases} 1 & \begin{aligned} & \text{if $d$ is square-free} \\ & \text{and is the product of an even} \\ & \text{number of prime factors} \end{aligned} \\ \\ -1 & \begin{aligned} & \text{if $d$ is square-free} \\ & \text{and is the product of an odd} \\ & \text{number of prime factors} \end{aligned} \\ \\ 0 & \text{if $d$ is not square-free} \end{cases}
for all d \in \mathbb{N}^{\star}. The function \mu is called Möbius function. |
| N.19: Summations and productions extended to the divisors of a positive integer number |
Let n \in \mathbb{N}^{\star}. The following symbols are defined:
where by “divisors” we mean, as usual, the positive divisors. |
| N.20: Summations and productions extended to couples of variables with constant product |
Let n be a positive integer. We define the following symbols:
|
| N.21: Summations and productions extended to couples of variables the product of which divides a constant |
Let n be a positive integer. We define the following symbols:
|
| N.22: Summations and productions starting from 1 |
Let x be a positive integer. We define the following symbols:
|
| N.23: Constants \alpha^{\prime} and \beta^{\prime} |
Given the variable \xi which can assume values inside the real interval [1, +\infty), we define the following constants: \alpha^{\prime} := \limsup_{\xi \to +\infty} |V(\log \xi)|
\beta^{\prime} := \limsup_{\xi \to +\infty} \frac{1}{\log \xi} \int_0^{\log \xi} |V(u)|\ du
|
| N.24: d function |
We define the function d: \mathbb{N}^{\star} \rightarrow \mathbb{N}^{\star} such that: d(n) := \textrm{number of positive divisors of $n$}
|
| N.25: \omega function |
We define the function \omega: \mathbb{N}^{\star} \rightarrow \mathbb{N}^{\star} such that: \omega(n) := \textrm{number of distinct prime factors of $n$}
|
| N.26: Euler’s \phi function |
We define the function \phi: \mathbb{N}^{\star} \rightarrow \mathbb{N}^{\star} such that: \phi(n) := \textrm{number of positive integers coprime to $n$ and less than it}
|
| N.27: Multiplicative function |
A function f: \mathbb{N}^{\star} \rightarrow \mathbb{R} is defined as multiplicative when it satisfies the following conditions at the same time:
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| N.28: Completely multiplicative function |
A function f: \mathbb{N}^{\star} \rightarrow \mathbb{R} is defined to be completely multiplicative when it satisfies the following conditions:
|