# Number theory definitions and symbols

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.

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
$n, m, x, d$ 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

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:

• P_n := \left\{ \begin{aligned} & \text{square-free divisors of n} \\ & \text{which are the product of an even number of prime factors} \end{aligned} \right\}
• D_n := \left\{ \begin{aligned} & \text{square-free divisors of n} \\ & \text{which are the product of an odd number of prime factors} \end{aligned} \right\}
• $Q_n := \left\{ \text{divisors of n which are not square-free} \right\}$
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:

• $\sum_{d \mid n} := \sum_{d \in \{\text{divisors of }n\}}$
• $\prod_{d \mid n} := \prod_{d \in \{\text{divisors of }n\}}$

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:

• $\sum_{ab = n} := \sum_{(a,b) \in \{ (a,b) \in \mathbb{N}^{\star} \times \mathbb{N}^{\star} \mid ab = n\}}$
• $\prod_{ab = n} := \prod_{(a,b) \in \{ (a,b) \in \mathbb{N}^{\star} \times \mathbb{N}^{\star} \mid ab = n\} }$
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:

• $\sum_{ab \mid n} := \sum_{(a,b) \in \{(a,b) \in \mathbb{N}^{\star} \times \mathbb{N}^{\star} \ \mid\ ab \mid n\} }$
• $\prod_{ab \mid n} := \prod_{((a,b) \in \{(a,b) \in \mathbb{N}^{\star} \times \mathbb{N}^{\star} \ \mid\ ab \mid n\} }$
N.22: Summations and productions starting from 1

Let $x$ be a positive integer. We define the following symbols:

• $\sum_{n \leq x} := \sum_{n = 1}^x$
• $\prod_{n \leq x} := \prod_{n = 1}^x$
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:

$$\alpha^{\prime} := \limsup_{\xi \to +\infty} |V(\log \xi)|$$
$$\beta^{\prime} := \limsup_{\xi \to +\infty} \frac{1}{\log \xi} \int_0^{\log \xi} |V(u)|\ du$$