Fungrim entry: e6deb7

\left|\sum_{n=0}^{N} \chi(n)\right| \le \varphi(q)

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; \chi \ne \chi_{q \, . \, 1}

TeX:

\left|\sum_{n=0}^{N} \chi(n)\right| \le \varphi(q)

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; \chi \ne \chi_{q \, . \, 1}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`Sum`	$\sum_{n} f(n)$	Sum
`Totient`	$\varphi(n)$	Euler totient function
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`ZZ`	$\mathbb{Z}$	Integers
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`DirichletCharacter`	$\chi_{q \, . \, \ell}$	Dirichlet character

Source code for this entry:

Entry(ID("e6deb7"),
    Formula(LessEqual(Abs(Sum(chi(n), For(n, 0, N))), Totient(q))),
    Variables(N, chi, q),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(N, ZZ), Element(chi, DirichletGroup(q)), NotEqual(chi, DirichletCharacter(q, 1)))))

Topics using this entry

Dirichlet characters