Fungrim entry: 4911bd

\left|\frac{1}{L\!\left(s, \chi\right)} - \prod_{p < N} \left(1 - \frac{\chi(p)}{{p}^{s}}\right)\right| \le \zeta\!\left(\operatorname{Re}(s), N\right)

Assumptions:

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

TeX:

\left|\frac{1}{L\!\left(s, \chi\right)} - \prod_{p < N} \left(1 - \frac{\chi(p)}{{p}^{s}}\right)\right| \le \zeta\!\left(\operatorname{Re}(s), N\right)

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; s \in \mathbb{C} \;\mathbin{\operatorname{and}}\; \operatorname{Re}(s) > 1 \;\mathbin{\operatorname{and}}\; N \in \mathbb{Z}_{\ge 1}

Definitions:

Fungrim symbol	Notation	Short description
`Abs`	$\left\|z\right\|$	Absolute value
`DirichletL`	$L\!\left(s, \chi\right)$	Dirichlet L-function
`PrimeProduct`	$\prod_{p} f(p)$	Product over primes
`Pow`	${a}^{b}$	Power
`HurwitzZeta`	$\zeta\!\left(s, a\right)$	Hurwitz zeta function
`Re`	$\operatorname{Re}(z)$	Real part
`ZZGreaterEqual`	$\mathbb{Z}_{\ge n}$	Integers greater than or equal to n
`DirichletGroup`	$G_{q}$	Dirichlet characters with given modulus
`CC`	$\mathbb{C}$	Complex numbers

Source code for this entry:

Entry(ID("4911bd"),
    Formula(LessEqual(Abs(Sub(Div(1, DirichletL(s, chi)), PrimeProduct(Parentheses(Sub(1, Div(chi(p), Pow(p, s)))), For(p), Less(p, N)))), HurwitzZeta(Re(s), N))),
    Variables(q, chi, s, N),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(s, CC), Greater(Re(s), 1), Element(N, ZZGreaterEqual(1)))))

Topics using this entry

Dirichlet L-functions