Fungrim entry: 5df909

\left|\sum_{n=M}^{N} \chi(n)\right| \le \frac{\sqrt{q} \log(q)}{2 \log(2)} + 3 \sqrt{q}

Assumptions:

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

Pólya-Vinogradov inequality, explicit form

References:

E. Dobrowolski and K. S. Williams, An upper bound for the sum ... for a certain class of functions f, Proceedings of the American Mathematical Society, Vol. 114, No. 1 (Jan., 1992), pp. 29-35, http://doi.org/10.2307/2159779

TeX:

\left|\sum_{n=M}^{N} \chi(n)\right| \le \frac{\sqrt{q} \log(q)}{2 \log(2)} + 3 \sqrt{q}

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; M \in \mathbb{Z} \;\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
`Sqrt`	$\sqrt{z}$	Principal square root
`Log`	$\log(z)$	Natural logarithm
`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("5df909"),
    Formula(LessEqual(Abs(Sum(chi(n), For(n, M, N))), Add(Div(Mul(Sqrt(q), Log(q)), Mul(2, Log(2))), Mul(3, Sqrt(q))))),
    Variables(q, chi, M, N),
    Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(M, ZZ), Element(N, ZZ), Element(chi, DirichletGroup(q)), NotEqual(chi, DirichletCharacter(q, 1)))),
    Description("Pólya-Vinogradov inequality, explicit form"),
    References("E. Dobrowolski and K. S. Williams, An upper bound for the sum ... for a certain class of functions f, Proceedings of the American Mathematical Society, Vol. 114, No. 1 (Jan., 1992), pp. 29-35, http://doi.org/10.2307/2159779"))

Topics using this entry

Dirichlet characters