# Fungrim entry: 5df909

$\left|\sum_{n=M}^{N} \chi\!\left(n\right)\right| \le \frac{\sqrt{q} \log\!\left(q\right)}{2 \log\!\left(2\right)} + 3 \sqrt{q}$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \,\mathbin{\operatorname{and}}\, M \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, N \in \mathbb{Z} \,\mathbin{\operatorname{and}}\, \chi \in G_{q} \,\mathbin{\operatorname{and}}\, \chi \ne \chi_{q}(1, \cdot)$
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\!\left(n\right)\right| \le \frac{\sqrt{q} \log\!\left(q\right)}{2 \log\!\left(2\right)} + 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, \cdot)
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
Sum$\sum_{n} f\!\left(n\right)$ Sum
Sqrt$\sqrt{z}$ Principal square root
Log$\log\!\left(z\right)$ 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$\chi_{q}(\ell, \cdot)$ Dirichlet character
Source code for this entry:
Entry(ID("5df909"),
Formula(LessEqual(Abs(Sum(chi(n), Tuple(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)), Unequal(chi, DirichletCharacter(q, 1)))),
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"))