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Fungrim entry: 5df909

n=MNχ ⁣(n)qlog ⁣(q)2log ⁣(2)+3q\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:qZ1andMZandNZandχGqandχχq(1,)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)
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\!\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
Absz\left|z\right| Absolute value
Sumnf ⁣(n)\sum_{n} f\!\left(n\right) Sum
Sqrtz\sqrt{z} Principal square root
Loglog ⁣(z)\log\!\left(z\right) Natural logarithm
ZZGreaterEqualZn\mathbb{Z}_{\ge n} Integers greater than or equal to n
ZZZ\mathbb{Z} Integers
DirichletGroupGqG_{q} Dirichlet characters with given modulus
DirichletCharacterχq(,)\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)))),
    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"))

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2019-08-17 11:32:46.829430 UTC