Estimates for character sums

Abstract

We give a number of estimates for character sums <disp-formula content-type="math/mathml"> \[ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="sigma-summation Underscript a element-of script upper A Endscripts sigma-summation Underscript b element-of script upper B Endscripts chi left-parenthesis a plus b right-parenthesis"> <mml:semantics> <mml:mrow> <mml:munder> <mml:mo movablelimits="false">∑<!-- ∑ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>a</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> </mml:mrow> </mml:munder> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:munder> <mml:mo movablelimits="false">∑<!-- ∑ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>b</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> </mml:mrow> </mml:munder> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>χ<!-- χ --></mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>a</mml:mi> <mml:mo>+</mml:mo> <mml:mi>b</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\sum \limits _{a \in \mathcal {A}} {\sum \limits _{b \in \mathcal {B}} {\chi (a + b)} }</mml:annotation> </mml:semantics> </mml:math> \] </disp-formula> for rather general sets <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="script upper A comma script upper B"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">A</mml:mi> </mml:mrow> <mml:mo>,</mml:mo> <mml:mspace width="thickmathspace" /> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi class="MJX-tex-caligraphic" mathvariant="script">B</mml:mi> </mml:mrow> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathcal {A},\;\mathcal {B}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. These give, in particular, a modified proof of the inequalities of Pólya-Vinogradov and of Burgess, which displays the latter as a generalization of the former.

Locations

• Proceedings of the American Mathematical Society - View - PDF