<p>We deal with moments of quadratic twists of the Möbius function of the form <Equation ID="Equ16"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_639_Article_Equ16.gif" Format="GIF" Height="58" Rendition="HTML" Resolution="72" Type="Linedraw" Width="273" /> </MediaObject> <EquationSource Format="TEX">\( S_k(X,Y) = \mathop {{\mathop {\sum }\nolimits ^*}}\limits _{d} \left( \sum _{n\le Y} \left( \frac{8d}{n}\right) \mu (n)\right) ^k, \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>S</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>X</mi> <mo>,</mo> <mi>Y</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <munder> <mrow> <msup> <mo>∑</mo> <mo>∗</mo> </msup> </mrow> <mi>d</mi> </munder> <msup> <mfenced close=")" open="("> <munder> <mo>∑</mo> <mrow> <mi>n</mi> <mo>≤</mo> <mi>Y</mi> </mrow> </munder> <mfenced close=")" open="("> <mfrac> <mrow> <mn>8</mn> <mi>d</mi> </mrow> <mi>n</mi> </mfrac> </mfenced> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfenced> <mi>k</mi> </msup> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40993_2025_639_Article_IEq1.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="32" /> </InlineMediaObject> <EquationSource Format="TEX">\(\left( \frac{8d}{\cdot }\right) \)</EquationSource> <EquationSource Format="MATHML"><math> <mfenced close=")" open="("> <mfrac> <mrow> <mn>8</mn> <mi>d</mi> </mrow> <mo>·</mo> </mfrac> </mfenced> </math></EquationSource> </InlineEquation> is the Kronecker symbol and <i>d</i> runs over positive, odd and square-free integers. We give unconditional results for their asymptotic behaviors.</p>