For a given integer <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$ m $</EquationSource> <EquationSource Format="MATHML"><math> <mi>m</mi> </math></EquationSource> </InlineEquation> and any residue <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$ a \ (\textrm{mod}\,m) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mspace width="4pt" /> <mo stretchy="false">(</mo> <mtext>mod</mtext> <mspace width="0.166667em" /> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> that can be written as a sum of 3 squares modulo m, we show the existence of infinitely many integers <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$ n \equiv a \ (\textrm{mod}\,m) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≡</mo> <mi>a</mi> <mspace width="4pt" /> <mo stretchy="false">(</mo> <mtext>mod</mtext> <mspace width="0.166667em" /> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such that the number of representations of n as a sum of three squares, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$ r_3(n) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>r</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, satisfies <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$r_3(n) \gg _m \sqrt{n} \log \log n $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>r</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mo>≫</mo> <mi>m</mi> </msub> <msqrt> <mi>n</mi> </msqrt> <mo>log</mo> <mo>log</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>. Consequently, we establish that there are infinitely many integers <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$ n \equiv a \ (\textrm{mod}\,m) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≡</mo> <mi>a</mi> <mspace width="4pt" /> <mo stretchy="false">(</mo> <mtext>mod</mtext> <mspace width="0.166667em" /> <mi>m</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> for which the Hurwitz class number <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$ H(n) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> also satisfies <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$ H(n) \gg _m \sqrt{n} \log \log n $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>H</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> <msub> <mo>≫</mo> <mi>m</mi> </msub> <msqrt> <mi>n</mi> </msqrt> <mo>log</mo> <mo>log</mo> <mi>n</mi> </mrow> </math></EquationSource> </InlineEquation>.

On extreme values of $r_3(n)$ in arithmetic progressions

期刊论文
发表日期：

2026年1月27日

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Michael Filaseta,
Jonah Klein,
Cihan Sabuncu

摘要

For a given integer $ m $ $m$ and any residue $ a \ (\textrm{mod}\,m) $ $a (mod m)$ that can be written as a sum of 3 squares modulo m, we show the existence of infinitely many integers $ n \equiv a \ (\textrm{mod}\,m) $ $n \equiv a (mod m)$ such that the number of representations of n as a sum of three squares, $ r_3(n) $ $r_{3} (n)$ , satisfies $r_3(n) \gg _m \sqrt{n} \log \log n $ $r_{3} (n) ≫_{m} \sqrt{n} log log n$ . Consequently, we establish that there are infinitely many integers $ n \equiv a \ (\textrm{mod}\,m) $ $n \equiv a (mod m)$ for which the Hurwitz class number $ H(n) $ $H (n)$ also satisfies $ H(n) \gg _m \sqrt{n} \log \log n $ $H (n) ≫_{m} \sqrt{n} log log n$ .

On extreme values of \(r_3(n)\) in arithmetic progressions

摘要