The sum <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$S(h,k):=\sum _{j=1}^{k-1}(-1)^{j+1+[hj/k]}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mrow> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <msup> <mrow> <mo stretchy="false">(</mo> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>j</mi> <mo>+</mo> <mn>1</mn> <mo>+</mo> <mo stretchy="false">[</mo> <mi>h</mi> <mi>j</mi> <mo stretchy="false">/</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> appears in the modular transformation formulae of the classical theta function <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\vartheta _3(z)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>ϑ</mi> <mn>3</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. The double sum <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$S(k):= \sum _{h=1}^{k-1}S(h,k)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>h</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <mi>k</mi> <mo>-</mo> <mn>1</mn> </mrow> </msubsup> <mi>S</mi> <mrow> <mo stretchy="false">(</mo> <mi>h</mi> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> has a remarkable distribution of values. Although properties for S(k) and a related sum can be established, several interesting conjectures are open.

An Arithmetic Sum Associated with the Classical Theta Function

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2025年12月29日

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Bruce C. Berndt,
Raghavendra Bhat,
Jeffrey L. Meyer,
Likun Xie,
Alexandru Zaharescu

摘要

The sum $S(h,k):=\sum _{j=1}^{k-1}(-1)^{j+1+[hj/k]}$ $S (h, k) : = \sum_{j = 1}^{k - 1} {(- 1)}^{j + 1 + [h j / k]}$ appears in the modular transformation formulae of the classical theta function $\vartheta _3(z)$ $ϑ_{3} (z)$ . The double sum $S(k):= \sum _{h=1}^{k-1}S(h,k)$ $S (k) : = \sum_{h = 1}^{k - 1} S (h, k)$ has a remarkable distribution of values. Although properties for S(k) and a related sum can be established, several interesting conjectures are open.