Higher Hickerson formula

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• 作者：Jungyun Lee^a ; ¹ ; ^{lee9311@ewha.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Byungheup Jun^b ; ² ; ^{bhjun@unist.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Hi-joon Chae^c ; ³ ; ^{hchae@hongik.ac.kr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Generalized Dedekind sums ; Siegel's formula ; Meyer's formula ; Partial zeta function ; Real quadratic fields
• 刊名：Journal of Number Theory
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：170
• 期：Complete
• 页码：191-210
• 全文大小：385 K

文摘

In [11], Hickerson made an explicit formula for Dedekind sums class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si1.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=a793f7facdb680f1d93c110e62574a92" title="Click to view the MathML source">s(p,q)class="mathContainer hidden">class="mathCode">$s(p,q)$ in terms of the continued fraction of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si2.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=c754417fa40160ef3561c82ad5e61478" title="Click to view the MathML source">p/qclass="mathContainer hidden">class="mathCode">$p/q$. We develop analogous formula for generalized Dedekind sums class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si3.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=50921a92e992d9bcb46d77112310dc01" title="Click to view the MathML source">s_i,j(p,q)class="mathContainer hidden">class="mathCode">$s_{i,j}(p,q)$ defined in association with the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si4.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=6d8bba80a3ff6f6fa5cbf0fd6f7131e9" title="Click to view the MathML source">xⁱy^jclass="mathContainer hidden">class="mathCode">$x^i{y}^j$-coefficient of the Todd power series of the lattice cone in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si5.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=a6e0cd47c5e9badb8a166515fc840d6b" title="Click to view the MathML source">R²class="mathContainer hidden">class="mathCode">${\mathbb{R}}^2$ generated by class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si6.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=92f11dee46d081ca88d5b5d14cd7c151" title="Click to view the MathML source">(1,0)class="mathContainer hidden">class="mathCode">$(1,0)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si7.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=3481483978028da6fd62d31f9d298748" title="Click to view the MathML source">(p,q)class="mathContainer hidden">class="mathCode">$(p,q)$. The formula generalizes Hickerson's original one and reduces to Hickerson's for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si8.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=956405bad31eb5d3a361ec5075979e01" title="Click to view the MathML source">i=j=1class="mathContainer hidden">class="mathCode">$i=j=1$. In the formula, generalized Dedekind sums are divided into two parts: the integral class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si9.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=099c6f0e102a6cedc9ffbf32a46641bb">class="imgLazyJSB inlineImage" height="21" width="55" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16301548-si9.gif">class="mathContainer hidden">class="mathCode">$s_{ij}^I(p,q)$ and the fractional class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si10.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=d90ac1223f8bb67d181b25c20727c386">class="imgLazyJSB inlineImage" height="21" width="55" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16301548-si10.gif">class="mathContainer hidden">class="mathCode">$s_{ij}^R(p,q)$. We apply the formula to Siegel's formula for partial zeta values at a negative integer and obtain a new expression which involves only class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si9.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=099c6f0e102a6cedc9ffbf32a46641bb">class="imgLazyJSB inlineImage" height="21" width="55" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16301548-si9.gif">class="mathContainer hidden">class="mathCode">$s_{ij}^I(p,q)$ the integral part of generalized Dedekind sums. This formula directly generalizes Meyer's formula for the special value at 0. Using our formula, we present the table of the partial zeta value at class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si11.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=689ff79aa1b80b4bb5d1d0ad78b952b6" title="Click to view the MathML source">s=−1class="mathContainer hidden">class="mathCode">$s=-1$ and −2 in more explicit form. Finally, we present another application on the equidistribution property of the fractional parts of the graph class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si12.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=02df000922d3a943c32a4fa5535f840f">class="imgLazyJSB inlineImage" height="29" width="167" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022314X16301548-si12.gif">class="mathContainer hidden">class="mathCode"> for a certain integer class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si13.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=285c25c2919bc817ce00b24b2077146a" title="Click to view the MathML source">R_i+jclass="mathContainer hidden">class="mathCode">$R_{i+j}$ depending on class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si14.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=feebb1991c9eb9545515f1bc4b33537f" title="Click to view the MathML source">i+jclass="mathContainer hidden">class="mathCode">$i+j$.