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 314X16301548&_mathId=si1.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=a793f7facdb680f1d93c110e62574a92" title="Click to view the MathML source">s(p,q)$s(p,q)$ in terms of the continued fraction of 314X16301548&_mathId=si2.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=c754417fa40160ef3561c82ad5e61478" title="Click to view the MathML source">p/q$p/q$. We develop analogous formula for generalized Dedekind sums i3" class="mathmlsrc">314X16301548&_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) defined in association with the 314X16301548&_mathId=si4.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=6d8bba80a3ff6f6fa5cbf0fd6f7131e9" title="Click to view the MathML source">xⁱy^j$x^i{y}^j$-coefficient of the Todd power series of the lattice cone in 314X16301548&_mathId=si5.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=a6e0cd47c5e9badb8a166515fc840d6b" title="Click to view the MathML source">R²${\mathbb{R}}^2$ generated by 314X16301548&_mathId=si6.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=92f11dee46d081ca88d5b5d14cd7c151" title="Click to view the MathML source">(1,0)$(1,0)$ and 314X16301548&_mathId=si7.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=3481483978028da6fd62d31f9d298748" title="Click to view the MathML source">(p,q)$(p,q)$. The formula generalizes Hickerson's original one and reduces to Hickerson's for 314X16301548&_mathId=si8.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=956405bad31eb5d3a361ec5075979e01" title="Click to view the MathML source">i=j=1$i=j=1$. In the formula, generalized Dedekind sums are divided into two parts: the integral 314X16301548&_mathId=si9.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=099c6f0e102a6cedc9ffbf32a46641bb">314X16301548-si9.gif">$s_{ij}^I(p,q)$ and the fractional 314X16301548&_mathId=si10.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=d90ac1223f8bb67d181b25c20727c386">314X16301548-si10.gif">$s_{ij}^R(p,q)$. We apply the formula to Siegel's formula for partial zeta values at a negative integer and obtain a new expression which involves only 314X16301548&_mathId=si9.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=099c6f0e102a6cedc9ffbf32a46641bb">314X16301548-si9.gif">$s_{ij}^I(p,q)$ 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 314X16301548&_mathId=si11.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=689ff79aa1b80b4bb5d1d0ad78b952b6" title="Click to view the MathML source">s=−1$s=-1$ and −2 in more explicit form. Finally, we present another application on the equidistribution property of the fractional parts of the graph 314X16301548&_mathId=si12.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=02df000922d3a943c32a4fa5535f840f">314X16301548-si12.gif">$(\frac{p}{q},R_{i+j}{q}^{i+j-2}{s}_{ij}(p,q))$ for a certain integer 314X16301548&_mathId=si13.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=285c25c2919bc817ce00b24b2077146a" title="Click to view the MathML source">R_i+j$R_{i+j}$ depending on 314X16301548&_mathId=si14.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=feebb1991c9eb9545515f1bc4b33537f" title="Click to view the MathML source">i+j$i+j$.