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 e62574a92" title="Click to view the MathML source">s(p,q)$s(p,q)$ in terms of the continued fraction of e61478" title="Click to view the MathML source">p/q$p/q$. We develop analogous formula for generalized Dedekind sums b46d77112310dc01" title="Click to view the MathML source">s_i,j(p,q)$s_{i,j}(p,q)$ defined in association with the a80a3ff6f6fa5cbf0fd6f7131e9" title="Click to view the MathML source">xⁱy^j$x^i{y}^j$-coefficient of the Todd power series of the lattice cone in R²${\mathbb{R}}^2$ generated by e46d081ca88d5b5d14cd7c151" title="Click to view the MathML source">(1,0)$(1,0)$ and 83978028da6fd62d31f9d298748" title="Click to view the MathML source">(p,q)$(p,q)$. The formula generalizes Hickerson's original one and reduces to Hickerson's for i=j=1$i=j=1$. In the formula, generalized Dedekind sums are divided into two parts: the integral 9ffbf32a46641bb">$s_{ij}^I(p,q)$ and the fractional $s_{ij}^R(p,q)$. We apply the formula to Siegel's formula for partial zeta values at a negative integer and obtain a new expression which involves only 9ffbf32a46641bb">$s_{ij}^I(p,q)$ 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 9ff79aa1b80b4bb5d1d0ad78b952b6" 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 a5535f840f">$(\frac{p}{q},R_{i+j}{q}^{i+j-2}{s}_{ij}(p,q))$ for a certain integer R_i+j$R_{i+j}$ depending on i+j$i+j$.