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 80f1d93c110e62574a92" title="Click to view the MathML source">s(p,q)$s(p,q)$ in terms of the continued fraction of p/q$p/q$. We develop analogous formula for generalized Dedekind sums s_i,j(p,q)$s_{i,j}(p,q)$ defined in association with the 80a3ff6f6fa5cbf0fd6f7131e9" 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 (1,0)$(1,0)$ and 8028da6fd62d31f9d298748" 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 $s_{ij}^I(p,q)$ 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 $s_{ij}^I(p,q)$ 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 80b4bb5d1d0ad78b952b6" 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 $(\frac{p}{q},R_{i+j}{q}^{i+j-2}{s}_{ij}(p,q))$ for a certain integer 85c25c2919bc817ce00b24b2077146a" title="Click to view the MathML source">R_i+j$R_{i+j}$ depending on i+j$i+j$.