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 a92e992d9bcb46d77112310dc01" title="Click to view the MathML source">s_i,j(p,q)$s_{i,j}(p,q)$ defined in association with the ba80a3ff6f6fa5cbf0fd6f7131e9" title="Click to view the MathML source">xⁱy^j$x^i{y}^j$-coefficient of the Todd power series of the lattice cone in badb8a166515fc840d6b" title="Click to view the MathML source">R²${\mathbb{R}}^2$ generated by (1,0)$(1,0)$ and (p,q)$(p,q)$. The formula generalizes Hickerson's original one and reduces to Hickerson's for bad31eb5d3a361ec5075979e01" 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 $s_{ij}^I(p,q)$ and the fractional 90ac1223f8bb67d181b25c20727c386">$s_{ij}^R(p,q)$. 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 b80b4bb5d1d0ad78b952b6" 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 a943c32a4fa5535f840f">$(\frac{p}{q},R_{i+j}{q}^{i+j-2}{s}_{ij}(p,q))$ for a certain integer R_i+j$R_{i+j}$ depending on i+j$i+j$.