In <
span id="bbr0110">[11]
span>, Hicker
son made an explicit formula for Dedekind
sum
s <
span id="mml
si1" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022314X16301548&_mathId=
si1.gif&_u
ser=111111111&_pii=S0022314X16301548&_rdoc=1&_i
ssn=0022314X&md5=a793f7facdb680f1d93c110e62574a92" title="Click to vie
w the MathML
source">
s(p,q)
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> in term
s of the continued fraction of <
span id="mml
si2" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022314X16301548&_mathId=
si2.gif&_u
ser=111111111&_pii=S0022314X16301548&_rdoc=1&_i
ssn=0022314X&md5=c754417fa40160ef3561c82ad5e61478" title="Click to vie
w the MathML
source">p/q
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>. We develop analogou
s formula for generalized Dedekind
sum
s <
span id="mml
si3" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022314X16301548&_mathId=
si3.gif&_u
ser=111111111&_pii=S0022314X16301548&_rdoc=1&_i
ssn=0022314X&md5=50921a92e992d9bcb46d77112310dc01" title="Click to vie
w the MathML
source">
s<
sub>i,j
sub>(p,q)
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> defined in a
ssociation
with the <
span id="mml
si4" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022314X16301548&_mathId=
si4.gif&_u
ser=111111111&_pii=S0022314X16301548&_rdoc=1&_i
ssn=0022314X&md5=6d8bba80a3ff6f6fa5cbf0fd6f7131e9" title="Click to vie
w the MathML
source">x<
sup>i
sup>y<
sup>j
sup>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>-coefficient of the Todd po
wer
serie
s of the lattice cone in <
span id="mml
si5" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022314X16301548&_mathId=
si5.gif&_u
ser=111111111&_pii=S0022314X16301548&_rdoc=1&_i
ssn=0022314X&md5=a6e0cd47c5e9badb8a166515fc840d6b" title="Click to vie
w the MathML
source">R<
sup>2
sup>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> generated by <
span id="mml
si6" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022314X16301548&_mathId=
si6.gif&_u
ser=111111111&_pii=S0022314X16301548&_rdoc=1&_i
ssn=0022314X&md5=92f11dee46d081ca88d5b5d14cd7c151" title="Click to vie
w the MathML
source">(1,0)
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> and <
span id="mml
si7" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022314X16301548&_mathId=
si7.gif&_u
ser=111111111&_pii=S0022314X16301548&_rdoc=1&_i
ssn=0022314X&md5=3481483978028da6fd62d31f9d298748" title="Click to vie
w the MathML
source">(p,q)
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>. The formula generalize
s Hicker
son'
s original one and reduce
s to Hicker
son'
s for <
span id="mml
si8" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022314X16301548&_mathId=
si8.gif&_u
ser=111111111&_pii=S0022314X16301548&_rdoc=1&_i
ssn=0022314X&md5=956405bad31eb5d3a361ec5075979e01" title="Click to vie
w the MathML
source">i=j=1
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>. In the formula, generalized Dedekind
sum
s are divided into t
wo part
s: the integral <
span id="mml
si9" cla
ss="mathml
src">
w 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">
ss="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">script>
style="vertical-align:bottom" width="55" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022314X16301548-si9.gif">script><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> and the fractional <
span id="mml
si10" cla
ss="mathml
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ss="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">script>
style="vertical-align:bottom" width="55" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022314X16301548-si10.gif">script><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>. We apply the formula to Siegel'
s formula for partial zeta value
s at a negative integer and obtain a ne
w expre
ssion
which involve
s only <
span id="mml
si9" cla
ss="mathml
src">
w 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">
ss="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">script>
style="vertical-align:bottom" width="55" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022314X16301548-si9.gif">script><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> the integral part of generalized Dedekind
sum
s. Thi
s formula directly generalize
s Meyer'
s formula for the
special value at 0. U
sing our formula,
we pre
sent the table of the partial zeta value at <
span id="mml
si11" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022314X16301548&_mathId=
si11.gif&_u
ser=111111111&_pii=S0022314X16301548&_rdoc=1&_i
ssn=0022314X&md5=689ff79aa1b80b4bb5d1d0ad78b952b6" title="Click to vie
w the MathML
source">
s=&minu
s;1
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> and &minu
s;2 in more explicit form. Finally,
we pre
sent another application on the equidi
stribution property of the fractional part
s of the graph <
span id="mml
si12" cla
ss="mathml
src">
w 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">
ss="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">script>
style="vertical-align:bottom" width="167" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S0022314X16301548-si12.gif">script><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> for a certain integer <
span id="mml
si13" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022314X16301548&_mathId=
si13.gif&_u
ser=111111111&_pii=S0022314X16301548&_rdoc=1&_i
ssn=0022314X&md5=285c25c2919bc817ce00b24b2077146a" title="Click to vie
w the MathML
source">R<
sub>i+j
sub>
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span> depending on <
span id="mml
si14" cla
ss="mathml
src"><
span cla
ss="formulatext
stixSupport mathImg" data-mathURL="/
science?_ob=MathURL&_method=retrieve&_eid=1-
s2.0-S0022314X16301548&_mathId=
si14.gif&_u
ser=111111111&_pii=S0022314X16301548&_rdoc=1&_i
ssn=0022314X&md5=feebb1991c9eb9545515f1bc4b33537f" title="Click to vie
w the MathML
source">i+j
span><
span cla
ss="mathContainer hidden"><
span cla
ss="mathCode">
span>
span>
span>.