In
[11], Hi
ckerson made an expli
ci
t formula for Dedekind sums
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si1.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=a793f7facdb680f1d93c110e62574a92" title="Click to view the MathML source">s(p,q)class="mathContainer hidden">class="mathCode">th altimg="si1.gif" overflow="scroll">stretchy="false">(p,qtretchy="false">)th> in
terms of
the
con
tinued fra
ction of
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si2.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=c754417fa40160ef3561c82ad5e61478" title="Click to view the MathML source">p/qclass="mathContainer hidden">class="mathCode">th altimg="si2.gif" overflow="scroll">ptretchy="false">/qth>. We develop analogous formula for generalized Dedekind sums
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si3.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=50921a92e992d9bcb46d77112310dc01" title="Click to view the MathML source">si,j(p,q)class="mathContainer hidden">class="mathCode">th altimg="si3.gif" overflow="scroll">si,jtretchy="false">(p,qtretchy="false">)th> defined in asso
cia
tion wi
th
the
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coeffi
cien
t of
the Todd power series of
the la
tti
ce
cone in
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si5.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=a6e0cd47c5e9badb8a166515fc840d6b" title="Click to view the MathML source">R2class="mathContainer hidden">class="mathCode">th altimg="si5.gif" overflow="scroll">thvariant="double-struck">R2th> genera
ted by
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si6.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=92f11dee46d081ca88d5b5d14cd7c151" title="Click to view the MathML source">(1,0)class="mathContainer hidden">class="mathCode">th altimg="si6.gif" overflow="scroll">tretchy="false">(1,0tretchy="false">)th> and
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si7.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=3481483978028da6fd62d31f9d298748" title="Click to view the MathML source">(p,q)class="mathContainer hidden">class="mathCode">th altimg="si7.gif" overflow="scroll">tretchy="false">(p,qtretchy="false">)th>. The formula generalizes Hi
ckerson's original one and redu
ces
to Hi
ckerson's for
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si8.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=956405bad31eb5d3a361ec5075979e01" title="Click to view the MathML source">i=j=1class="mathContainer hidden">class="mathCode">th altimg="si8.gif" overflow="scroll">i=j=1th>. In
the formula, generalized Dedekind sums are divided in
to
two par
ts:
the in
tegral
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the fra
ctional
class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si10.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=d90ac1223f8bb67d181b25c20727c386">class="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">cript>t="21" border="0" 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">cript>class="mathContainer hidden">class="mathCode">th altimg="si10.gif" overflow="scroll">sijRtretchy="false">(p,qtretchy="false">)th>. We apply
the formula
to Siegel's formula for par
tial ze
ta values a
t a nega
tive in
teger and ob
tain a new expression whi
ch involves only
class="mathmlsrc">title="View 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">class="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">cript>t="21" border="0" 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">cript>class="mathContainer hidden">class="mathCode">th altimg="si9.gif" overflow="scroll">sijItretchy="false">(p,qtretchy="false">)th> the in
tegral par
t of generalized Dedekind sums. This formula dire
ctly generalizes Meyer's formula for
the spe
cial value a
t 0. Using our formula, we presen
t the
table of
the par
tial ze
ta value a
t class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si11.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=689ff79aa1b80b4bb5d1d0ad78b952b6" title="Click to view the MathML source">s=−1class="mathContainer hidden">class="mathCode">th altimg="si11.gif" overflow="scroll">s=−1th> and −2 in more expli
ci
t form. Finally, we presen
t ano
ther appli
ca
tion on
the equidis
tribu
tion proper
ty of
the fra
ctional par
ts of
the graph
class="mathmlsrc">title="View 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">class="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">cript>t="29" border="0" 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">cript>class="mathContainer hidden">class="mathCode">th altimg="si12.gif" overflow="scroll">tretchy="true" maxsize="3.8ex" minsize="3.8ex">(c>pqc>,Ri+jqi+j−2sijtretchy="false">(p,qtretchy="false">)tretchy="true" maxsize="3.8ex" minsize="3.8ex">)th> for a
cer
tain in
teger
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si13.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=285c25c2919bc817ce00b24b2077146a" title="Click to view the MathML source">Ri+jclass="mathContainer hidden">class="mathCode">th altimg="si13.gif" overflow="scroll">Ri+jth> depending on
class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022314X16301548&_mathId=si14.gif&_user=111111111&_pii=S0022314X16301548&_rdoc=1&_issn=0022314X&md5=feebb1991c9eb9545515f1bc4b33537f" title="Click to view the MathML source">i+jclass="mathContainer hidden">class="mathCode">th altimg="si14.gif" overflow="scroll">i+jth>.