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<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN lang=EN-US>I wonder if
N(2) goes infinity.<?xml:namespace prefix = o ns =
"urn:schemas-microsoft-com:office:office" /><o:p></o:p></SPAN></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US
style="FONT-SIZE: 12pt; FONT-FAMILY: 'ＭＳ Ｐゴシック'; mso-bidi-font-family: 'ＭＳ Ｐゴシック'; mso-font-kerning: 0pt"><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN>Where </SPAN><SPAN lang=EN-US
style="FONT-FAMILY: 'MS UI Gothic'"><FONT size=3>N(s)= Sum_{m=1 to
infinity} (-1)Sigma(m)/m^s<o:p></o:p></FONT></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US style="FONT-FAMILY: 'MS UI Gothic'"><FONT
size=3><SPAN style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN>Mathar computed up to
m=1000000<o:p></o:p></FONT></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US style="FONT-FAMILY: 'MS UI Gothic'"><o:p><FONT
size=3> </FONT></o:p></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US style="FONT-FAMILY: 'MS UI Gothic'"><FONT
size=3><SPAN style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN>s m N(s) <SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P>
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style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US style="FONT-FAMILY: 'MS UI Gothic'"><FONT
size=3>> 2 1 1.000000000000000000000000000<BR>> 2 2
1.250000000000000000000000000<BR>> 2 3 1.472222222222222222222222222<BR>>
2 4 1.784722222222222222222222222<BR>> 2 5
1.944722222222222222222222222<BR>> 2 6 2.000277777777777777777777778<BR>>
2 7 2.122726757369614512471655329<BR>> 2 8
2.325851757369614512471655329<BR>> 2 9 2.461654226505416981607457798<BR>>
2 10000 8.619813177975507784214344698<BR>> 2 20000
9.240635095871089404115753095<BR>> 2 30000
9.603791897573761358292380061<BR>> 2 40000
9.861458278261452185262848858<BR>> 2 50000
10.06133043757656995196303496<BR>> 2 60000
10.22462154552329778519365923<BR>> 2 70000
10.36270038005636494621123847<BR>> 2 80000
10.48231016697477680498962023<BR>> 2 90000
10.58780522013752853231672862<BR>> 2 100000
10.68217402235127296288785820<BR>> 2 110000
10.76754463607533705360724580<BR>> 2 120000
10.84547789729425765394094918<BR>> 2 130000
10.91717268365027978460835188<BR>> 2 140000
10.98355652885066079792761960<BR>> 2 150000
11.04534545539722125239381859<BR>> 2 160000
11.10315629233238669062836862<BR>> 2 170000
11.15746025373895516747888760<BR>> 2 180000
11.20865775307751148407150182<BR>> 2 190000
11.25708463818456648077093060<BR>> 2 200000
11.30303121795283056763425981<BR>> 2 210000
11.34672871948779384291766276<BR>> 2 220000
11.38839827063844308711114745<BR>> 2 230000
11.42821240662906506074836109<BR>> 2 240000
11.46633165190016188934526342<BR>> 2 250000
11.50290004647923375585461222<BR>> 2 260000
11.53802980091084626273394586<BR>> 2 270000
11.57183602344116902681335304<BR>> 2 280000
11.60440829561909536077171934<BR>> 2 290000
11.63584045378869260629085740<BR>> 2 300000
11.66620539683007648663354504<BR>> 2 310000
11.69557551635064846235471407<BR>> 2 320000
11.72401391041322342493145640<BR>> 2 330000
11.75157395832622563415629569<BR>> 2 340000
11.77831576251751436463182945<BR>> 2 350000
11.80427986574616427809478607<BR>> 2 360000
11.82951337960016320803463309<BR>> 2 370000
11.85405377158818097821428014<BR>> 2 380000
11.87794149361683265940241605<BR>> 2 390000
11.90120806565142413868457075<BR>> 2 400000
11.92388559358277288937420440<BR>> 2 410000
11.94600303970164868968234173<BR>> 2 420000
11.96758569454234998141373716<BR>> 2 430000
11.98866287480559328733313634<BR>> 2 440000
12.00925544388189350449936925<BR>> 2 450000
12.02938482345842327394291528<BR>> 2 460000
12.04907101416111155199518539<BR>> 2 470000
12.06833481184766752219695603<BR>> 2 480000
12.08719139777266186606623586<BR>> 2 490000
12.10566116650605692754670219<BR>> 2 500000
12.12375840131307130932177405<BR>> 2 510000
12.14149351071909408890611230<BR>> 2 520000
12.15888769016783248485026059<BR>> 2 530000
12.17594975401341942285000014<BR>> 2 540000
12.19269280126406301090981599<BR>> 2 550000
12.20912705098196862278557687<BR>> 2 560000
12.22526702472305560991043337<BR>> 2 570000
12.24111996896626519121471994<BR>> 2 580000
12.25669798166764128539680162<BR>> 2 590000
12.27200952384435815319104573<BR>> 2 600000
12.28706299338740013449483108<BR>> 2 610000
12.30187012252190226181814868<BR>> 2 620000
12.31643453756242668558457904<BR>> 2 630000
12.33076670711890682146429120<BR>> 2 640000
12.34487182648867321005497632<BR>> 2 650000
12.35875920337989470739102627<BR>> 2 660000
12.37243430644503889140869087<BR>> 2 670000
12.38590421689421153982279985<BR>> 2 680000
12.39917431914306915108546756<BR>> 2 690000
12.41224965443445935379058168<BR>> 2 700000
12.42513867918653815467126754<BR>> 2 710000
12.43784365088579118428209464<BR>> 2 720000
12.45037185509338005878377820<BR>> 2 730000
12.46272650898502725121177200<BR>> 2 740000
12.47491268376206272216237202<BR>> 2 750000
12.48693594951832440956683550<BR>> 2 760000
12.49880013638169511883229569<BR>> 2 770000
12.51050884124405914256853750<BR>> 2 780000
12.52206537108093533871333277<BR>> 2 790000
12.53347692259745252540675575<BR>> 2 800000
12.54474459716977128685280547<BR>> 2 810000
12.55587125001645305770022096<BR>> 2 820000
12.56686130746829341595469366<BR>> 2 830000
12.57771842011223665163037795<BR>> 2 840000
12.58844534265072464040923243<BR>> 2 850000
12.59904566706465055360427300<BR>> 2 860000
12.60952226922408394518254911<BR>> 2 870000
12.61987642201166975414741555<BR>> 2 880000
12.63011418303496833362065684<BR>> 2 890000
12.64023571583693921823484395<BR>> 2 900000
12.65024408926657285403820450<BR>> 2 910000
12.66014049405210257196085418<BR>> 2 920000
12.66993013960070545251999449<BR>> 2 930000
12.67961329658082585267888159<BR>> 2 940000
12.68919341076487592672709130<BR>> 2 950000
12.69867217785123447166660908<BR>> 2 960000
12.70805040456023104531710849<BR>> 2 970000
12.71733330696500792660758365<BR>> 2 980000
12.72652037909829933006161382<BR>> 2 990000
12.73561385879130711822105048<BR>> 2 1000000
12.74461639944508312563225494<o:p></o:p></FONT></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US style="FONT-FAMILY: 'MS UI Gothic'"><o:p><FONT
size=3> </FONT></o:p></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US style="FONT-FAMILY: 'MS UI Gothic'"><FONT
size=3><SPAN style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN>UnitaryPhi(m) < (-1)Sigma(m) <
Sigma(m)<o:p></o:p></FONT></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US style="FONT-FAMILY: 'MS UI Gothic'"><FONT
size=3><SPAN style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN>So, U(2) < N(2) <
S(2)=Zeta(1):*Zeta(2)<o:p></o:p></FONT></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US style="FONT-FAMILY: 'MS UI Gothic'"><FONT
size=3><SPAN style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN>Where,U(s) = Sum_{m=1 to infinity}
UnitaryPhi(m)/m^s , S(s) = Sum_{m=1 to infinity}
Sigma(m)/m^s<o:p></o:p></FONT></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US style="FONT-FAMILY: 'MS UI Gothic'"><FONT
size=3><SPAN
style="mso-spacerun: yes"> </SPAN><o:p></o:p></FONT></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US style="FONT-FAMILY: 'MS UI Gothic'"><FONT
size=3><SPAN style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN>I suppose that U(2) is already
known.<o:p></o:p></FONT></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US style="FONT-FAMILY: 'MS UI Gothic'"><FONT
size=3><SPAN style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN>Does anyone know
it?<o:p></o:p></FONT></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US style="FONT-FAMILY: 'MS UI Gothic'"><o:p><FONT
size=3> </FONT></o:p></SPAN></P>
<P class=MsoNormal
style="MARGIN: 0mm 0mm 0pt; TEXT-ALIGN: left; mso-pagination: widow-orphan"
align=left><SPAN lang=EN-US style="FONT-FAMILY: 'MS UI Gothic'"><FONT
size=3><SPAN style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN><SPAN
style="mso-spacerun: yes"> </SPAN>Yasutoshi<o:p></o:p></FONT></SPAN></P></FONT></DIV></BODY></HTML>