N(2)

[kohmoto]

    I wonder if N(2) goes infinity.

    Where N(s)=   Sum_{m=1 to infinity} (-1)Sigma(m)/m^s

    Mathar computed up to m=1000000



  s m N(s)

> 2 1 1.000000000000000000000000000
> 2 2 1.250000000000000000000000000
> 2 3 1.472222222222222222222222222
> 2 4 1.784722222222222222222222222
> 2 5 1.944722222222222222222222222
> 2 6 2.000277777777777777777777778
> 2 7 2.122726757369614512471655329
> 2 8 2.325851757369614512471655329
> 2 9 2.461654226505416981607457798
> 2 10000 8.619813177975507784214344698
> 2 20000 9.240635095871089404115753095
> 2 30000 9.603791897573761358292380061
> 2 40000 9.861458278261452185262848858
> 2 50000 10.06133043757656995196303496
> 2 60000 10.22462154552329778519365923
> 2 70000 10.36270038005636494621123847
> 2 80000 10.48231016697477680498962023
> 2 90000 10.58780522013752853231672862
> 2 100000 10.68217402235127296288785820
> 2 110000 10.76754463607533705360724580
> 2 120000 10.84547789729425765394094918
> 2 130000 10.91717268365027978460835188
> 2 140000 10.98355652885066079792761960
> 2 150000 11.04534545539722125239381859
> 2 160000 11.10315629233238669062836862
> 2 170000 11.15746025373895516747888760
> 2 180000 11.20865775307751148407150182
> 2 190000 11.25708463818456648077093060
> 2 200000 11.30303121795283056763425981
> 2 210000 11.34672871948779384291766276
> 2 220000 11.38839827063844308711114745
> 2 230000 11.42821240662906506074836109
> 2 240000 11.46633165190016188934526342
> 2 250000 11.50290004647923375585461222
> 2 260000 11.53802980091084626273394586
> 2 270000 11.57183602344116902681335304
> 2 280000 11.60440829561909536077171934
> 2 290000 11.63584045378869260629085740
> 2 300000 11.66620539683007648663354504
> 2 310000 11.69557551635064846235471407
> 2 320000 11.72401391041322342493145640
> 2 330000 11.75157395832622563415629569
> 2 340000 11.77831576251751436463182945
> 2 350000 11.80427986574616427809478607
> 2 360000 11.82951337960016320803463309
> 2 370000 11.85405377158818097821428014
> 2 380000 11.87794149361683265940241605
> 2 390000 11.90120806565142413868457075
> 2 400000 11.92388559358277288937420440
> 2 410000 11.94600303970164868968234173
> 2 420000 11.96758569454234998141373716
> 2 430000 11.98866287480559328733313634
> 2 440000 12.00925544388189350449936925
> 2 450000 12.02938482345842327394291528
> 2 460000 12.04907101416111155199518539
> 2 470000 12.06833481184766752219695603
> 2 480000 12.08719139777266186606623586
> 2 490000 12.10566116650605692754670219
> 2 500000 12.12375840131307130932177405
> 2 510000 12.14149351071909408890611230
> 2 520000 12.15888769016783248485026059
> 2 530000 12.17594975401341942285000014
> 2 540000 12.19269280126406301090981599
> 2 550000 12.20912705098196862278557687
> 2 560000 12.22526702472305560991043337
> 2 570000 12.24111996896626519121471994
> 2 580000 12.25669798166764128539680162
> 2 590000 12.27200952384435815319104573
> 2 600000 12.28706299338740013449483108
> 2 610000 12.30187012252190226181814868
> 2 620000 12.31643453756242668558457904
> 2 630000 12.33076670711890682146429120
> 2 640000 12.34487182648867321005497632
> 2 650000 12.35875920337989470739102627
> 2 660000 12.37243430644503889140869087
> 2 670000 12.38590421689421153982279985
> 2 680000 12.39917431914306915108546756
> 2 690000 12.41224965443445935379058168
> 2 700000 12.42513867918653815467126754
> 2 710000 12.43784365088579118428209464
> 2 720000 12.45037185509338005878377820
> 2 730000 12.46272650898502725121177200
> 2 740000 12.47491268376206272216237202
> 2 750000 12.48693594951832440956683550
> 2 760000 12.49880013638169511883229569
> 2 770000 12.51050884124405914256853750
> 2 780000 12.52206537108093533871333277
> 2 790000 12.53347692259745252540675575
> 2 800000 12.54474459716977128685280547
> 2 810000 12.55587125001645305770022096
> 2 820000 12.56686130746829341595469366
> 2 830000 12.57771842011223665163037795
> 2 840000 12.58844534265072464040923243
> 2 850000 12.59904566706465055360427300
> 2 860000 12.60952226922408394518254911
> 2 870000 12.61987642201166975414741555
> 2 880000 12.63011418303496833362065684
> 2 890000 12.64023571583693921823484395
> 2 900000 12.65024408926657285403820450
> 2 910000 12.66014049405210257196085418
> 2 920000 12.66993013960070545251999449
> 2 930000 12.67961329658082585267888159
> 2 940000 12.68919341076487592672709130
> 2 950000 12.69867217785123447166660908
> 2 960000 12.70805040456023104531710849
> 2 970000 12.71733330696500792660758365
> 2 980000 12.72652037909829933006161382
> 2 990000 12.73561385879130711822105048
> 2 1000000 12.74461639944508312563225494



         UnitaryPhi(m) < (-1)Sigma(m) < Sigma(m)

         So, U(2) < N(2) < S(2)=Zeta(1):*Zeta(2)

         Where,U(s) = Sum_{m=1 to infinity} UnitaryPhi(m)/m^s , S(s) = 
Sum_{m=1 to infinity} Sigma(m)/m^s



    I suppose that U(2) is already known.

    Does anyone know it?



    Yasutoshi
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