Close Encounters of the Rademacher kind



                              By Shalosh B. Ekhad



              The great analytic number theorist, Hans Rademacher

                       in his posthumously published book

              "Topics in Analytic Number Theory", Springer, 1973

                   (pp. 299-302) conjectured that C_{hkl}(N),

                  the coefficient of (x-exp(2*Pi*I*h/k)^(-l)

                    in the partial fraction decomposition of

                                       1
                            ------------------------
                                      i
                            Prod(1 - x , i = 1 .. N)

             converges to a certain number defined in his book and

                 implemented in procedure ChklInfinity(h,k,l) in

             The Maple package HANS, available, free of charge from

             http://www.math.rutgers.edu/~zeilberg/tokhniot/HANS  .

         It turns out (empirically), that these sequences diverge very

        badly, and as N goes to infinity, sooner or later they oscillate

        widely with peaks that go to infinity (exponentially fast!) and

        valleys that go to negative infinity (also exponentially fast!)



               But it seems that one can tweak Hans's conjecture

        if instead of going all the way to infinity, you only go so far.

             In this article we will find, for h and k, 1<=h<k<=, 3

                        with gcd(h,k)=1, and l<=, 5, ,

                       that integer N, between 1 and, 250

         such that C_{hkl}(N) gets closest to Rademacher's conjectured

          value for the "limit", what he called, with wishful thinking

                               C_{hkl}(infinity)

                 For , C_[0, 1, 1](N), N=, 25, gets closest to

C_[0, 1, 1](infinity) =

    -0.292927573960492036753087343838045066374149102115197502668739

the difference being,

    0.000031774176649089292491989138228042994317501145538447599461

and the ratio being,

    0.999891528898357056803436843179971734516485565025459202417126

                 For , C_[0, 1, 2](N), N=, 47, gets closest to

C_[0, 1, 2](infinity) =

    0.189767068844053274636865530367944871904086821544934201892043

the difference being,

    0.000143430727451381457860561106795039790143270735809573840837

and the ratio being,

    0.999244174827987410616628366363844909273685798816303917476115

                 For , C_[0, 1, 3](N), N=, 71, gets closest to

C_[0, 1, 3](infinity) =

    -0.0884388439516741449247178862412908618092570333358781842656905

the difference being,

                                                                 -5
    0.82850161521644477925861766461397445451046328000497169213 10

and the ratio being,

    0.999906319262193299931066735040470876007279944194375676223574

                 For , C_[0, 1, 4](N), N=, 95, gets closest to

C_[0, 1, 4](infinity) =

    0.0321582884564354825905112410606580451596021620544700229579815

the difference being,

                                                                 -5
    0.46155962709467538480127022688504383401999986980586157995 10

and the ratio being,

    1.00014352742302188926400697147827985332834403907340621948388

                 For , C_[0, 1, 5](N), N=, 149, gets closest to

C_[0, 1, 5](infinity) =

    -0.00958335596037266461101556644232869376227827782357426722015693

the difference being,

                                                                -7
    0.8588291545473063201628712907403349191787065067865868221 10

and the ratio being,

    1.00000896167436645971492479783111147410469051716309820888834

                     The closest, C_[1, 2, 1](N),  gets to

C_[1, 2, 1](infinity) =

    0.0938828534841788418970622452008482335837100682141156422696275

                             is when , N = 49,  ,

the abs. value of the difference being,

                                                                 -5
    0.49947593262253066450508654037992014768331582794299466305 10

and the absolute value of the ratio being,

    1.00005320203999835842794513816499108148878611506438053777883

                     The closest, C_[1, 2, 2](N),  gets to

C_[1, 2, 2](infinity) =

    0.0153092470856487319413671298887912050278788860536494972345423

                             is when , N = 193,  ,

the abs. value of the difference being,

                                                                -6
    0.3281156184107299534566174735981901973711756853182812593 10

and the absolute value of the ratio being,

    0.999978567488226229382953012454260188741373957027391841227259

                     The closest, C_[1, 2, 3](N),  gets to

C_[1, 2, 3](infinity) =

    0.00179016653079753465441680536377589719983985169846709088414268

                             is when , N = 82,  ,

the abs. value of the difference being,

                                                                 -6
    0.13368255561096980643039118052606103793523287642744653646 10

and the absolute value of the ratio being,

    1.00007467604455291266843463441074312024491308481903948926630

                     The closest, C_[1, 2, 4](N),  gets to

C_[1, 2, 4](infinity) =

    0.000163100317149382813121064375698460601151684627848399791659191

                             is when , N = 124,  ,

the abs. value of the difference being,

                                                                 -7
    0.13434818660661076554551194034443692217854588512014685054 10

and the absolute value of the ratio being,

    0.999917628494564138772633254919229667492822338710168401082768

                     The closest, C_[1, 2, 5](N),  gets to

C_[1, 2, 5](infinity) =

    0.0000121695799538861365577107485572221466588402495687900005406284

                             is when , N = 210,  ,

the abs. value of the difference being,

                                                                -9
    0.1688624649339375575551950204894882203843289867177588477 10

and the absolute value of the ratio being,

    0.999986124215825541754593762410528893487828170756282878175738

                     The closest, C_[1, 3, 1](N),  gets to

C_[1, 3, 1](infinity) =

    0.0241723788203328355952799035767262518948404601159547157945541 -

    0.0288075193006798476768593210161179248072389705480761722736956 I

                             is when , N = 35,  ,

the abs. value of the difference being,

    0.000273961914079265682597526860860594165252174757808584617677558

and the absolute value of the ratio being,

    1.00328762789512932994905293115723557248242053114840146068038

                     The closest, C_[1, 3, 2](N),  gets to

C_[1, 3, 2](infinity) =

    -0.000936414954163097390551865522870913490829587107391118384140164 -

    0.00257277894180274522396720097610150604341366208673355461278711 I

                             is when , N = 50,  ,

the abs. value of the difference being,

                                                                     -5
    0.553039059226744935869216160248247981912190299509004871610625 10

and the absolute value of the ratio being,

    0.998113237877081584769388177377400995052193976935125613436692

                     The closest, C_[1, 3, 3](N),  gets to

C_[1, 3, 3](infinity) =

    -0.000140478381853147871574938441697004979055586949543707936580219 -

    0.0000247701289269763807610888511645622093279707049272021415026639 I

                             is when , N = 217,  ,

the abs. value of the difference being,

                                                                     -7
    0.964452480660499744873836265025989733012152313386569814523484 10

and the absolute value of the ratio being,

    0.999324126700054727778148651671281702666879663304521942328296

                     The closest, C_[1, 3, 4](N),  gets to

C_[1, 3, 4](infinity) =

                                                                      -5
    -0.371866331222294935796086845242833549021386546605269589382543 10   +

                                                                     -5
    0.443173036198998417537890164889305350676988210004494919686277 10   I

                             is when , N = 233,  ,

the abs. value of the difference being,

                                                                     -8
    0.202565871285376093949985145637195924279888266265291708260924 10

and the absolute value of the ratio being,

    1.00031739105696837966287806940088700328009246802847124799844

                     The closest, C_[1, 3, 5](N),  gets to

C_[1, 3, 5](infinity) =

                                                                     -7
    0.656867525307423676397659969568264896617339595630888432756529 10   +

                                                                     -6
    0.180472869335518419516832999710528824327650065092522166119516 10   I

                             is when , N = 216,  ,

the abs. value of the difference being,

                                                                     -10
    0.468311089489933417956520711419385725407909895422142930933716 10

and the absolute value of the ratio being,

    0.999924588869401612732415394516931820101662099289142740662197

                     The closest, C_[2, 3, 1](N),  gets to

C_[2, 3, 1](infinity) =

    0.0241723788203328355952799035767262518948404601159547157945541 +

    0.0288075193006798476768593210161179248072389705480761722736956 I

                             is when , N = 35,  ,

the abs. value of the difference being,

    0.000273961914079265682597526860860594165252174757808584617677558

and the absolute value of the ratio being,

    1.00328762789512932994905293115723557248242053114840146068038

                     The closest, C_[2, 3, 2](N),  gets to

C_[2, 3, 2](infinity) =

    -0.000936414954163097390551865522870913490829587107391118384139348 +

    0.00257277894180274522396720097610150604341366208673355461278741 I

                             is when , N = 50,  ,

the abs. value of the difference being,

                                                                     -5
    0.553039059226744935869216160248247981912190299509004871612459 10

and the absolute value of the ratio being,

    0.998113237877081584769388177377400995052193976935125613436685

                     The closest, C_[2, 3, 3](N),  gets to

C_[2, 3, 3](infinity) =

    -0.000140478381853147871574938441697004979055586949543707936580219 +

    0.0000247701289269763807610888511645622093279707049272021415021067 I

                             is when , N = 217,  ,

the abs. value of the difference being,

                                                                     -7
    0.964452480660499744873836265025989733012152313386569813394856 10

and the absolute value of the ratio being,

    0.999324126700054727778148651671281702666879663304521942329217

                     The closest, C_[2, 3, 4](N),  gets to

C_[2, 3, 4](infinity) =

                                                                      -5
    -0.371866331222294935796086845242833549021386546605269589456576 10   -

                                                                     -5
    0.443173036198998417537890164889305350676988210004494919627050 10   I

                             is when , N = 233,  ,

the abs. value of the difference being,

                                                                     -8
    0.202565871285376093949985145637195924279888266265291896692328 10

and the absolute value of the ratio being,

    1.00031739105696837966287806940088700328009246802847124833520

                     The closest, C_[2, 3, 5](N),  gets to

C_[2, 3, 5](infinity) =

                                                                     -7
    0.656867525307423676397659969568264896617339595630888430531641 10   -

                                                                     -6
    0.180472869335518419516832999710528824327650065092522165340804 10   I

                             is when , N = 216,  ,

the abs. value of the difference being,

                                                                     -10
    0.468311089489933417956520711419385725407909895422153856724647 10

and the absolute value of the ratio being,

    0.999924588869401612732415394516931820101662099289142735959949

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