# Estimate term in Ramanujan Lost Notebook (classic analytic number theory)

This is the fault of Igor Rivin, who asked about sums of divisor functions. I will put in links eventually. What I would like to know is the size of the right hand side in Ramanujan's formula (381), see original and a rendition FROM HERE in the second jpeg below. The quotes from Ramanujan are in (including some sign corrections) Nicolas and Robin, The Ramanujan Journal, Volume 1, Issue 2, June 1997, pages 119-153, Highly Composite Numbers by Srinivasa Ramanujan.

The questions are about the surprising size of the thing. For $s = 1/2$, formula 380, as a function of $N$ it is still larger than any power of $\log N$. Note that the construction "generalised superior highly composite numbers" always gives something smaller than any root $N^\alpha$ for $\alpha > 0$.

Question (A): is there a subinterval of the given $1/2 < s < 1$ for which the right-hand side of (381) has comparable growth to some $(\log N)^\beta$ for some $0 < \beta < \infty$?

Question (B): how do we get down to the really tiny growth for $s=1$, dominant term $\log \log N?$ Note that this was later improved in Robin's Criterion.

Between (330) and (332) we find $$S_s(x) = - s \sum \; \frac{x^{\rho - s}}{\rho (\rho - s)}$$ where the sum is over the complex roots of $\zeta.$''

With that in mind, the right hand side of (381) is $$| \zeta(s) | \exp \left\{ \mbox{Li} \left( (\log N)^{1-s} \right) - \frac{2s(2^{1/(2s)} -1)}{2s-1} \frac{(\log N)^{\frac{1}{2}-s}}{\log \log N} \right\} + \frac{S_s(\log N)}{\log \log N} + O \left\{ \frac{(\log N)^{\frac{1}{2}-s}}{(\log \log N)^2} \right\}$$  • mathoverflow.net/questions/71453/… – Will Jagy Jul 26 '13 at 21:09
• It may be a good idea to also write the relevant formulas in the body of the question. – Ricardo Andrade Jul 26 '13 at 22:52
• @RicardoAndrade working on it – Will Jagy Jul 26 '13 at 23:05
• @RicardoAndrade, got it. There are notes by Nicolas and Robin emphasizing the function (parenthesis) order in the use of the logarithmic integral function. – Will Jagy Jul 26 '13 at 23:21

## 1 Answer

EDIT, Monday: while specific constants depend on RH, the following is unconditional: for $s=1$ the "ratio" grows as $\log \log n,$ but for fixed $0 < s < 1$ the "ratio" grows faster than any power of $\log n.$

I heard from Jean-Louis Nicolas. For fixed $\frac{1}{2} < s < 1$ and $n \rightarrow \infty$ the growth of Ramanujan's "generalised superior highly composite numbers" is still swifter than any power of $\log n.$ My own fiddling with formula (381) shows that the single dominant term is an awfully good estimate. Note that $\zeta(3/4) \approx -3.441285$

==================================

t  0.75

ratio = sigma_{3/4}(n) / n^{3/4}

Ram = exp( Li ( (log n)^{1/4} )  )

==================================

ratio   1.594603558                                                         2 bump   2^1
ratio   2.294142325                                  log n   1.791759469    6 bump   3^1
ratio   2.802796526                                  log n   2.48490665     12 bump   2^2
ratio   3.641028199                                  log n   4.094344562    60 bump   5^1
ratio   4.033928726                                  log n   4.787491743    120 bump   2^3
ratio   4.573537136                                  log n   5.886104031    360 bump   3^2
ratio   5.63628118                                   log n   7.832014181    2520 bump   7^1
ratio   5.962699514                                  log n   8.525161361    5040 bump   2^4
ratio   6.949884201                                  log n   10.92305663    55440 bump   11^1
ratio   7.965010441                                  log n   13.48800599    720720 bump   13^1
ratio   8.224276385                                  log n   14.18115317    1441440 bump   2^5
ratio   8.649956395                                  log n   15.27976546    4324320 bump   3^3
ratio   9.245517654    over  Ram   3.028600308       log n   16.88920337    21621600 bump   5^2
ratio   10.34983665    over  Ram   3.165867659       log n   19.72241672    367567200 bump   17^1
ratio   11.48711898    over  Ram   3.083261736       log n   22.6668557     6983776800 bump   19^1
ratio   12.58086187    over  Ram   3.172163117       log n   25.80234991    160626866400 bump   23^1
ratio   12.81668461    over  Ram   3.231623923       log n   26.49549709    321253732800 bump   2^6
ratio   13.84228269    over  Ram   3.095238829       log n   29.86279292     bump   29^1
ratio   14.4487679    over  Ram   3.230853497       log n   31.80870307     bump   7^2
ratio   15.54855733    over  Ram   3.101387246       log n   35.24269028     bump   31^1
ratio   15.88423195    over  Ram   3.168342458       log n   36.34130256     bump   3^4
ratio   16.94303437    over  Ram   3.19811678       log n   39.95222048     bump   37^1
ratio   17.9887252    over  Ram   3.217053505       log n   43.66579254     bump   41^1
ratio   18.18553099    over  Ram   3.084784213       log n   44.35893972     bump   2^7
ratio   19.26852148    over  Ram   3.103456721       log n   48.12013984     bump   43^1
ratio   20.34195462    over  Ram   3.114010326       log n   51.97028744     bump   47^1
ratio   21.37753969    over  Ram   3.113314008       log n   55.94057936     bump   53^1
ratio   21.78937675    over  Ram   3.173291822       log n   57.55001727     bump   5^3
ratio   22.81291905    over  Ram   3.163514705       log n   61.62755471     bump   59^1
ratio   23.85808075    over  Ram   3.152929072       log n   65.73842858     bump   61^1
ratio   24.87685838    over  Ram   3.135532889       log n   69.9431212     bump   67^1
ratio   25.46187993    over  Ram   3.209270267       log n   72.34101647     bump   11^2
ratio   26.50287028    over  Ram   3.188425241       log n   76.60369635     bump   71^1
ratio   26.67341265    over  Ram   3.208942324       log n   77.29684353     bump   2^8
ratio   27.74144999    over  Ram   3.187825713       log n   81.58730297     bump   73^1
ratio   28.78835877    over  Ram   3.162018062       log n   85.95675082     bump   79^1
ratio   29.05524627    over  Ram   3.191332103       log n   87.05536311     bump   3^5
ratio   30.11185937    over  Ram   3.163394697       log n   91.47420372     bump   83^1
ratio   31.15104963    over  Ram   3.132063376       log n   95.96284009     bump   89^1
ratio   31.73094398    over  Ram   3.190368501       log n   98.52778944     bump   13^2
ratio   32.75755091    over  Ram   3.154083607       log n   103.1025004     bump   97^1
ratio   33.78573385    over  Ram   3.253082917       log n   107.7176209     bump   101^1
ratio   34.83070774    over  Ram   3.213503967       log n   112.3523499     bump   103^1
ratio   35.87765418    over  Ram   3.173480786       log n   117.0251788     bump   107^1
ratio   36.94119501    over  Ram   3.267554004       log n   121.7165266     bump   109^1
ratio   38.00705978    over  Ram   3.224796113       log n   126.4439145     bump   113^1
ratio   38.15155234    over  Ram   3.237055915       log n   127.1370616     bump   2^9
ratio   39.16001407    over  Ram   3.188810947       log n   131.9812487     bump   127^1
ratio   40.17133587    over  Ram   3.271163165       log n   136.8564461     bump   131^1
ratio   41.1745089    over  Ram   3.21940471       log n   141.776427     bump   137^1
ratio   41.57610945    over  Ram   3.250805563       log n   143.7223371     bump   7^3
ratio   42.60313855    over  Ram   3.200038518       log n   148.6568111     bump   139^1
ratio   43.60210727    over  Ram   3.275073797       log n   153.6607574     bump   149^1
ratio   44.61432694    over  Ram   3.22071367       log n   158.6780372     bump   151^1
ratio   45.18291626    over  Ram   3.261760202       log n   161.5112505     bump   17^2
ratio   46.20162379    over  Ram   3.206931917       log n   166.5674964     bump   157^1
ratio   47.21440711    over  Ram   3.277230899       log n   171.6612466     bump   163^1
ratio   48.23074282    over  Ram   3.220290278       log n   176.7792404     bump   167^1
ratio   49.24183179    over  Ram   3.287799086       log n   181.932532     bump   173^1
ratio   50.24805522    over  Ram   3.228541517       log n   187.1199178     bump   179^1
ratio   51.26631908    over  Ram   3.293967076       log n   192.3184148     bump   181^1
ratio   51.82404921    over  Ram   3.205575163       log n   195.2628538     bump   19^2
ratio   52.83273596    over  Ram   3.267967455       log n   200.5151272     bump   191^1
ratio   53.04563115    over  Ram   3.281136081       log n   201.6137395     bump   3^6
ratio   54.07005962    over  Ram   3.220955494       log n   206.8764297     bump   193^1
ratio   55.09832965    over  Ram   3.282209579       log n   212.1596334     bump   197^1
ratio   56.13824653    over  Ram   3.22181287       log n   217.4529382     bump   199^1
ratio   56.45557713    over  Ram   3.240024693       log n   219.0623761     bump   5^4
ratio   57.47533017    over  Ram   3.298549027       log n   224.4142343     bump   211^1
ratio   57.60476235    over  Ram   3.305977229       log n   225.1073815     bump   2^10
ratio   58.60299034    over  Ram   3.241380692       log n   230.5145532     bump   223^1
ratio   59.6050657    over  Ram   3.296806323       log n   235.9395033     bump   227^1
ratio   60.61759257    over  Ram   3.232422062       log n   241.3732253     bump   229^1
ratio   61.63403256    over  Ram   3.286623539       log n   246.8242637     bump   233^1
ratio   62.64799568    over  Ram   3.221820568       log n   252.3007273     bump   239^1
ratio   63.67221839    over  Ram   3.27449363       log n   257.7855242     bump   241^1
ratio   64.68192381    over  Ram   3.209099212       log n   263.3109771     bump   251^1
ratio   65.68962805    over  Ram   3.259094987       log n   268.8600532     bump   257^1
ratio   66.69547057    over  Ram   3.308998395       log n   274.4322072     bump   263^1
ratio   67.69958251    over  Ram   3.241375141       log n   280.0269186     bump   269^1
ratio   68.71316482    over  Ram   3.289904251       log n   285.6290375     bump   271^1
ratio   69.28195086    over  Ram   3.317137047       log n   288.7645317     bump   23^2
ratio   70.30232697    over  Ram   3.24929488       log n   294.3885492     bump   277^1
ratio   71.32665713    over  Ram   3.296638274       log n   300.0269038     bump   281^1
ratio   72.36039885    over  Ram   3.229428647       log n   305.6723507     bump   283^1
ratio   73.382162    over  Ram   3.275029711       log n   311.3525233     bump   293^1
ratio   74.38270722    over  Ram   3.319683824       log n   317.0793711     bump   307^1
ratio   75.38709562    over  Ram   3.249769843       log n   322.819164     bump   311^1
ratio   76.400164    over  Ram   3.293440965       log n   328.5653672     bump   313^1
ratio   77.41711457    over  Ram   3.224374375       log n   334.324269     bump   317^1
ratio   78.41473649    over  Ram   3.265924703       log n   340.1263873     bump   331^1
ratio   79.41169082    over  Ram   3.307447227       log n   345.9464703     bump   337^1
ratio   80.39941875    over  Ram   3.236182228       log n   351.7957951     bump   347^1
ratio   81.39513097    over  Ram   3.276260954       log n   357.650867     bump   349^1
ratio   82.39459553    over  Ram   3.316490715       log n   363.517335     bump   353^1
ratio   83.39362406    over  Ram   3.244889968       log n   369.4006574     bump   359^1
ratio   84.38818935    over  Ram   3.283589029       log n   375.3060193     bump   367^1
ratio   85.38244953    over  Ram   3.322276218       log n   381.2275977     bump   373^1
ratio   86.37645598    over  Ram   3.24984013       log n   387.1651339     bump   379^1
ratio   87.37414754    over  Ram   3.28737742       log n   393.1131689     bump   383^1
ratio   87.49088072    over  Ram   3.291769406       log n   393.8063161     bump   2^11
ratio   88.48973144    over  Ram   3.329350308       log n   399.7698954     bump   389^1
ratio   89.48467856    over  Ram   3.256298948       log n   405.7538317     bump   397^1
ratio   90.48327589    over  Ram   3.292637363       log n   411.7477931     bump   401^1
ratio   91.47816766    over  Ram   3.328840935       log n   417.7615083     bump   409^1
ratio   92.46594017    over  Ram   3.255165011       log n   423.7993792     bump   419^1
ratio   93.46081907    over  Ram   3.290188663       log n   429.842012     bump   421^1
ratio   94.01493491    over  Ram   3.309695722       log n   433.2093079     bump   29^2
ratio   95.00882623    over  Ram   3.23649366       log n   439.2754159     bump   431^1
ratio   96.00974315    over  Ram   3.270590084       log n   445.3461537     bump   433^1
ratio   97.01081885    over  Ram   3.304691917       log n   451.4306531     bump   439^1
ratio   98.01547483    over  Ram   3.338915714       log n   457.5242229     bump   443^1
ratio   99.02034486    over  Ram   3.264797057       log n   463.6312457     bump   449^1
ratio   100.0221593    over  Ram   3.297827853       log n   469.7559291     bump   457^1
ratio   101.0275168    over  Ram   3.330975468       log n   475.8893272     bump   461^1
ratio   102.0396879    over  Ram   3.257021972       log n   482.0270542     bump   463^1
ratio   103.0554254    over  Ram   3.289443466       log n   488.1733835     bump   467^1
ratio   103.2368708    over  Ram   3.295235055       log n   489.2719958     bump   3^7
ratio   103.6295806    over  Ram   3.307770024       log n   491.6698911     bump   11^3
ratio   104.6417008    over  Ram   3.34007606       log n   497.8415917     bump   479^1
ratio   105.2050828    over  Ram   3.251655773       log n   501.2755789     bump   31^2
ratio   106.2199051    over  Ram   3.283021679       log n   507.463843     bump   487^1
ratio   107.2382498    over  Ram   3.314496454       log n   513.6602871     bump   491^1
ratio   108.2539705    over  Ram   3.240575218       log n   519.8728932     bump   499^1
ratio   109.2731903    over  Ram   3.271085492       log n   526.0934834     bump   503^1
ratio   110.292897    over  Ram   3.301610344       log n   532.3259314     bump   509^1
ratio   111.3042884    over  Ram   3.331886276       log n   538.5816814     bump   521^1
ratio   112.3220256    over  Ram   3.257208542       log n   544.8412629     bump   523^1
ratio   113.3233319    over  Ram   3.286245266       log n   551.1346822     bump   541^1
ratio   114.3252422    over  Ram   3.315299503       log n   557.439131     bump   547^1
ratio   115.3223698    over  Ram   3.344215048       log n   563.7616962     bump   557^1
ratio   116.320144    over  Ram   3.268343789       log n   570.0949758     bump   563^1
ratio   117.3185812    over  Ram   3.296397707       log n   576.4388563     bump   569^1
ratio   118.3229419    over  Ram   3.324618066       log n   582.7862455     bump   571^1
ratio   119.3279906    over  Ram   3.249339042       log n   589.1440878     bump   577^1
ratio   120.3285981    over  Ram   3.276585902       log n   595.5191126     bump   587^1
ratio   121.3299295    over  Ram   3.303852473       log n   601.904307     bump   593^1
ratio   122.3319989    over  Ram   3.331139143       log n   608.2995686     bump   599^1
ratio   123.3398218    over  Ram   3.255529759       log n   614.6981635     bump   601^1
ratio   124.3484052    over  Ram   3.282151114       log n   621.1066923     bump   607^1
ratio   125.3577623    over  Ram   3.308792893       log n   627.5250572     bump   613^1
ratio   126.370361    over  Ram   3.335520231       log n   633.9499263     bump   617^1
ratio   127.3886645    over  Ram   3.259857576       log n   640.3780315     bump   619^1
ratio   128.4004973    over  Ram   3.285750232       log n   646.8253374     bump   631^1
ratio   129.4084107    over  Ram   3.311542589       log n   653.2883669     bump   641^1
ratio   130.4218652    over  Ram   3.337476745       log n   659.7545116     bump   643^1
ratio   131.4385169    over  Ram   3.261534446       log n   666.2268579     bump   647^1
ratio   132.4560248    over  Ram   3.286783033       log n   672.708435     bump   653^1
ratio   133.4743996    over  Ram   3.312053134       log n   679.1991585     bump   659^1
ratio   134.4982745    over  Ram   3.337459713       log n   685.6929124     bump   661^1
ratio   135.5161752    over  Ram   3.261385529       log n   692.2046577     bump   673^1
ratio   135.8203463    over  Ram   3.268705832       log n   694.1505679     bump   7^4
ratio   136.8436943    over  Ram   3.293334126       log n   700.6682391     bump   677^1
ratio   137.4137237    over  Ram   3.307052677       log n   704.279157     bump   37^2
ratio   137.5227394    over  Ram   3.309676293       log n   704.9723042     bump   2^12
ratio   138.5520797    over  Ram   3.334448801       log n   711.4987991     bump   683^1
ratio   139.5801067    over  Ram   3.258552335       log n   718.0369389     bump   691^1
ratio   140.6046611    over  Ram   3.282470958       log n   724.5894468     bump   701^1
ratio   141.6279895    over  Ram   3.306360962       log n   731.1533023     bump   709^1
ratio   142.6479948    over  Ram   3.330173386       log n   737.7311637     bump   719^1
ratio   143.6668556    over  Ram   3.353959091       log n   744.3200902     bump   727^1
ratio   144.6866876    over  Ram   3.277153174       log n   750.9172359     bump   733^1
ratio   145.7074983    over  Ram   3.300274535       log n   757.5225338     bump   739^1
ratio   146.7313577    over  Ram   3.323464947       log n   764.1332298     bump   743^1
ratio   147.754163    over  Ram   3.346631486       log n   770.7546355     bump   751^1
ratio   148.0025434    over  Ram   3.352257301       log n   772.3640734     bump   5^5
ratio   149.0280708    over  Ram   3.275506404       log n   778.9934366     bump   757^1
ratio   149.4258815    over  Ram   3.284249934       log n   781.558386     bump   13^3
ratio   150.457187    over  Ram   3.306917125       log n   788.1930194     bump   761^1
ratio   151.4874976    over  Ram   3.32956245       log n   794.8381103     bump   769^1
ratio   152.5208351    over  Ram   3.352274302       log n   801.4883894     bump   773^1
ratio   153.5473095    over  Ram   3.275430973       log n   808.1566176     bump   787^1
ratio   154.5709524    over  Ram   3.297267055       log n   814.8374723     bump   797^1
ratio   155.5899343    over  Ram   3.319003711       log n   821.5332712     bump   809^1
ratio   156.613736    over  Ram   3.340843181       log n   828.2315393     bump   811^1
ratio   157.6348459    over  Ram   3.264122817       log n   834.9420624     bump   821^1
ratio   158.6607396    over  Ram   3.285365855       log n   841.6550186     bump   823^1
ratio   159.2299653    over  Ram   3.297152732       log n   845.3685906     bump   41^2
ratio   160.2624786    over  Ram   3.318532842       log n   852.0863953     bump   827^1
ratio   161.2998063    over  Ram   3.340012641       log n   858.8066155     bump   829^1
ratio   162.3345013    over  Ram   3.361437928       log n   865.5388262     bump   839^1
ratio   163.3629889    over  Ram   3.284177807       log n   872.2875857     bump   853^1
ratio   164.3943674    over  Ram   3.304912187       log n   879.0410237     bump   857^1
ratio   165.4304444    over  Ram   3.325741025       log n   885.7967926     bump   859^1
ratio   166.4694248    over  Ram   3.34662823       log n   892.5572073     bump   863^1
ratio   167.5023879    over  Ram   3.269806211       log n   899.3337143     bump   877^1
ratio   168.5382193    over  Ram   3.290026627       log n   906.1147719     bump   881^1
ratio   169.5786852    over  Ram   3.310337514       log n   912.8980971     bump   883^1
ratio   170.6220317    over  Ram   3.330704631       log n   919.6859421     bump   887^1
ratio   171.1931289    over  Ram   3.341852992       log n   923.4471422     bump   43^2
ratio   172.2289407    over  Ram   3.362073026       log n   930.2572846     bump   907^1
ratio   173.2675862    over  Ram   3.284840511       log n   937.0718275     bump   911^1
ratio   174.3056659    over  Ram   3.304520627       log n   943.8951137     bump   919^1
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ratio   179.4946651    over  Ram   3.305301405       log n   978.1317628     bump   953^1
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ratio   184.6865287    over  Ram   3.303867566       log n   1012.558098     bump   991^1
ratio   185.7274403    over  Ram   3.322488492       log n   1019.462849     bump   997^1
ratio   186.7648676    over  Ram   3.341047087       log n   1026.379564     bump   1009^1
ratio   187.8049988    over  Ram   3.359654051       log n   1033.300235     bump   1013^1
ratio   188.8463003    over  Ram   3.282372532       log n   1040.226812     bump   1019^1
ratio   189.8918367    over  Ram   3.300545195       log n   1047.15535     bump   1021^1
ratio   190.9355046    over  Ram   3.318685379       log n   1054.093635     bump   1031^1
ratio   191.9833843    over  Ram   3.336898771       log n   1061.033857     bump   1033^1
ratio   193.0324483    over  Ram   3.355132747       log n   1067.979871     bump   1039^1
ratio   194.0796943    over  Ram   3.278040755       log n   1074.935464     bump   1049^1
ratio   195.1311187    over  Ram   3.295799501       log n   1081.892961     bump   1051^1
ratio   196.1807578    over  Ram   3.313528093       log n   1088.859928     bump   1061^1
ratio   197.2345536    over  Ram   3.331326893       log n   1095.828779     bump   1063^1
ratio   198.289547    over  Ram   3.349145919       log n   1102.803258     bump   1069^1
ratio   198.8724667    over  Ram   3.358991537       log n   1106.653405     bump   47^2
ratio   199.9229823    over  Ram   3.281810856       log n   1113.644582     bump   1087^1
ratio   200.9761418    over  Ram   3.299098865       log n   1120.639432     bump   1091^1
ratio   202.0333959    over  Ram   3.316454089       log n   1127.636114     bump   1093^1
ratio   203.093304    over  Ram   3.333852877       log n   1134.636448     bump   1097^1
ratio   204.1544227    over  Ram   3.35127154       log n   1141.642237     bump   1103^1
ratio   205.2167544    over  Ram   3.368710113       log n   1148.653451     bump   1109^1
ratio   206.2788728    over  Ram   3.291415684       log n   1155.671853     bump   1117^1
ratio   207.3422073    over  Ram   3.308382405       log n   1162.695612     bump   1123^1
ratio   208.4067603    over  Ram   3.325368567       log n   1169.724699     bump   1129^1
ratio   209.4614029    over  Ram   3.342196597       log n   1176.773086     bump   1151^1
ratio   210.5200032    over  Ram   3.359087778       log n   1183.823208     bump   1153^1
ratio   211.577085    over  Ram   3.375954729       log n   1190.881966     bump   1163^1
ratio   212.6340265    over  Ram   3.298354947       log n   1197.94758     bump   1171^1
ratio   212.7979734    over  Ram   3.300898072       log n   1199.046192     bump   3^8
ratio   213.8542558    over  Ram   3.317283006       log n   1206.120309     bump   1181^1
ratio   214.9117545    over  Ram   3.333686807       log n   1213.199493     bump   1187^1
ratio   215.9704714    over  Ram   3.350109503       log n   1220.28372     bump   1193^1
ratio   217.0290842    over  Ram   3.366530584       log n   1227.37463     bump   1201^1
ratio   218.0849831    over  Ram   3.289163364       log n   1234.475481     bump   1213^1
ratio   219.1434026    over  Ram   3.305126474       log n   1241.579626     bump   1217^1
ratio   219.2466955    over  Ram   3.30668434       log n   1242.272773     bump   2^13
ratio   220.3068355    over  Ram   3.322673399       log n   1249.381835     bump   1223^1
ratio   221.3681987    over  Ram   3.338680907       log n   1256.495791     bump   1229^1
ratio   222.4333755    over  Ram   3.354745931       log n   1263.611373     bump   1231^1
ratio   223.4997818    over  Ram   3.370829498       log n   1270.731817     bump   1237^1
ratio   224.5635702    over  Ram   3.29345261       log n   1277.861916     bump   1249^1
ratio   225.6260484    over  Ram   3.30903493       log n   1284.999989     bump   1259^1
ratio   226.6822481    over  Ram   3.324525172       log n   1292.152258     bump   1277^1
ratio   227.7421474    over  Ram   3.34006967       log n   1299.306092     bump   1279^1
ratio   228.8045116    over  Ram   3.355650319       log n   1306.463048     bump   1283^1
ratio   229.8681032    over  Ram   3.371248969       log n   1313.62467     bump   1289^1
ratio   230.9353971    over  Ram   3.293907657       log n   1320.787842     bump   1291^1
ratio   232.0039242    over  Ram   3.309148411       log n   1327.955652     bump   1297^1
ratio   233.074919    over  Ram   3.324424362       log n   1335.12654     bump   1301^1
ratio   234.1496189    over  Ram   3.339753162       log n   1342.298965     bump   1303^1
ratio   235.2267951    over  Ram   3.355117282       log n   1349.474454     bump   1307^1
ratio   236.3015346    over  Ram   3.370446645       log n   1356.659084     bump   1319^1
ratio   237.3799583    over  Ram   3.293283751       log n   1363.845228     bump   1321^1
ratio   238.4596279    over  Ram   3.308262515       log n   1371.035904     bump   1327^1
ratio   239.0477073    over  Ram   3.316421218       log n   1375.006196     bump   53^2
ratio   240.114527    over  Ram   3.331221707       log n   1382.222171     bump   1361^1
ratio   241.1825781    over  Ram   3.346039283       log n   1389.442545     bump   1367^1
ratio   242.251862    over  Ram   3.360873961       log n   1396.667298     bump   1373^1
ratio   243.3212169    over  Ram   3.375709625       log n   1403.897861     bump   1381^1
ratio   244.3849109    over  Ram   3.298208543       log n   1411.141374     bump   1399^1
ratio   245.4475631    over  Ram   3.312550053       log n   1418.39201     bump   1409^1
ratio   246.5069511    over  Ram   3.326847509       log n   1425.652532     bump   1423^1
ratio   247.568674    over  Ram   3.341176477       log n   1432.915862     bump   1427^1
ratio   248.6338503    over  Ram   3.355552052       log n   1440.180592     bump   1429^1
ratio   249.7013693    over  Ram   3.369959244       log n   1447.448117     bump   1433^1
ratio   250.7701174    over  Ram   3.292696017       log n   1454.719821     bump   1439^1
ratio   251.8389861    over  Ram   3.306730623       log n   1461.997069     bump   1447^1
ratio   252.9101907    over  Ram   3.320795899       log n   1469.277077     bump   1451^1
ratio   253.9848409    over  Ram   3.334906418       log n   1476.558463     bump   1453^1
ratio   255.060727    over  Ram   3.349033165       log n   1483.843969     bump   1459^1
ratio   256.1345535    over  Ram   3.363132868       log n   1491.137667     bump   1471^1
ratio   257.2074353    over  Ram   3.377220167       log n   1498.43814     bump   1481^1
ratio   258.2837212    over  Ram   3.299876046       log n   1505.739962     bump   1483^1
ratio   259.3623296    over  Ram   3.313656527       log n   1513.044478     bump   1487^1
ratio   260.4443511    over  Ram   3.327480615       log n   1520.350338     bump   1489^1
ratio   261.5287026    over  Ram   3.341334471       log n   1527.658881     bump   1493^1
ratio   262.6142983    over  Ram   3.355204224       log n   1534.971435     bump   1499^1
ratio   263.6979009    over  Ram   3.369048512       log n   1542.291961     bump   1511^1
ratio   264.7795384    over  Ram   3.382867694       log n   1549.620399     bump   1523^1
ratio   265.8613535    over  Ram   3.305462762       log n   1556.954075     bump   1531^1
ratio   266.9412466    over  Ram   3.318889108       log n   1564.295559     bump   1543^1
ratio   268.0223746    over  Ram   3.332330807       log n   1571.640924     bump   1549^1
ratio   269.1057836    over  Ram   3.345800867       log n   1578.988868     bump   1553^1
ratio   270.1904307    over  Ram   3.359286319       log n   1586.340668     bump   1559^1
ratio   271.275277    over  Ram   3.372774248       log n   1593.697586     bump   1567^1
ratio   272.3623985    over  Ram   3.295729417       log n   1601.057054     bump   1571^1
ratio   273.4497264    over  Ram   3.308886662       log n   1608.421601     bump   1579^1
ratio   274.5393257    over  Ram   3.322071391       log n   1615.788678     bump   1583^1
ratio   275.6260663    over  Ram   3.335221528       log n   1623.16456     bump   1597^1
ratio   276.7150635    over  Ram   3.348398972       log n   1630.542943     bump   1601^1
ratio   277.8053005    over  Ram   3.361591417       log n   1637.925068     bump   1607^1
ratio   278.8988123    over  Ram   3.37482349       log n   1645.308436     bump   1609^1
ratio   279.9945861    over  Ram   3.297853938       log n   1652.694287     bump   1613^1
ratio   280.4154458    over  Ram   3.302810941       log n   1655.5275     bump   17^3
ratio   281.5141146    over  Ram   3.315751367       log n   1662.917064     bump   1619^1
ratio   282.6160672    over  Ram   3.328730471       log n   1670.307863     bump   1621^1
ratio   283.7192721    over  Ram   3.341724324       log n   1677.702356     bump   1627^1
ratio   284.8217054    over  Ram   3.35470909       log n   1685.102977     bump   1637^1
ratio   285.9183887    over  Ram   3.367626129       log n   1692.515741     bump   1657^1
ratio   287.0163142    over  Ram   3.380557801       log n   1699.932119     bump   1663^1
ratio   287.6212267    over  Ram   3.387682628       log n   1704.009656     bump   59^2