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Consider the sum of $x$-th powers of the divisor function -- in other words, $$\sigma_x(n) =\sum_{d|n} d^x.$$ There is a lot of results on $\sigma_0$ and $\sigma_1,$ but I am interested in $\sigma_x$ for $x < 0.$ (for example, if $x<-1,$ the sums are obviously uniformly bounded, but one can still say something, the case I am actually interested in has $-1 < x < 0$).

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3  
I think we can understand these functions pretty well - but it's impossible to contribute much without knowing what you actually want to know about $\sigma_x$. Can you ask a specific question? –  Greg Martin Jul 18 '13 at 0:01

5 Answers 5

Preliminary. Part 1: for nonnegative integer $x,$ the Ramanujan procedure is to find the maximum of $$ \frac{\sigma_x(n)}{n^{x+\delta}} $$ for $0 < \delta < 1,$ which occurs at a single value of $n$ for all but a countable set of $\delta'$s.

Part 2: In a procedure done repeatedly by J.-L. Nicolas and Guy Robin, certain envelope curves can then be drawn around the extremal pairs $(n, \sigma_x(n)).$ This gives the first few terms in an asymptotic expansion.

I can do part 1, and have posted that plenty of times on MO as answers, Which $n$ maximize $G(n)=\frac{\sigma(n)}{n \log \log n}$? . Part 2, which I've never done, gives effective versions of, for example, Gronwall's inequality. http://en.wikipedia.org/wiki/Colossally_abundant_number Theorem 323 in Hardy and Wright. There is a complete discussion for $\sigma_0$ in a survey by Nicolas, email me if you'd like a pdf; especially pages 229-230 in On Highly Composite Numbers (1988) in the book Ramanujan Revisited.

EDIT 1 am. I noticed that, for real $t \geq 0,$ we get $$ \frac{\sigma_t(n)}{n^{t+\delta}} = \frac{\sigma_{-t}(n)}{n^{\delta}} $$ because it works when $n$ is a prime power and the $\sigma$ functions are number-theoretic multiplicative. So, perhaps the quantity to optimize for given $\delta$ is not quite what I first thought. If that still seems true by daylight, then I already know what happens for $t=1,0,$ and might fiddle with $t = 1/2.$

The workhorse here is Theorem 316 in Hardy and Wright, that a multiplicative function that goes to zero on prime powers goes to zero over the positive integers. Which is why the displayed quantities, which are equal to $1$ for $n=1,$ have a maximum that is achieved at a finite number of positive integers. The next thing is you choose the largest of this finite set of integers. Then you work out the prime factorization of this, for your case both $x,\delta$ will be involved in finding the exponent for any given prime. But it will still be true that the exponents will (non-strictly) decrease, the number will be the product of primorials.

I did a simple computer run. Beginning with n=1, I computed $\sigma_{1/2}(n)$ and $\frac{\sigma_{1/2}(n)}{\sqrt n}.$ I told it to print out only when $\frac{\sigma_{1/2}(n)}{\sqrt n}$ achieved a new record. This is analogous to the "superabundant" numbers of Alaoglu and Erdos. A subsequence of these give the analogy to the colossally abundant numbers, that is when you throw in the $\delta.$

sigma_(1/2)(n)/sqrt(n)   sigma_(1/2)(n)        n
1.000000000000000     1.000000000000000      1 =  1 
1.707106781186547     2.414213562373095      2 = 2
2.207106781186547     4.414213562373095      4 = 2^2
2.692705340840036     6.59575411272515       6 = 2 * 3
3.48138047543485     12.05985572786291      12 = 2^2 * 3
4.039058011260055    19.78726233817545      24 = 2^3 * 3
4.217082735830365    25.30249641498219      36 = 2^2 * 3^2
4.433395578557461    30.71546556845096      48 = 2^4 * 3
5.038301155157422    39.02651293420457      60 = 2^2 * 3 * 5
5.845379666908572    64.03292601495718     120 = 2^3 * 3 * 5
6.103019468641861    81.88059839932708     180 = 2^2 * 3^2 * 5
6.416070355517759    99.39733454006152     240 = 2^4 * 3 * 5
7.080653738260245   134.3459588153286      360 = 2^3 * 3^2 * 5
7.771945559827761   208.5431830704304      720 = 2^4 * 3^2 * 5
8.054725512250524   233.4481239703321      840 = 2^3 * 3 * 5 * 7
8.409744005872138   298.5162989651        1260 = 2^2 * 3^2 * 5 * 7
8.841117006231155   362.3779627157551     1680 = 2^4 * 3 * 5 * 7
9.756889297002594   489.7919554872137     2520 = 2^3 * 3^2 * 5 * 7
10.70946486760447   760.2965830926046     5040 = 2^4 * 3^2 * 5 * 7
10.73963401633403   933.7924958813286     7560 = 2^3 * 3^3 * 5 * 7
11.3830375131697   1142.847896136832     10080 = 2^5 * 3^2 * 5 * 7
11.7881560083082   1449.51185086302      15120 = 2^4 * 3^3 * 5 * 7
11.85932529847063  1683.857151347613     20160 = 2^6 * 3^2 * 5 * 7
12.18947651217511  1935.0194077967       25200 = 2^4 * 3^2 * 5^2 * 7
12.69870210783823  2114.248097172755     27720 = 2^3 * 3^2 * 5 * 7 * 11
12.95613460796823  2908.645005596511     50400 = 2^5 * 3^2 * 5^2 * 7
13.93849001954421  3281.915078600041     55440 = 2^4 * 3^2 * 5 * 7 * 11
13.97775550887199  4030.831836769185     83160 = 2^3 * 3^3 * 5 * 7 * 11
14.81515245914461  4933.245560069761    110880 = 2^5 * 3^2 * 5 * 7 * 11
15.34241877646565  6256.998789349251    166320 = 2^4 * 3^3 * 5 * 7 * 11
15.4350464149976   7268.579522924334    221760 = 2^6 * 3^2 * 5 * 7 * 11
15.8647419650601   8352.752745514061    277200 = 2^4 * 3^2 * 5^2 * 7 * 11
jagy@phobeusjunior:~$ 
    jagy@phobeusjunior:~$ date
Wed Jul 17 20:28:17 PDT 2013
jagy@phobeusjunior:~$ 
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How do you chose $\delta$ for CA numbers? –  joro Jul 20 '13 at 11:24
    
@joro, posted a second answer with what you requested. –  Will Jagy Jul 21 '13 at 4:34

Given a fixed $t \geq 0$ and $\delta > 0,$ the maximum of $$ \frac{\sigma_t(n)}{n^{t+\delta}} $$ occurs when $n = \prod p^{k_p}$ and $$ k_p = \left\lfloor \frac{\log \left(p^{t + \delta} -1 \right) - \log \left(p^{ \delta} -1 \right) }{t \log p} \right\rfloor - 1. $$ This is correct, it appears as formula (311) on journal page 130 in The Ramanujan Journal, Volume 1, Issue 2, June 1997, pages 119-153, annotations by Nicolas and Robin on Ramanujan's Lost Notebook. The bad news is that I am unable to extract an elementary bound for my $t=1/2,$ even though Ramanujan put something elaborate as formula (380) on journal page 143. Also, it would appear he is assuming RH in (380). He says pretty much that in the paragraph including formula (383).

It was possible to invert the thing, for a prime factor $p$ and desired exponent $k$ the optimal (largest) value of $\delta$ that works is $$ \delta = \left( \frac{\log \left(p^{kt + t} -1 \right) - \log \left(p^{ kt} -1 \right) }{ \log p} \right) - t. $$

For $t = \frac{1}{2}$ as in the other answer, $$ f(0.9) = 1, $$ $$ f(0.7) = 2, $$ $$ f(0.4) = 6 = 2 \cdot 3, $$ $$ f(0.3) = 12 = 2^2 \cdot 3, $$ $$ f(0.22) = 60 = 2^2 \cdot 3 \cdot 5, $$ $$ f(0.2) = 120 = 2^3 \cdot 3 \cdot 5, $$ $$ f(0.17) = 360 = 2^3 \cdot 3^2 \cdot 5, $$ $$ f(0.15) = 2520 = 2^3 \cdot 3^2 \cdot 5 \cdot 7, $$ $$ f(0.12) = 5040 = 2^4 \cdot 3^2 \cdot 5 \cdot 7, $$ $$ f(0.1) = 55440 = 2^4 \cdot 3^2 \cdot 5 \cdot 7 \cdot 11. $$ What you do is make a list of the $\delta$ values down to some bound. If you then take actual values of $\delta$ to be rational and between consecutive borderline values, the next "CA" is what we want, the previous one multiplied by a single prime, so that only one exponent increases.

We chose things so that, if there should be more than one $n$ achieving the maximum of $ \frac{\sigma_t(n)}{n^{t+\delta}}, $ we pick the largest. With that detail cared for, it follows that these numbers automatically give new maxima of $ \frac{\sigma_t(n)}{n^{t}}. $ So, the $n$ values immediately above are readily found in the C++ list I posted in my first answer.

I figured out a good way to program the increasing "CA" type numbers so that I would not need to keep typing in values of $\delta$ by hand. The results are quite different from the actual CA numbers $t=1.$ With my $t=1/2,$ the ratio $ \frac{\sigma_t(n)}{n^{t}}$ is not only much larger than $\log \log n,$ it grows faster than $\log n.$ Go figure. Thinking again, all we really know is that $ \frac{\sigma_t(n)}{n^{t}} = o(n^\delta)$ for all $\delta > 0.$ That still leaves a variety of slowly growing possibilities.

Wednesday, July 24: from Ramanujan's (380) and comments at (383) about RH or not, I get $$ e^{2 \sqrt {\log n} / \log \log n} < \frac{\sigma_{1/2}(n)}{\sqrt n} < e^{\sqrt {\log n}} $$ for these "generalised superior highly composite numbers," and the upper bound holds for all $n.$

==================================
 ratio   1.0                                               1 
 ratio   1.707106781186547                                 2 bump   2^1
 ratio   2.692705340840036    log n   1.791759469228055    6 bump   3^1
 ratio   3.481380475434849    log n   2.484906649788    12 bump   2^2
 ratio   5.038301155157422    log n   4.0943445622221    60 bump   5^1
 ratio   5.845379666908575    log n   4.787491742782046    120 bump   2^3
 ratio   7.080653738260244    log n   5.886104031450156    360 bump   3^2
 ratio   9.756889297002592    log n   7.832014180505469    2520 bump   7^1
 ratio   10.70946486760446    log n   8.525161361065415    5040 bump   2^4
 ratio   13.9384900195442    log n   10.92305663386379    55440 bump   11^1
 ratio   17.80433159400648    log n   13.48800599132532    720720 bump   13^1
 ratio   18.9241364472417    log n   14.18115317188527    1441440 bump   2^5
 ratio   20.8302352657604    log n   15.27976546055338    4324320 bump   3^3
 ratio   23.70890298012282    log n   16.88920337298748    21621600 bump   5^2
 ratio   29.45915658333269    log n   19.72241671704369    367567200 bump   17^1
 ratio   36.21755061010575    log n   22.66685569621013    6983776800 bump   19^1
 ratio   43.76943195415439    log n   25.80234991213928    160626866400 bump   23^1
 ratio   45.60082764139564    log n   26.49549709269923    321253732800 bump   2^6
 ratio   50.3283834502508    log n   28.44140724175454     bump   7^2
 ratio   59.67412963494946    log n   31.80870307174101     bump   29^1
 ratio   70.39191971121947    log n   35.24269027622616     bump   31^1
 ratio   74.11081045086605    log n   36.34130256489427     bump   3^4
 ratio   86.29455263736772    log n   39.9522204775385     bump   37^1
 ratio   88.74518320255322    log n   40.64536765809844     bump   2^7
 ratio   102.6048523127895    log n   44.35893972480275     bump   41^1
 ratio   118.2519455996779    log n   48.12013984049631     bump   43^1
 ratio   135.5007646391844    log n   51.97028744220637     bump   47^1
 ratio   142.858375614814    log n   53.57972535464047     bump   5^3
 ratio   162.4814826540585    log n   57.55001726819259     bump   53^1
 ratio   183.63476993534    log n   61.62755471209831     bump   59^1
 ratio   207.146792925056    log n   65.73842857627162     bump   61^1
 ratio   221.6157614115811    log n   68.13632384907001     bump   11^2
 ratio   225.94307832489    log n   68.82947102962996     bump   2^8
 ratio   253.546418659995    log n   73.03416364902093     bump   67^1
 ratio   283.6368425766372    log n   77.29684352606225     bump   71^1
 ratio   291.8542320483043    log n   78.39545581473034     bump   3^5
 ratio   326.0131861812835    log n   82.68591525587873     bump   73^1
 ratio   362.6925353127741    log n   87.05536310834574     bump   79^1
 ratio   402.5032130551653    log n   91.47420371614234     bump   83^1
 ratio   426.742288673805    log n   94.03915307360388     bump   13^2
 ratio   471.9768808041347    log n   98.52778944333602     bump   89^1
 ratio   519.8988723997552    log n   103.1025004218394     bump   97^1
 ratio   571.6307437183547    log n   107.7176209386807     bump   101^1
 ratio   627.9551945268071    log n   112.3523499269103     bump   103^1
 ratio   688.6619145302903    log n   117.0251787613722     bump   107^1
 ratio   754.6237628670738    log n   121.7165266436014     bump   109^1
 ratio   764.8433920960285    log n   122.4096738241613     bump   2^9
 ratio   836.7938060950318    log n   127.1370616428736     bump   113^1
 ratio   866.5031540495891    log n   129.082971791929     bump   7^3
 ratio   943.3928754671388    log n   133.9271588783876     bump   127^1
 ratio   1025.817493699215    log n   138.8023562015887     bump   131^1
 ratio   1113.458996815389    log n   143.7223371274168     bump   137^1
 ratio   1207.901356290527    log n   148.6568110605475     bump   139^1
 ratio   1265.0852466568    log n   151.4900244046037     bump   17^2
 ratio   1368.725068280695    log n   156.4939707105492     bump   149^1
 ratio   1480.110334660455    log n   161.5112505473641     bump   151^1
 ratio   1598.235999226204    log n   166.5674963527124     bump   157^1
 ratio   1723.419504020291    log n   171.6612465535192     bump   163^1
 ratio   1751.43499592212    log n   172.7598588421873     bump   3^6
 ratio   1886.965104933907    log n   177.8778526546041     bump   167^1
 ratio   1967.746528300427    log n   180.8222916337705     bump   19^2
 ratio   2117.351526509843    log n   185.9755832282683     bump   173^1
 ratio   2166.119990568261    log n   187.5850211407024     bump   5^4
 ratio   2328.023400174482    log n   192.7724069465432     bump   179^1
 ratio   2501.064016742093    log n   197.970903977809     bump   181^1
 ratio   2524.69449062345    log n   198.6640511583689     bump   2^10
 ratio   2707.374972577132    log n   203.9163245864156     bump   191^1
 ratio   2902.256087421516    log n   209.1790147753204     bump   193^1
 ratio   3109.033271959751    log n   214.4622185040584     bump   197^1
 ratio   3329.426797183535    log n   219.7555233287829     bump   199^1
 ratio   3558.633990417844    log n   225.107381462259     bump   211^1
 ratio   3796.937748395296    log n   230.5145532337191     bump   223^1
 ratio   4048.949358025683    log n   235.9395032512005     bump   227^1
 ratio   4316.511462834089    log n   241.3732252547547     bump   229^1
 ratio   4599.295531024331    log n   246.8242637083204     bump   233^1
 ratio   4896.79922729686    log n   252.300727260252     bump   239^1
 ratio   5212.229794338636    log n   257.7855241937426     bump   241^1
 ratio   5399.748192195057    log n   260.9210184096717     bump   23^2
 ratio   5740.577275900156    log n   266.4464713488035     bump   251^1
 ratio   6098.664647912183    log n   271.9955474336987     bump   257^1
 ratio   6474.724420321691    log n   277.5677014658765     bump   263^1
 ratio   6869.495335582333    log n   283.1624128454783     bump   269^1
 ratio   7286.787459253727    log n   288.764531666358     bump   271^1
 ratio   7724.608085351445    log n   294.3885491725454     bump   277^1
 ratio   8185.419570078661    log n   300.0269038418791     bump   281^1
 ratio   8671.992258071368    log n   305.6723507395224     bump   283^1
 ratio   9178.615455235302    log n   311.3525233485394     bump   293^1
 ratio   9702.466708693246    log n   317.0793710961266     bump   307^1
 ratio   10252.64312072074    log n   322.8191640083058     bump   311^1
 ratio   10832.15677315473    log n   328.565367198846     bump   313^1
 ratio   10903.84759147919    log n   329.2585143794059     bump   2^11
 ratio   11516.26868611524    log n   335.0174161532832     bump   317^1
 ratio   12149.26006039437    log n   340.8195345286603     bump   331^1
 ratio   12811.07242681962    log n   346.6396174590126     bump   337^1
 ratio   13498.80682851644    log n   352.4889422389595     bump   347^1
 ratio   14221.3813917337    log n   358.344014161162     bump   349^1
 ratio   14978.30914787333    log n   364.2104822180953     bump   353^1
 ratio   15768.8340694372    log n   370.0938046065835     bump   359^1
 ratio   16591.96050763988    log n   375.9991664546381     bump   367^1
 ratio   17451.05965781483    log n   381.9207448742819     bump   373^1
 ratio   17958.576773283    log n   385.2880407042684     bump   29^2
 ratio   18881.04689953689    log n   391.2255769093509     bump   379^1
 ratio   19845.82332762504    log n   397.1736118985315     bump   383^1
 ratio   20029.10233435938    log n   398.2722241871996     bump   3^7
 ratio   21044.61813441118    log n   404.2358035308181     bump   389^1
 ratio   22100.81724216603    log n   410.2197398115053     bump   397^1
 ratio   23204.47938775229    log n   416.2137012388118     bump   401^1
 ratio   23661.26499660484    log n   418.6115965116102     bump   11^3
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This was studied by Ramanujan in 1915, but the relevant portion of the article was left out due to paper shortages. The whole thing, with annotation by Jean-Louis Nicolas and Guy Robin, appeared in Volume 1, Issue 2, June 1997 of the Ramanujan Journal, pages 119-153. I now have a copy of that, email me if you would like to see it. Meanwhile, additional explication is in part 3 of Ramanujan's Lost Notebook by Bruce Berndt and George Andrews. The call number says 2005 but part 3 did not really appear until 2012 or the like.

Some explanation is HERE

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Alright, still doing $t=1/2.$ Ramanujan's upper bound in this case is essentially a constant times $$ e^{\operatorname{Li}(\sqrt {\log n})} $$ where $\operatorname{Li}$ refers to the Logarithmic Integral function. That works very very well, here is about as much as is likely to fit.

Ram(n) = exp( LogIntegral (sqrt(log n)))
==================================
 sigma_{1/2}(n) / sqrt(n)                              log n                 n
 ratio   1.0                                                                 1 
 ratio   1.707106781                                                         2 bump   2^1
 ratio   2.692705341                                  log n   1.791759469    6 bump   3^1
 ratio   3.481380475    over  Ram   3.481380475       log n   2.48490665     12 bump   2^2
 ratio   5.038301155    over  Ram   1.771639117       log n   4.094344562    60 bump   5^1
 ratio   5.845379667    over  Ram   1.568960458       log n   4.787491743    120 bump   2^3
 ratio   7.080653738    over  Ram   1.412339983       log n   5.886104031    360 bump   3^2
 ratio   9.756889297    over  Ram   1.353008034       log n   7.832014181    2520 bump   7^1
 ratio   10.70946487    over  Ram   1.349844052       log n   8.525161361    5040 bump   2^4
 ratio   13.93849002    over  Ram   1.232899175       log n   10.92305663    55440 bump   11^1
 ratio   17.80433159    over  Ram   1.18876701        log n   13.48800599    720720 bump   13^1
 ratio   18.92413645    over  Ram   1.170551948       log n   14.18115317    1441440 bump   2^5
 ratio   20.83023527    over  Ram   1.152137835       log n   15.27976546    4324320 bump   3^3
 ratio   23.70890298    over  Ram   1.135153953       log n   16.88920337    21621600 bump   5^2
 ratio   29.45915658    over  Ram   1.108375519       log n   19.72241672    367567200 bump   17^1
 ratio   36.21755061    over  Ram   1.119404161       log n   22.6668557     6983776800 bump   19^1
 ratio   43.76943195    over  Ram   1.08610104        log n   25.80234991    160626866400 bump   23^1
 ratio   45.60082764    over  Ram   1.097447439       log n   26.49549709    321253732800 bump   2^6
 ratio   50.32838345    over  Ram   1.073591889       log n   28.44140724     bump   7^2
 ratio   59.67412963    over  Ram   1.067513819       log n   31.80870307     bump   29^1
 ratio   70.39191971    over  Ram   1.061652756       log n   35.24269028     bump   31^1
 ratio   74.11081045    over  Ram   1.057066907       log n   36.34130256     bump   3^4
 ratio   86.29455264    over  Ram   1.07290295       log n   39.95222048     bump   37^1
 ratio   88.7451832    over  Ram   1.045265396       log n   40.64536766     bump   2^7
 ratio   102.6048523    over  Ram   1.057693019       log n   44.35893972     bump   41^1
 ratio   118.2519456    over  Ram   1.042381522       log n   48.12013984     bump   43^1
 ratio   135.5007646    over  Ram   1.051151424       log n   51.97028744     bump   47^1
 ratio   142.8583756    over  Ram   1.053673584       log n   53.57972535     bump   5^3
 ratio   162.4814827    over  Ram   1.032100959       log n   57.55001727     bump   53^1
 ratio   183.6347699    over  Ram   1.032202175       log n   61.62755471     bump   59^1
 ratio   207.1467929    over  Ram   1.032272747       log n   65.73842858     bump   61^1
 ratio   221.6157614    over  Ram   1.028280049       log n   68.13632385     bump   11^2
 ratio   225.9430783    over  Ram   1.023845032       log n   68.82947103     bump   2^8
 ratio   253.5464187    over  Ram   1.021753702       log n   73.03416365     bump   67^1
 ratio   283.6368426    over  Ram   1.018109822       log n   77.29684353     bump   71^1
 ratio   291.854232    over  Ram   1.023825767       log n   78.39545581     bump   3^5
 ratio   326.0131862    over  Ram   1.020509877       log n   82.68591526     bump   73^1
 ratio   362.6925353    over  Ram   1.014505832       log n   87.05536311     bump   79^1
 ratio   402.5032131    over  Ram   1.029940706       log n   91.47420372     bump   83^1
 ratio   426.7422887    over  Ram   1.021972025       log n   94.03915307     bump   13^2
 ratio   471.9768808    over  Ram   1.013154643       log n   98.52778944     bump   89^1
 ratio   519.8988724    over  Ram   1.023369759       log n   103.1025004     bump   97^1
 ratio   571.6307437    over  Ram   1.010705245       log n   107.7176209     bump   101^1
 ratio   627.9551945    over  Ram   1.019758993       log n   112.3523499     bump   103^1
 ratio   688.6619145    over  Ram   1.027848207       log n   117.0251788     bump   107^1
 ratio   754.6237629    over  Ram   1.014482034       log n   121.7165266     bump   109^1
 ratio   764.8433921    over  Ram   1.028220841       log n   122.4096738     bump   2^9
 ratio   836.7938061    over  Ram   1.014259317       log n   127.1370616     bump   113^1
 ratio   866.503154    over  Ram   1.028853624       log n   129.0829718     bump   7^3
 ratio   943.3928755    over  Ram   1.03192572       log n   133.9271589     bump   127^1
 ratio   1025.817494    over  Ram   1.013547025       log n   138.8023562     bump   131^1
 ratio   1113.458997    over  Ram   1.014787307       log n   143.7223371     bump   137^1
 ratio   1207.901356    over  Ram   1.016000069       log n   148.6568111     bump   139^1
 ratio   1265.085247    over  Ram   1.022465256       log n   151.4900244     bump   17^2
 ratio   1368.725068    over  Ram   1.021751705       log n   156.4939707     bump   149^1
 ratio   1480.110335    over  Ram   1.021037044       log n   161.5112505     bump   151^1
 ratio   1598.235999    over  Ram   1.019338315       log n   166.5674964     bump   157^1
 ratio   1723.419504    over  Ram   1.016727162       log n   171.6612466     bump   163^1
 ratio   1751.434996    over  Ram   1.01338148       log n   172.7598588     bump   3^6
 ratio   1886.965105    over  Ram   1.010481656       log n   177.8778527     bump   167^1
 ratio   1967.746528    over  Ram   1.01391026       log n   180.8222916     bump   19^2
 ratio   2117.351527    over  Ram   1.010410186       log n   185.9755832     bump   173^1
 ratio   2166.119991    over  Ram   1.014110208       log n   187.5850211     bump   5^4
 ratio   2328.0234    over  Ram   1.009942177       log n   192.7724069     bump   179^1
 ratio   2501.064017    over  Ram   1.025020452       log n   197.970904     bump   181^1
 ratio   2524.694491    over  Ram   1.015324731       log n   198.6640512     bump   2^10
 ratio   2707.374973    over  Ram   1.009735141       log n   203.9163246     bump   191^1
 ratio   2902.256087    over  Ram   1.023186675       log n   209.1790148     bump   193^1
 ratio   3109.033272    over  Ram   1.017197193       log n   214.4622185     bump   197^1
 ratio   3329.426797    over  Ram   1.030215169       log n   219.7555233     bump   199^1
 ratio   3558.63399    over  Ram   1.022557846       log n   225.1073815     bump   211^1
 ratio   3796.937748    over  Ram   1.013542737       log n   230.5145532     bump   223^1
 ratio   4048.949358    over  Ram   1.022952412       log n   235.9395033     bump   227^1
 ratio   4316.511463    over  Ram   1.01371957       log n   241.3732253     bump   229^1
 ratio   4599.295531    over  Ram   1.022769297       log n   246.8242637     bump   233^1
 ratio   4896.799227    over  Ram   1.012810118       log n   252.3007273     bump   239^1
 ratio   5212.229794    over  Ram   1.02124556       log n   257.7855242     bump   241^1
 ratio   5399.748192    over  Ram   1.020590162       log n   260.9210184     bump   23^2
 ratio   5740.577276    over  Ram   1.0281473       log n   266.4464713     bump   251^1
 ratio   6098.664648    over  Ram   1.016911652       log n   271.9955474     bump   257^1
 ratio   6474.72442    over  Ram   1.023456423       log n   277.5677015     bump   263^1
 ratio   6869.495336    over  Ram   1.011472914       log n   283.1624128     bump   269^1
 ratio   7286.787459    over  Ram   1.017504896       log n   288.7645317     bump   271^1
 ratio   7724.608085    over  Ram   1.023103218       log n   294.3885492     bump   277^1
 ratio   8185.41957    over  Ram   1.028482906       log n   300.0269038     bump   281^1
 ratio   8671.992258    over  Ram   1.015937852       log n   305.6723507     bump   283^1
 ratio   9178.615455    over  Ram   1.020467512       log n   311.3525233     bump   293^1
 ratio   9702.466709    over  Ram   1.02387066       log n   317.0793711     bump   307^1
 ratio   10252.64312    over  Ram   1.027083921       log n   322.819164     bump   311^1
 ratio   10832.15677    over  Ram   1.012657618       log n   328.5653672     bump   313^1
 ratio   10903.84759    over  Ram   1.019359723       log n   329.2585144     bump   2^11
 ratio   11516.26869    over  Ram   1.022392289       log n   335.0174162     bump   317^1
 ratio   12149.26006    over  Ram   1.024417235       log n   340.8195345     bump   331^1
 ratio   12811.07243    over  Ram   1.026115233       log n   346.6396175     bump   337^1
 ratio   13498.80683    over  Ram   1.027190785       log n   352.4889422     bump   347^1
 ratio   14221.38139    over  Ram   1.010924009       log n   358.3440142     bump   349^1
 ratio   14978.30915    over  Ram   1.011870382       log n   364.2104822     bump   353^1
 ratio   15768.83407    over  Ram   1.012525211       log n   370.0938046     bump   359^1
 ratio   16591.96051    over  Ram   1.01275914       log n   375.9991665     bump   367^1
 ratio   17451.05966    over  Ram   1.012720867       log n   381.9207449     bump   373^1
 ratio   17958.57677    over  Ram   1.007730322       log n   385.2880407     bump   29^2
 ratio   18881.0469    over  Ram   1.007515705       log n   391.2255769     bump   379^1
 ratio   19845.82333    over  Ram   1.007171854       log n   397.1736119     bump   383^1
 ratio   20029.10233    over  Ram   1.01647323       log n   398.2722242     bump   3^7
 ratio   21044.61813    over  Ram   1.015871757       log n   404.2358035     bump   389^1
 ratio   22100.81724    over  Ram   1.01490067       log n   410.2197398     bump   397^1
 ratio   23204.47939    over  Ram   1.013812709       log n   416.2137012     bump   401^1
 ratio   23661.265    over  Ram   1.016777275       log n   418.6115965     bump   11^3
 ratio   24831.23925    over  Ram   1.015376378       log n   424.6253117     bump   409^1
 ratio   26044.32467    over  Ram   1.013524801       log n   430.6631826     bump   419^1
 ratio   27313.64739    over  Ram   1.011683822       log n   436.7058154     bump   421^1
 ratio   28060.57977    over  Ram   1.005740132       log n   440.1398026     bump   31^2
 ratio   29412.21034    over  Ram   1.020147918       log n   446.2059107     bump   431^1
 ratio   30825.67097    over  Ram   1.017909089       log n   452.2766484     bump   433^1
 ratio   32296.89997    over  Ram   1.015470916       log n   458.3611479     bump   439^1
 ratio   33831.372    over  Ram   1.012943254       log n   464.4547176     bump   443^1
 ratio   35427.97315    over  Ram   1.010226526       log n   470.5617405     bump   449^1
 ratio   37085.22384    over  Ram   1.007228192       log n   476.6864239     bump   457^1
 ratio   38812.45491    over  Ram   1.020533436       log n   482.8198219     bump   461^1
 ratio   40616.22266    over  Ram   1.017391791       log n   488.957549     bump   463^1
 ratio   42495.7173    over  Ram   1.014174244       log n   495.1038783     bump   467^1
 ratio   44437.39618    over  Ram   1.010509411       log n   501.2755789     bump   479^1
 ratio   46451.04681    over  Ram   1.006599572       log n   507.463843     bump   487^1
 ratio   48547.35321    over  Ram   1.018821121       log n   513.6602871     bump   491^1
 ratio   50720.63121    over  Ram   1.014520055       log n   519.8728932     bump   499^1
 ratio   52982.15238    over  Ram   1.010166651       log n   526.0934834     bump   503^1
 ratio   55330.545    over  Ram   1.021827206       log n   532.3259314     bump   509^1
 ratio   57754.62018    over  Ram   1.016854264       log n   538.5816814     bump   521^1
 ratio   60280.0534    over  Ram   1.011923509       log n   544.8412629     bump   523^1
 ratio   60560.30131    over  Ram   1.016628042       log n   545.5344101     bump   2^12
 ratio   63163.99203    over  Ram   1.011085455       log n   551.8278294     bump   541^1
 ratio   65864.68931    over  Ram   1.021465342       log n   558.1322782     bump   547^1
 ratio   66718.23466    over  Ram   1.018471301       log n   560.0781883     bump   7^4
 ratio   69545.17699    over  Ram   1.012508114       log n   566.4007535     bump   557^1
 ratio   72476.15675    over  Ram   1.006455931       log n   572.7340332     bump   563^1
 ratio   75514.5153    over  Ram   1.016165906       log n   579.0779136     bump   569^1
 ratio   78674.69941    over  Ram   1.00996093       log n   585.4253028     bump   571^1
 ratio   81949.97    over  Ram   1.003675567       log n   591.7831451     bump   577^1
 ratio   83852.12216    over  Ram   1.011018428       log n   595.394063     bump   37^2
 ratio   87313.06963    over  Ram   1.00450419       log n   601.7690878     bump   587^1
 ratio   90898.58766    over  Ram   1.013606909       log n   608.1542822     bump   593^1
 ratio   94612.60326    over  Ram   1.006823734       log n   614.5495438     bump   599^1
 ratio   98471.93185    over  Ram   1.01577967       log n   620.9481388     bump   601^1
 ratio   99463.412    over  Ram   1.010178335       log n   622.5575767     bump   5^5
 ratio   103500.5074    over  Ram   1.003333395       log n   628.9661055     bump   607^1
 ratio   107680.8535    over  Ram   1.011985255       log n   635.3844704     bump   613^1
 ratio   112015.9214    over  Ram   1.00495314       log n   641.8093394     bump   617^1
 ratio   116518.2214    over  Ram   1.013523888       log n   648.2374447     bump   619^1
 ratio   118353.7981    over  Ram   1.01371413       log n   650.802394     bump   13^3
 ratio   123065.3885    over  Ram   1.006405511       log n   657.2496999     bump   631^1
 ratio   127926.179    over  Ram   1.014423153       log n   663.7127294     bump   641^1
 ratio   132971.0951    over  Ram   1.006887327       log n   670.1788741     bump   643^1
 ratio   138198.7282    over  Ram   1.014822511       log n   676.6512204     bump   647^1
 ratio   143606.8624    over  Ram   1.007127229       log n   683.1327975     bump   653^1
 ratio   149200.9915    over  Ram   1.014804599       log n   689.623521     bump   659^1
 ratio   155004.2375    over  Ram   1.007014379       log n   696.1172749     bump   661^1
 ratio   160979.2118    over  Ram   1.014383159       log n   702.6290202     bump   673^1
 ratio   167166.1455    over  Ram   1.006281168       log n   709.1466915     bump   677^1
 ratio   173562.5803    over  Ram   1.013456762       log n   715.6731863     bump   683^1
 ratio   180165.212    over  Ram   1.020500459       log n   722.2113262     bump   691^1
 ratio   186969.9581    over  Ram   1.011886487       log n   728.7638341     bump   701^1
 ratio   190914.212    over  Ram   1.01768071       log n   732.4774061     bump   41^2
 ratio   198084.1458    over  Ram   1.008985804       log n   739.0412617     bump   709^1
 ratio   205471.4379    over  Ram   1.015421792       log n   745.619123     bump   719^1
 ratio   213091.9517    over  Ram   1.006419022       log n   752.2080495     bump   727^1
 ratio   220962.6825    over  Ram   1.012573629       log n   758.8051952     bump   733^1
 ratio   229090.9262    over  Ram   1.018651884       log n   765.4104931     bump   739^1
 ratio   237495.4571    over  Ram   1.009404529       log n   772.0211892     bump   743^1
 ratio   246161.7894    over  Ram   1.015257689       log n   778.6425948     bump   751^1
 ratio   255108.6919    over  Ram   1.021035065       log n   785.2719581     bump   757^1
 ratio   264356.3747    over  Ram   1.011511878       log n   791.9065914     bump   761^1
 ratio   273889.3094    over  Ram   1.017069915       log n   798.5516824     bump   769^1
 ratio   283740.4232    over  Ram   1.007429186       log n   805.2019614     bump   773^1
 ratio   289465.9092    over  Ram   1.012519612       log n   808.9631616     bump   43^2
 ratio   299784.2545    over  Ram   1.017771887       log n   815.6313898     bump   787^1
 ratio   310403.1576    over  Ram   1.007735471       log n   822.3122445     bump   797^1
 ratio   321316.3515    over  Ram   1.012564193       log n   829.0080434     bump   809^1
 ratio   332599.2946    over  Ram   1.017404701       log n   835.7063114     bump   811^1
 ratio   344207.09    over  Ram   1.022087757       log n   842.4168346     bump   821^1
 ratio   356205.3949    over  Ram   1.011666913       log n   849.1297908     bump   823^1
 ratio   368591.8702    over  Ram   1.016275949       log n   855.8475955     bump   827^1
 ratio   381393.595    over  Ram   1.020895199       log n   862.5678156     bump   829^1
 ratio   394560.7641    over  Ram   1.010319514       log n   869.3000263     bump   839^1
 ratio   408070.2674    over  Ram   1.014504392       log n   876.0487859     bump   853^1
 ratio   422009.6822    over  Ram   1.018657792       log n   882.8022238     bump   857^1
 ratio   436408.4667    over  Ram   1.007867082       log n   889.5579927     bump   859^1
 ratio   451263.9837    over  Ram   1.011950678       log n   896.3184074     bump   863^1
 ratio   466502.0857    over  Ram   1.015812118       log n   903.0949144     bump   877^1
 ratio   482218.9403    over  Ram   1.019641897       log n   909.875972     bump   881^1
 ratio   498446.8997    over  Ram   1.008581453       log n   916.6592972     bump   883^1
 ratio   515183.1086    over  Ram   1.012344246       log n   923.4471422     bump   887^1
 ratio   532289.483    over  Ram   1.015783765       log n   930.2572846     bump   907^1
 ratio   535101.6331    over  Ram   1.021150274       log n   931.3558969     bump   3^8
 ratio   552830.3411    over  Ram   1.00971372       log n   938.1704398     bump   911^1
 ratio   563095.3791    over  Ram   1.013550542       log n   942.0205874     bump   47^2
 ratio   581670.1817    over  Ram   1.016865256       log n   948.8438736     bump   919^1
 ratio   600754.1613    over  Ram   1.020043185       log n   955.6779823     bump   929^1
 ratio   602719.947    over  Ram   1.008577787       log n   956.3711295     bump   2^13
 ratio   622409.9494    over  Ram   1.01163399       log n   963.2138128     bump   937^1
 ratio   642699.9339    over  Ram   1.014658841       log n   970.0607559     bump   941^1
 ratio   663584.8755    over  Ram   1.017618421       log n   976.914055     bump   947^1
 ratio   685080.4981    over  Ram   1.020512922       log n   983.7736699     bump   953^1
 ratio   707111.2017    over  Ram   1.023209855       log n   990.6478684     bump   967^1
 ratio   729803.4801    over  Ram   1.011124941       log n   997.5261949     bump   971^1
 ratio   753151.9619    over  Ram   1.013702509       log n   1004.410682     bump   977^1
 ratio   777173.7789    over  Ram   1.016217182       log n   1011.301291     bump   983^1
 ratio   801861.5176    over  Ram   1.018637504       log n   1018.200005     bump   991^1
 ratio   827256.7266    over  Ram   1.020995549       log n   1025.104756     bump   997^1
 ratio   853299.9493    over  Ram   1.008558995       log n   1032.021471     bump   1009^1
 ratio   880109.9606    over  Ram   1.010713155       log n   1038.942142     bump   1013^1
 ratio   907680.7909    over  Ram   1.012806975       log n   1045.868719     bump   1019^1
 ratio   936087.4574    over  Ram   1.014900879       log n   1052.797257     bump   1021^1
 ratio   965240.7152    over  Ram   1.016874958       log n   1059.735542     bump   1031^1
 ratio   995272.7994    over  Ram   1.018848877       log n   1066.675764     bump   1033^1
 ratio   1026149.747    over  Ram   1.020763287       log n   1073.621778     bump   1039^1
 ratio   1057832.506    over  Ram   1.022559918       log n   1080.577371     bump   1049^1
 ratio   1090462.392    over  Ram   1.00980891       log n   1087.534868     bump   1051^1
 ratio   1123939.89    over  Ram   1.011477339       log n   1094.501835     bump   1061^1
 ratio   1158412.682    over  Ram   1.013145116       log n   1101.470686     bump   1063^1
 ratio   1193842.951    over  Ram   1.014755687       log n   1108.445165     bump   1069^1
 ratio   1230053.276    over  Ram   1.016142865       log n   1115.436342     bump   1087^1
 ratio   1267293.437    over  Ram   1.017501725       log n   1122.431192     bump   1091^1
 ratio   1305625.934    over  Ram   1.018859661       log n   1129.427873     bump   1093^1
 ratio   1345045.828    over  Ram   1.020189332       log n   1136.428207     bump   1097^1
 ratio   1385545.297    over  Ram   1.021463697       log n   1143.433997     bump   1103^1
 ratio   1427151.2    over  Ram   1.022683088       log n   1150.44521     bump   1109^1
 ratio   1469852.726    over  Ram   1.023821177       log n   1157.463612     bump   1117^1
 ratio   1513714.273    over  Ram   1.02490512       log n   1164.487371     bump   1123^1
 ratio   1558764.495    over  Ram   1.011518021       log n   1171.516459     bump   1129^1
 ratio   1584623.182    over  Ram   1.013853794       log n   1175.486751     bump   53^2
 ratio   1631330.867    over  Ram   1.01463861       log n   1182.535137     bump   1151^1
 ratio   1679373.567    over  Ram   1.015422387       log n   1189.58526     bump   1153^1
 ratio   1728618.035    over  Ram   1.016105007       log n   1196.644018     bump   1163^1
 ratio   1779133.063    over  Ram   1.016712368       log n   1203.709631     bump   1171^1
 ratio   1830903.696    over  Ram   1.017220846       log n   1210.783748     bump   1181^1
 ratio   1884045.971    over  Ram   1.017679965       log n   1217.862932     bump   1187^1
 ratio   1938593.022    over  Ram   1.018090099       log n   1224.947159     bump   1193^1
 ratio   1994532.079    over  Ram   1.01842781       log n   1232.038069     bump   1201^1
 ratio   2051799.896    over  Ram   1.018646873       log n   1239.138921     bump   1213^1
 ratio   2110615.114    over  Ram   1.018842035       log n   1246.243065     bump   1217^1
 ratio   2170967.691    over  Ram   1.018990219       log n   1253.352127     bump   1223^1
 ratio   2232894.317    over  Ram   1.019091811       log n   1260.466083     bump   1229^1
 ratio   2296535.631    over  Ram   1.019192973       log n   1267.581665     bump   1231^1
 ratio   2361831.896    over  Ram   1.019247993       log n   1274.702109     bump   1237^1
 ratio   2428661.327    over  Ram   1.019189652       log n   1281.832208     bump   1249^1
 ratio   2497108.276    over  Ram   1.019041956       log n   1288.970281     bump   1259^1
 ratio   2566986.51    over  Ram   1.018718881       log n   1296.12255     bump   1277^1
 ratio   2638764.008    over  Ram   1.018396253       log n   1303.276384     bump   1279^1
 ratio   2667692.78    over  Ram   1.015309151       log n   1306.109597     bump   17^3
 ratio   2742169.833    over  Ram   1.014977287       log n   1313.266553     bump   1283^1
 ratio   2818547.764    over  Ram   1.01460303       log n   1320.428175     bump   1289^1
 ratio   2896992.224    over  Ram   1.014229225       log n   1327.591348     bump   1291^1
 ratio   2977433.202    over  Ram   1.013813494       log n   1334.759157     bump   1297^1
 ratio   3059980.599    over  Ram   1.013377221       log n   1341.930045     bump   1301^1
 ratio   3144751.434    over  Ram   1.012941465       log n   1349.10247     bump   1303^1
 ratio   3231737.268    over  Ram   1.012485308       log n   1356.27796     bump   1307^1
 ratio   3320721.621    over  Ram   1.011926271       log n   1363.462589     bump   1319^1
 ratio   3412086.874    over  Ram   1.025467085       log n   1370.648733     bump   1321^1
 ratio   3505753.445    over  Ram   1.024849868       log n   1377.839409     bump   1327^1
 ratio   3600781.602    over  Ram   1.023910178       log n   1385.055384     bump   1361^1
 ratio   3698171.189    over  Ram   1.022933065       log n   1392.275758     bump   1367^1
 ratio   3797976.061    over  Ram   1.021918933       log n   1399.500511     bump   1373^1
 ratio   3900177.117    over  Ram   1.020848798       log n   1406.731075     bump   1381^1
 ratio   3958667.07    over  Ram   1.021964509       log n   1410.808612     bump   59^2
 ratio   4064504.704    over  Ram   1.020752722       log n   1418.052125     bump   1399^1
 ratio   4172785.673    over  Ram   1.019468561       log n   1425.30276     bump   1409^1
 ratio   4283403.12    over  Ram   1.01807618       log n   1432.563283     bump   1423^1
 ratio   4396793.696    over  Ram   1.01666922       log n   1439.826613     bump   1427^1
 ratio   4513104.477    over  Ram   1.015266108       log n   1447.091343     bump   1429^1
 ratio   4632325.349    over  Ram   1.013848561       log n   1454.358868     bump   1433^1
 ratio   4754440.249    over  Ram   1.012398626       log n   1461.630572     bump   1439^1
 ratio   4879427.33    over  Ram   1.010898789       log n   1468.90782     bump   1447^1
 ratio   5007523.206    over  Ram   1.023312601       log n   1476.187828     bump   1451^1
 ratio   5138891.38    over  Ram   1.02177252       log n   1483.469214     bump   1453^1
 ratio   5273428.397    over  Ram   1.020200957       log n   1490.75472     bump   1459^1
 ratio   5410923.337    over  Ram   1.018545365       log n   1498.048418     bump   1471^1
 ratio   5551526.099    over  Ram   1.016824733       log n   1505.348891     bump   1481^1
 ratio   5695685.115    over  Ram   1.015109213       log n   1512.650713     bump   1483^1
 ratio   5778458.905    over  Ram   1.015894231       log n   1516.761587     bump   61^2
 ratio   5928308.835    over  Ram   1.014174883       log n   1524.066103     bump   1487^1
 ratio   6081941.467    over  Ram   1.012460554       log n   1531.371963     bump   1489^1
 ratio   6239344.223    over  Ram   1.010734093       log n   1538.680506     bump   1493^1
 ratio   6400497.125    over  Ram   1.022811978       log n   1545.993059     bump   1499^1
 ratio   6565154.609    over  Ram   1.020942918       log n   1553.313586     bump   1511^1
 ratio   6733381.34    over  Ram   1.018995544       log n   1560.642023     bump   1523^1
 ratio   6905467.37    over  Ram   1.017004461       log n   1567.9757     bump   1531^1
 ratio   7081263.826    over  Ram   1.014937655       log n   1575.317184     bump   1543^1
 ratio   7261186.155    over  Ram   1.012845127       log n   1582.662549     bump   1549^1
 ratio   7445442.242    over  Ram   1.024548281       log n   1590.010492     bump   1553^1
 ratio   7634010.002    over  Ram   1.022382921       log n   1597.362292     bump   1559^1
 ratio   7826859.373    over  Ram   1.020176505       log n   1604.719211     bump   1567^1
 ratio   8024328.605    over  Ram   1.017961565       log n   1612.078678     bump   1571^1
 ratio   8226266.414    over  Ram   1.015706584       log n   1619.443225     bump   1579^1
 ratio   8433024.41    over  Ram   1.013443587       log n   1626.810302     bump   1583^1
 ratio   8644047.947    over  Ram   1.011095052       log n   1634.186184     bump   1597^1
 ratio   8860081.646    over  Ram   1.022456365       log n   1641.564568     bump   1601^1
 ratio   9081100.736    over  Ram   1.020036944       log n   1648.946692     bump   1607^1
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Is the bound unconditional? For x=1 the bound is equivalent to RH. –  joro Jul 29 '13 at 12:05
    
@joro, it is conditional as written. Ramanujan makes a small comment that amounts to this: if you do not want to assume RH, replace every log N by log N + O(1) or some similar comment. –  Will Jagy Jul 29 '13 at 16:49
    
@joro, however, see mathoverflow.net/questions/137865/… as the behavior below 1 is quite different from the behavior at 1. –  Will Jagy Jul 29 '13 at 17:01

If $f (n)$is a polynomial with integer coefficients, then for $0<\alpha<1$ $$\sum_{n\le x,f(n)\ne 0}\sigma_{-\alpha}(f(n))=c_f(\alpha)x+O(x^{1-\alpha}(\log x)^{c_0}).$$ For more results see Sándor, Jó.; Mitrinović, D. S. & Crstici, B. Handbook of number theory. I Springer, 2006, $\S$ III.7.

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