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$\begingroup$ 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? $\endgroup$– Greg MartinJul 18, 2013 at 0:01
5 Answers
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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$\begingroup$ @joro, posted a second answer with what you requested. $\endgroup$ Jul 21, 2013 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
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
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)))
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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
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$\begingroup$ Is the bound unconditional? For x=1 the bound is equivalent to RH. $\endgroup$– joroJul 29, 2013 at 12:05
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$\begingroup$ @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. $\endgroup$ Jul 29, 2013 at 16:49
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$\begingroup$ @joro, however, see mathoverflow.net/questions/137865/… as the behavior below 1 is quite different from the behavior at 1. $\endgroup$ Jul 29, 2013 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.