# sum of powers of divisors function

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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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

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:~$ - 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  - 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))) ================================== 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  - 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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