Extended comment. Robin's criterion, equivalent to RH, is fairly widely known. First, however, his adviser, J.L. Nicolas, came up with [THIS as pdf][1]. The description of this on wikipedia is poor.
First, in a procedure invented by Ramanujan for his "superior highly composite numbers," it is easy to show that the smallest value of $\frac{\phi(n)}{n^\delta}$ for $0 < \delta < 1$ occurs when $n = n_\delta$ is a primorial, the product of consecutive primes beginning with 2.
Let's see, Rosser and Schoenfeld gave some effective bounds, the way I am writing this comes out
$$ \frac{\phi(n)}{n} > \frac{1}{e^\gamma \log \log n + \frac{3}{\log \log n}} $$
So the reasonable question comes, we know we get surprisingly small values of $\frac{\phi(n)}{n}$ when $n$ is a primorial. In that case, is it possible to replace the $3$ by a $0,$ giving
$$ \frac{\phi(n)}{n} > \frac{1}{e^\gamma \log \log n } ? $$
And here is the answer: If RH is true, we can drop the 3. On the other hand, if RH is false, the inequality (no 3) is true for infinitely many primorials and false for infinitely many primorials. So, we have a statement equivalent to RH.
Next, it is much easier to compute this comparison than Robin's. All you do is let $P$ be a primorial, and calculate
$$ \frac{e^\gamma \log \log P \phi(P)}{P} $$
which can be update fairly nicely as each $P$ is multiplied by the next prime. For all known primorials, this quantity strictly increases with $P.$ Since RH says it is below 1, we see the ratio increasing to 1, very pretty. i wrote out my own C++ program. There is, however, an amusing catch. [Michael Planat and colleagues][2] showed that Cramer's conjecture on prime gaps would be violated if the sequence increased forever.
Enough for now, let me see if I can find the C++ program and post some early output.
p phi(P) / P log(P) exp(gamma) * loglog(P) * phi(P) / P
2 0.5 0.6931471805599453 -0.3263930268425172
3 0.3333333333333333 1.791759469228055 0.3462393386356046
5 0.2666666666666667 3.401197381662155 0.5814026130255172
7 0.2285714285714286 5.347107530717468 0.6825296531368984
11 0.2077922077922078 7.745002803515839 0.7575980132430825
13 0.1918081918081918 10.30995216097738 0.7970469070005248
17 0.1805253569959452 13.14316550503359 0.8282264738589192
19 0.1710240224172113 16.08760448420003 0.8462108841813194
23 0.1635881953555934 19.22309870012918 0.8613001326390455
29 0.1579472231019522 22.59039453011566 0.8770078674522567
31 0.1528521513889861 26.0243817346008 0.8872418245081134
37 0.1487210121622567 29.63529964724503 0.8976791721551992
41 0.1450936704022016 33.34887171394934 0.9062933133274125
43 0.1417193989974993 37.1100718296429 0.9121906541657894
47 0.1387040926358503 40.96021943135295 0.9171685846758989
53 0.1360870342842305 44.93051134490508 0.9222875682673444
59 0.133780474381108 49.0080487888108 0.9273537165832041
61 0.1315873518502701 53.11892265298411 0.9310291166644691
67 0.1296233615241467 57.32361527237508 0.9347206020080276
71 0.1277976803759193 61.58629514941639 0.9378817320183697
73 0.1260470272200848 65.87675459056479 0.94015320110454
79 0.1244514952299571 70.24620244303181 0.9424874828967286
83 0.1229520796247769 74.6650430508284 0.9444916553953424
89 0.1215705955840491 79.15367942056054 0.9465200179408284
97 0.1203172904749352 83.72839039906393 0.9488025715708337
101 0.1191260301732031 88.34351091590518 0.9507925028376726
103 0.1179694667734633 92.97823990413482 0.9523051177952127
107 0.1168669483924029 97.65106873859673 0.9536116556434188
109 0.1157947745539405 102.3424166208259 0.9545403985983615
113 0.114770042035764 107.0698044395382 0.9553238060578663
127 0.1138663409173722 111.9139915259968 0.956775584934687
131 0.1129971322080793 116.789188849198 0.9580534856815344
137 0.1121723356226188 121.7091697750261 0.9593043754635934
139 0.1113653404023122 126.6436437081568 0.9602858875196447
149 0.1106179220103504 131.6475900141022 0.9614757581574626
151 0.1098853529904143 136.6648698509172 0.9624286793532265
157 0.109185446283469 141.7211156562655 0.9633634421768695
163 0.1085155969197667 146.8148658570722 0.9642779835121893
167 0.1078658029262352 151.932859669489 0.9650870024015282
173 0.1072423011752165 157.0861512639868 0.9658796096962042
179 0.1066431821742376 162.2735370698275 0.9666545741957177
181 0.1060539933224463 167.4720341010934 0.9672701979733297
191 0.1054987368129047 172.72430752914 0.9680084010810988
193 0.1049521112335632 177.9869977180449 0.9686032079042936
197 0.1044193593998903 183.2702014467829 0.9691265093804561
199 0.1038946390008959 188.5635062715074 0.9695253328152903
211 0.1034022473468632 193.9153644049834 0.9700846945685031
223 0.1029385601390297 199.3225361764435 0.9707768755899375
227 0.1024850863058181 204.7474861939249 0.9714019207735292
229 0.1020375531778451 210.1812081974792 0.9719201381994437
233 0.1015996237650647 215.6322466510449 0.9723820734575654
239 0.101174520736759 221.1087102029764 0.9728329395456006
241 0.1007547094473948 226.5935071364671 0.9731934250226916
251 0.1003532962623454 232.1189600755988 0.9736223342629629
257 0.09996281651035187 237.6680361604941 0.9740401207640036
263 0.09958272975555965 243.2401901926718 0.9744468690918663
269 0.09921253373416351 248.8349015722737 0.9748426972126225
271 0.09884643582370535 254.4370203931534 0.9751650771767573
277 0.09848958948499162 260.0610378993407 0.9754797735926634
281 0.09813909272525856 265.6993925686745 0.9757574820666572
283 0.09779231147887955 271.3448394663177 0.9759715944322618
293 0.09745854932366153 277.0250120753348 0.9762367602929667
307 0.09714109476560401 282.751859822922 0.9765970572556302
311 0.09682874397857634 288.4916527351012 0.9769226892226303
313 0.09651938696906012 294.2378559256413 0.9771919557217343
317 0.09621490940764353 299.9967576995186 0.9774309460927161
331 0.09592422992302829 305.7988760748956 0.9777507384622824
337 0.09563958829120922 311.618959005248 0.9780609321048802
347 0.09536396988114811 317.4682837851949 0.9784009755317026
349 0.09509072068378092 323.3233557073974 0.9786926522404308
353 0.09482134187164556 329.1898237643306 0.9789569560335052
359 0.09455721557116745 335.0731461528189 0.9792133789754222
367 0.09429956648241768 340.9785080008735 0.9794794850413051
373 0.09404675263125838 346.9000864205173 0.9797375117033696
379 0.09379860816521284 352.8376226255997 0.9799876894087147
383 0.09355370318305824 358.7856576147804 0.980214493257318
389 0.09331320523143084 364.7492369583989 0.9804344186018505
397 0.09307815937442472 370.7331732390861 0.9806624481373041
401 0.0928460442637653 376.7271346663926 0.9808691292388929
409 0.09261903682057761 382.7408498224354 0.9810834041462074
419 0.09239798900000344 388.7787207423575 0.9813177619639269
421 0.09217851634204619 394.8213535760399 0.9815189454652563
431 0.09196464507443122 400.8874616661436 0.9817390896696049
433 0.09175225559388982 406.9581993941461 0.9819279137990317
439 0.09154325273376707 413.0426988072213 0.9821108474573144
443 0.09133660882240416 419.1362685772664 0.9822763181354403
449 0.09113318653104023 425.2432914650086 0.9824365632620818
457 0.09093377036795261 431.3679748559028 0.9826028407018458
461 0.0907365170699744 437.5013728988994 0.9827530235125199
463 0.09054054187111917 443.6390999529856 0.9828770382747256
467 0.09034666490779771 449.7854292106546 0.9829864320086854
479 0.09015804974097558 455.9571298080655 0.9831226489806625
487 0.08997292027538836 462.1453939311481 0.9832641846193747
491 0.08978967603857495 468.3418380589426 0.9833915947744915
499 0.08960973680803672 474.5544441546941 0.9835240816730877
503 0.08943158623784181 480.7750343247939 0.9836431393887107
509 0.08925588567548849 487.0074823413444 0.983758191990933
521 0.08908456919626491 493.2632323830977 0.9838951123805503
523 0.08891423541195083 499.5228138471626 0.9840108606660501
541 0.08874988377532984 505.8162331260091 0.9841710475019165
547 0.08858763535892156 512.1206819284312 0.9843262453744257
557 0.08842859113026999 518.4432471683584 0.9844915911353739
563 0.08827152436094447 524.7765267964982 0.9846518699155999
569 0.08811638987173367 531.1204072306244 0.9848072237098611
571 0.08796207044989177 537.4677964402805 0.9849437339724121
577 0.08780962318741362 543.8256387067886 0.9850759084765289
587 0.08766003268794613 550.2006635266167 0.9852173398341821
593 0.08751220801224976 556.5858579256144 0.9853543650721274
599 0.08736611083693716 562.9811195237298 0.9854871032269296
601 0.0872207429320504 569.3797144582651 0.9856029987989381
607 0.08707705142804373 575.7882432493246 0.9857151077609236
613 0.08693500077318558 582.2066081852608 0.985823528987842
617 0.08679410125815611 588.6314772091663 0.9859223369209658
619 0.08665388461638203 595.0595824818508 0.9860058580723636
631 0.08651655674852722 601.506888344392 0.9861038169112781
641 0.08638158552115043 607.9699178013127 0.9862097122681003
643 0.08624724401956232 614.4360625255504 0.9863010892038749
647 0.0861139407057763 620.9084088200512 0.9863838403154985
653 0.08598206637085168 627.3899859493276 0.9864636225316135
659 0.08585159282552414 633.8807094838302 0.9865405091177107
661 0.08572171144454756 640.3744633236819 0.986604141082272
673 0.08559433891639816 646.8862086533267 0.9866805411118564
677 0.08546790710115976 653.4038799262389 0.9867491665098705
683 0.0853427710731932 659.9303747858097 0.9868151677733565
691 0.08521926489219002 666.4685146095774 0.9868834196787549
701 0.08509769675397007 673.021022496612 0.986958460538422
709 0.08497767179380931 679.5848780231441 0.9870353693798222
719 0.0848594830986858 686.1627393808652 0.9871184736732537
727 0.08474275753733959 692.7516658583987 0.9872031068557536
733 0.08462714668121771 699.3488115602854 0.9872849065505048
739 0.08451263092116193 705.9541094812336 0.9873639402573243
743 0.08439888579206212 712.5648055259513 0.9874361338383814
751 0.08428650378701277 719.1862111777154 0.9875098358700443
757 0.08417516098148171 725.8155744311529 0.9875809626382508
761 0.08406454973183457 732.4502077890146 0.9876456311855002
769 0.08395523302217028 739.0952987585202 0.987711790840558
773 0.08384662340635894 745.7455778071077 0.9877717305077077
787 0.08374008385946395 752.4138060555251 0.9878443212700446
797 0.08363501474546212 759.0946607343153 0.9879216805403214
809 0.08353163401030085 765.7904596513738 0.9880070803250892
811 0.08342863569462847 772.4887277054892 0.988082889689585
821 0.0833270173807495 779.1992508149416 0.9881630444319862
823 0.08322576948599768 785.9122070156187 0.9882339312495667
827 0.08312513373087556 792.6300117106424 0.9882991057044151
829 0.08302486215822071 799.3502318657777 0.9883553889023328
839 0.08292590523073773 806.0824425722449 0.9884160837979863
853 0.08282868846024448 812.8312021197365 0.9884873028373118
857 0.08273203888211117 819.5846400383343 0.9885530948845231
859 0.08263572684615994 826.3404089603185 0.9886104979809445
863 0.08253997281736949 833.1008236514019 0.9886627620273367
877 0.08244585654277728 839.8773306437741 0.9887250314787478
881 0.08235227441276277 846.6583882697104 0.988782237854162
883 0.08225901022882985 853.4417134703143 0.9888315764337965
887 0.082166271773104 860.2295584526239 0.9888761146355637
907 0.08207568051425823 867.039700902739 0.9889385649856043
911 0.08198558646319977 873.8542437999989 0.9889961951120146
919 0.08189637472602546 880.6775299223547 0.9890545459234467
929 0.08180821931727839 887.5116386611685 0.9891162269207064
937 0.08172091065205184 894.3543219434069 0.9891784946642675
941 0.08163406590109323 901.2012650829923 0.9892361713865228
947 0.08154786308599177 908.0545641761784 0.9892919089854316
953 0.08146229345001486 914.9141790798326 0.9893457471049893
967 0.08137805116102829 921.7883775752858 0.9894075747948702
971 0.08129424266343713 928.6667040435772 0.9894650271534431
977 0.081211034636146 935.55119069562 0.9895205939024816
983 0.08112841913804208 942.4417998157671 0.9895743128440608
991 0.08104655393205011 949.3405143500971 0.9896285495143424
997 0.08096526350684244 956.245265120059 0.9896809802538771
1009 0.08088502043101801 963.1619801404127 0.9897384065167235
1013 0.08080517342170802 970.0826516446614 0.9897917889704071
1019 0.08072587491982215 977.0092286778842 0.9898434119538752
1021 0.0806468094203904 983.9377664960488 0.9898889517047108
1031 0.0805685874907877 990.8760509800658 0.9899371621272057
1033 0.08049059273039004 997.8162734491854 0.9899794579479165
1039 0.08041312343998543 1004.762287440285 0.9900201806963382
1049 0.08033646650629621 1011.717880048681 0.9900635169842441
1051 0.08026002838402571 1018.675377419558 0.9901011792835088
1061 0.08018438273993143 1025.642344558172 0.9901414177744228
1063 0.08010895058307355 1032.611194936514 0.9901761349706515
1069 0.08003401236924466 1039.585673847539 0.9902094231372823
1087 0.07996038402299881 1046.57685073466 0.9902529983116087
1091 0.07988709311188698 1053.571700720493 0.9902931452725218
1093 0.07981400336521553 1060.56838220867 0.9903280285636298
1097 0.07974124675321442 1067.568716668945 0.9903596306731417
1103 0.0796689518785515 1074.574505688199 0.9903898866951015
1109 0.07959711332861592 1081.585719675549 0.9904188213715832
1117 0.07952585360316505 1088.604121474618 0.9904482839989819
1123 0.07945503806122101 1095.627880429357 0.9904764515947025
1129 0.07938466158818183 1102.656967993506 0.990503347669047
1151 0.07931569142172816 1109.705354402228 0.9905429194949587
1153 0.07924690070930689 1116.755476922497 0.9905776942164042
1163 0.07917876063990938 1123.814235075016 0.9906145205192787
1171 0.07911114427727922 1130.879848438614 0.9906516706232278
1181 0.07904415770295468 1137.953965254811 0.990690760831116
1187 0.07897756616318809 1145.033149649421 0.9907285049804925
1193 0.07891136535332791 1152.117376071519 0.9907649266154878
1201 0.07884566063613113 1159.208285893599 0.9908016279474779
1213 0.07878066009150118 1166.309137802543 0.990841695487324
1217 0.07871592659923207 1173.41328189553 0.9908789071354395
1223 0.07865156361754831 1180.522344031218 0.9909148353475631
1229 0.07858756722729807 1187.636300140784 0.9909495020928574
1231 0.0785237267990062 1194.751882266968 0.9909799396588371
1237 0.07846024763425356 1201.872326639361 0.9910091903420785
1249 0.07839742918138387 1209.002425149486 0.9910416635958519
1259 0.07833515957917467 1216.140498183531 0.991075819391518
1277 0.0782738164628245 1223.292767039563 0.9911172157595045
1279 0.07821261723181369 1230.446600841142 0.9911545693310034
1283 0.07815165650131345 1237.603557205758 0.9911893226196458
1289 0.07809102682210374 1244.765179208697 0.9912228885305447
1291 0.07803053803293093 1251.928351599544 0.9912525697229118
1297 0.0779703757059973 1259.09616078386 0.9912811300185953
Wed Jul 10 15:50:20 PDT 2013
[1]: http://math.univ-lyon1.fr/~nicolas/petitsphi83.pdf
[2]: http://arxiv.org/abs/1012.3613