8
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I am studying Sieve theory from Iwaniec's notes. I have come across a theorem which estimates $\varphi(x,N)=\#\{1\leq n \leq x:(n,N)=1\}$, where $N$ is product of distinct primes.

Let's define $R(x,N)=\sum_{d|N}\mu(d)\{\frac{x}{d}\}=x\sum_{d|N}\frac{\mu(d)}{d}-\sum_{d|N}\mu(d)[\frac{x}{d}]=x\frac{\varphi(N)}{N}-\varphi(x,N)$.

Also say, $K(N)=\displaystyle\sum_{n=1}^{\infty}(\frac{R(n.N)}{n})^2$.

Now let's introduce a weighted $l^2$ space $\mathcal{H}$ with norm defined as $||x||^2:=\sum_{n=1}^{\infty}\frac{x(n)^2}{n^2}$ (provided convergent). Consider a Hilbert space $\mathcal{M}$ generated by $\left\{\gamma_n|\gamma_n(k)=\left\{\frac{k}{n}\right\},k=1,2,...;n>1\right\}$ inside $\mathcal{H}$.

It is known from Bagchi's result that the RH is true if and only if $\gamma=(1,1,...)\in\mathcal{M}$. It can be deduced that, above is also equivalent to the statement that if $x_m=\sum_{n=2}^{m}\mu(n)\gamma_n$ then $x_m\to_{strongly} \gamma$ as $m\to\infty$.

If we define, $\bar x_m=\sum_{n|m}\mu(n)\gamma_n$ we see $K(N)=||x_N||^2$.

I think that it will be easy to show, $\bar x_m\to_{strongly} \gamma$ if and only if $x_m\to_{strongly} \gamma$.

My questions are:

  1. Can this methodology create some other reformulation of RH? Like, "RH is true if and only if $K(N)\to1$"?

  2. Does there exist any sieve method which gives a good bound of $K(N)$? Honestly, I know a very little of sieve methods. Whatever sieve bounds I have seen those force upper bounds of $K(N)$ to go to infinity.

  3. Has any work been done in this way? If not, will it be a good reformulation? If yes, how one can approch further? As, $K(N)=\displaystyle\sum_{n=1}^{\infty}[\frac{\varphi(n,N)}{n}-\frac{\varphi(N)}{N}]^2$ looks like a 'good' arithmetical as well as a probabilistic function.

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  • 3
    $\begingroup$ It seems to me that $|R(x,N)|\le \tau(N)$ by the first expression. So the $n^2$ in the denominator dominates, making only small $n$ relevant in $K(N)$. But for those, you can take $N$ a product of primes up to some point $P$, then $R(x,N)=x\prod_{p\le P}(1-1/p)-1$ for $x\le P$, which is essentially $x{e^{-\gamma}\over\log P}$ (say for $x\ge P/2$ and $P$ large). Dividing by $x$ and squaring in the $K(N)$ sum, truncating this sum at $P$, you have $K(N)\ge \sum_{x\le P} e^{-2\gamma}{1\over (\log P)^2}=e^{-2\gamma}{P\over (\log P)^2}$, so at least on these special $N$ the $K(N)$ must be big. $\endgroup$
    – v08ltu
    Jul 8, 2013 at 21:42
  • $\begingroup$ I think that $|R(x,N)|\leq\tau(N)$ is a crude bound, as you are losing 'cancellation opportunity' by change of sign of $\mu(n)$. $|R(n,N)|=O(2^{\omega(N)})=O(2^{\log N})$ $\endgroup$ Jul 8, 2013 at 21:57

1 Answer 1

5
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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. 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 cannot 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 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
$\endgroup$
1
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    $\begingroup$ Thank you very much for this resourceful explanation. I really didn't expect that, Crammer's conjecture could come here. Still, I am wondering whether or not it is possible to find a bound of $K(N)$ using (or not using) Sieve methods. Or does it always diverge! $\endgroup$ Jul 11, 2013 at 21:14

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