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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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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. –  v08ltu Jul 8 '13 at 21:42
    
I think that $|R(x,N)|\leq\tau(N)$ is a crude bound, as you are losing 'cancellation opportunity' by change of sign of $\mu(n)$. $|R(n,N)|=O(2^{\omega(N)})=O(2^{\log N})$ –  Kunnysan Jul 8 '13 at 21:57
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1 Answer

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
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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
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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! –  Kunnysan Jul 11 '13 at 21:14
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