The following are simple deductions using easy calculations for inequalities and limits.

I define the following sequences, whose shape is inspired in Nicolas, Robin and Lagarias, respectively, equivalences for the Riemann Hypothesis (see below the reference for each one) for $n\geq 5041$

Robin's sequence is defined as $$\mathcal{R}_n:=\frac{\sigma(n)}{n\log\log n},$$ where $\sigma(n)=\sum_{d\mid n}d$ is the sum of divisors function;

Lagarias sequence is defined by $$\mathcal{L}_n:=\frac{e^{\sigma(n)}}{n\left(H_n\right)^{e^{H_n}}},$$ where $H_n=1+1/2+\cdots+1/n$ is the nth harmonic number;

and Nicolas sequence is defined (here there was a mistake/typo that was fixed) as $$\mathcal{N}_n:=\frac{N(n)}{\phi(N(n))}\frac{1}{\log\log N(n)},$$ where $\phi(m)$ is the Euler's totient function, and the arithmetic function $N(n)=\prod_{k=1}^n p_k$ is defined to be the product of the first $n$ primes.

Claim 1. On assumption of the Riemann hypothesis thus it is possible to write that $$\mathcal{R_n}<e^{\gamma}<\mathcal{N_n},$$ for $n>5040$.

It is possible to show, if my calculations were right, (you are welcome to ask in comments about them, if you want; I've used $\lim_{n\to\infty}\sigma(n)-\log n-e^{H_n}\log H_n\leq \gamma\leq \lim_{n\to\infty}\log(\mathcal{N}_n)$, and after I take the exponential)

Claim 2. On assumption of the Riemann hypothesis $$\lim_{n\to \infty}\mathcal{L}_n\leq \lim_{n\to \infty}\mathcal{N}_n.$$

I've used Gronwall's Theorem to deduce

Claim 3. On assumption of the Riemann hypothesis one has $$\gamma<\limsup_{n\to\infty}\mathcal{N}_n\frac{\log H_n}{\log(\log H_n)}.$$

My question if that if it possible state relationships between the general term of such sequences:

Question. It is known (in such case reference it) or can you deduce if on assumption of the Riemann hypothesis is there an integer $N>5040$ such that for $n\geq N$ one has the relationship $$\mathcal{R}_n\leq\mathcal{L}_n\leq \mathcal{N}_n?$$ If previous relationship is wrong, what is the first counterexample? Thanks in advance.

Appendix: (A remark, and the references of the cited papers for previous equivalences): The remark is that it is possible write this kind of relationships for arithmetics functions from different equivalences, for example one that involves the least common multiple, if one define $\text{Num}(n)$ by $$H_n=\frac{\text{Num}(n)}{\text{l.c.m}(1,2,\ldots,n)}$$ (I don't write the reference because I don't write such equivalence).

Previous references to justify the shape of the sequences that I've typed were

G. ROBIN, Grandes valeurs de la fonction somme des diviseurs et hypothése de Riemann, J. Math. Pures Appl. (9) 63 (1984), 187-213.

J. NICOLAS, Petites valeurs de la fonction d'Euler, J. Number Theory 17 (1983), 375-388.

J.C. LAGARIAS, An elementary problem equivalent to the Riemann Hypothesis, Amer. Math. Monthly 109 (2002), 534-543.

  • $\begingroup$ For Claim 3. I've deduced $$e^{H_n}\log(H_n)<\frac{N(n)}{\phi(N(n))}ne^{-O(1/n)}\frac{\log H_n}{\log(\log H_n)},$$ and after I've applied Gronwall's Theorem. $\endgroup$ – user243301 Jul 1 '16 at 7:57
  • $\begingroup$ Can you write all the steps for showing how RH is implied/equivalent to $R_n < e^{\gamma}$ for $n$ large enough ? (I think the $\implies$ part is not very complicated) $\endgroup$ – reuns Jul 9 '16 at 20:31
  • $\begingroup$ I'm searching for proofs of that, but I didn't find any yet $\endgroup$ – reuns Jul 9 '16 at 21:14
  • $\begingroup$ I'm sorry @user1952009 there was a great mistake/typo, that I've fixed, Nicolas sequence is $\mathcal{N}_n:=\frac{N(n)}{\phi(N(n))}\frac{1}{\log\log N(n)}.$ Then **Claim 1 is obvious, that is the conjunction of Nicolas and Robin equivalences. Claim 2 follows from a similar comparison between Lagarias and Nicolas: from $\lim_{n\to\infty}\sigma(n)-\log n-e^{H_n}\log H_n\leq \gamma\leq \lim_{n\to\infty}\log(\mathcal{N}_n)$ then we will take exponentials for $\log\lim_{n\to\infty} \mathcal{L}_n \leq \log\lim_{n\to\infty} \mathcal{N}_n$. Now I am thinking what was about the third claim. $\endgroup$ – user243301 Jul 10 '16 at 7:48
  • 1
    $\begingroup$ I think the interesting part is to prove $R_n <e^\gamma \implies $ RH, once you got it, the rest should follow easily. $\ \ $ Tks to will Jagy there is the original Robin paper, and 3 others that should help you (us) : paris8.free.fr/Zeta_Riemann/RobinCriterion $\endgroup$ – reuns Jul 10 '16 at 18:05

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy