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II. REDUCTIONTOPRIMORIALNUMBERS

Define the primorial number Nn of index n as the product of the first n primes n

Nn = pk, kY=1 so that N0 = 1, N1 = 2, N2 = 6, ··· and so on. The primorial numbers (OEIS sequence A002110 [11]) play the role here of superabundant numbers in [12] or in [8]. They are champion numbers (ie left to right maxima) of the function x 7→ Ψt(x)/x : Ψ (m) Ψ (n) t < t for any m < n. (1) m n We give a rigorous proof of this fact. Proposition 1: The primorial numbers and their multiples are exactly the champion numbers of the function x 7→ Ψt(x)/x. Proof: The proof is by induction on n. The induction hypothesis Hn is that the statement is true up to Nn. Sloane sequence A002110 begins 1, 2, 4, 6 ... so that H2 is true. Assume Hn true. Let Nn ≤ m< Nn+1 denote a generic integer. The prime of m are ≤ pn. Therefore Ψt(m)/m ≤ Ψt(Nn)/Nn with equality iff m is a multiple of Nn. Further Ψt(Nn)/Nn < Ψt(Nn+1)/Nn+1. The proof of Hn+1 follows.

In this section we reduce the maximization of Rt(n) over all integers n to the maximization over primorials. Proposition 2: Let n be an integer ≥ 2. For any m in the range Nn ≤ m < Nn+1 one has Rt(m) < Rt(Nn). Proof: Like in the preceding proof we have

Ψt(m)/m ≤ Ψt(Nn)/Nn.

Since 0 < log log Nn ≤ log log m, the result follows.

III. Ψt AT PRIMORIAL NUMBERS We begin by an easy application of Mertens formula. Proposition 3: For n going to ∞ we have eγ lim R (N )= . t n ζ(t)

Proof: Writing 1+1/p = (1−1/p2)/(1−1/p) in the definition of Ψ(n) we can combine the Eulerian product for ζ(t) with Mertens formula (1 − 1/p)−1 ∼ eγ log(x) pY≤x to obtain eγ Ψ(N ) ∼ log(p ). n ζ(t) n Now the Theorem [6, Th. 6, Th. 420] shows that x ∼ θ(x) for x large, where θ(x) stands for Chebyshev’s first summatory function: θ(x)= log p. Xp≤x 3

This shows that, taking x = pn we have

pn ∼ θ(pn) = log(Nn). The result follows. eγ This motivates the search for explicit upper bounds on Rt(Nn) of the form ζ(t) (1 + o(1)). In that direction we have the following bound. Proposition 4: For n large enough to have pn ≥ 20000, we have

Ψt(Nn) exp(γ +2/pn) 1.1253 ≤ (log log Nn + ). Nn ζ(t) log pn We prepare for the proof of the preceding Proposition by some Lemmas. First an upper bound on a partial Eulerian product from [13, (3.30) p.70]. Lemma 1: For x ≥ 2, we have 1 (1 − 1/p)−1 ≤ eγ(log x + ). log x pY≤x Next an upper bound on the tail of the Eulerian product for ζ(t). Lemma 2: For n ≥ 2 we have

t −1 (1 − 1/p ) ≤ exp(2/pn).

p>pYn

t 1−t Proof: Use Lemma 6.4 in [3] with x = pn. Bound t−1 x above by 2/x. Lemma 3: For n ≥ 2263, we have 0.1253 log pn < log log Nn + . log pn

Proof: If n ≥ 2263, then pn ≥ 20000. By [13], we know then that 1 log Nn >pn(1 − ). 8pn On taking log’s we obtain 0.1253 log log Nn > log pn − , pn upon using x log(1 − ) > −0.1253x 8 for x small enough. In particular x< 1/20000 is enough. We are now ready for the proof of Proposition 4. Proof: Write Ψ (N ) n 1 − 1/p t (1 − 1/pt)−1 t n k p>pn − −1 = = Q (1 1/p) Nn 1 − 1/pk ζ(t) kY=1 pY≤pn and use both Lemmas to derive

Ψt(Nn) exp(γ +2/pn) 1 ≤ (log pn + ). Nn ζ(t) log pn Now we get rid of the first log in the RHS by Lemma 3. 4

The result follows.

So, armed with this powerful tool, we derive the following significant Corollaries. For convenience let 1.1253 f(n)=(1+ ). log pn log log Nn

2/pn Corollary 1: Let n0 = 2263. Let n1(t) denote the least n ≥ n0 such that e f(n) < ζ(t). For γ n ≥ n1(t) we have Rt(Nn) < e . Proof: Let n ≥ n0. We need to check that 1.1253 exp(2/pn)(1 + ) ≤ ζ(t). log pn log log Nn which, for fixed t holds for n large enough. Indeed ζ(t) > 1 and the LHS goes monotonically to 1+ for n large.

We give a numerical illustration of Corollary 1 in Table 1.

t n1(t) Nn1(t) 3 10 6.5 × 109 4 24 2.4 × 1034 5 79 4.1 × 10163 6 509 5.8 × 101551 7 10 596 2.5 × 1048337 TABLE I: The numbers in Corollary 1.

We can extend this Corollary to all integers ≥ n0 by using the reduction of preceding section. γ Corollary 2: For all N ≥ Nn such that n ≥ n1(t) we have Rt(N) < e .

Proof: Combine Corollary 1 with Proposition 2. We are now in a position to derive the main result of this note. Theorem 1: If N is a 7−free integer, then σ(N) < Neγ log log N. Proof: If N is ≥ Nn with n ≥ n1(7), then the above upper bound holds for Ψ7(N) by Corollary 2, hence for σ(N) by the remark in the Introduction. If not, we invoke the results of [2], who checked 10 Robin inequality for 5040 < N ≤ 1010 , and observe that all 7−free integers are > 5040.

IV. VARYING t We begin with an easy Lemma. 1 1 Lemma 4: Let t be a real variable. For t large, we have ζ(t)=1+ 2t + o( 2t ). Proof: By definition, for t> 1 we may write ∞ 1 ζ(t)= nt Xn=1 so that 1 ζ(t) ≥ 1+ . 2t In the other direction, we write 1 1 ∞ 1 ζ(t)=1+ + + , 2t 3t nt Xn=4 5 and compare the remainder of the series expansion of the ζ function with an integral: ∞ 1 ∞ du 3 1 < = = O( ). nt Z ut (t − 1)3t 3t Xn=4 3 The result follows. We can derive a result when t grows slowly with n. Theorem 2: Let Sn be a sequence of integers such that Sn ≥ Nn for n large, and such that Sn is t−free with t = o(log log n). For n large enough, Robin inequality holds for Sn. Proof: For Corollary 2 to hold we need e2/pn f(n) <ζ(t) to hold, or , taking logs, the exact bound

2/pn + log f(n) < log ζ(t), or up to o(1) terms 1.1253 2/pn + ≤ log ζ(t). log pn log log Nn 2 In the LHS, the dominant term is of order 1/(log pn) , since, like in the proof of Proposition 3, we may 2 2 write pn ∼ log Nn . Now pn ∼ n log n by [6, Th. 8], entailing log pn ∼ log n and (log pn) ∼ (log n) . ∼ 1 In the RHS, with the hypothesis made on t we have, by Lemma 4, the estimate log ζ(t) 2t . The result follows after comparing logarithms of both sides.

V. CONCLUSION In this article we have proposed a technique to check Robin inequality for t−free integers for some values of t. The main idea has been to investigate the complex structure of the σ though the sequence of Dedekind psi functions ψt. The latter are simpler for the following reasons • Ψt(n) solely depends on the prime divisors of n and not on their multiplicity • the champions of Ψt are the primorials instead of the colossally abundant numbers • Ψt is easier to bound for n large because of connections with Eulerian products Further, σ(n) ≤ Ψt(n) for t−free integers n. We checked Robin inequality for t−free integers for t = 6, 7 and t = o(log log n). It is an interesting and difficult open problem to apply Theorem 2 to superabundant numbers or colossally abundant numbers for instance. We do not believe it is possible. New ideas are required to prove Robin inequality in full generality.

REFERENCES [1] Peter Borwein, Stephen Choi, Brendan Rooney and Andrea Weirathmueller The . A resource for the afficionado and virtuoso alike. CMS Books in Mathematics. Springer, New York, 2008. [2] Keith Briggs, Abundant numbers and the Riemann hypothesis. Experiment. Math. 15 (2006), no. 2, 251–256. [3] YoungJu Choie, Nicolas Lichiardopol, Pieter Moree, Patrick Sol´e, On Robin’s criterion for the Riemann hypothesis, J. Th´eor. Nombres Bordeaux 19 (2007), no. 2, 357–372. [4] J. A. Csirik, M. Zieve and J. Wetherell, On the genera of X0(N), unpublished manuscript (2001); available online at http://www.csirik.net/papers.html [5] Brian J. Conrey, The Riemann hypothesis. Notices Amer. Math. Soc. 50 (2003), no. 3, 341–353. [6] G.H. Hardy, E.M. Wright, An introduction to the theory of numbers, Oxford (1979). [7] Gilles Lachaud, L’hypoth`ese de Riemann : le Graal des math´ematiciens. La Recherche Hors-S´erie no 20, August 2005. [8] Jean-Louis Nicolas, Petites valeurs de la fonction d’Euler. J. 17 (1983), no. 3, 375–388. [9] Patrick Sol´e, Michel Planat, Extreme values of the Dedekind Ψ function, http://fr.arxiv.org/abs/1011.1825 [10] Michel Planat, Riemann hypothesis from the Dedekind psi function, hal.archives-ouvertes.fr/docs/00/52/64/54/PDF/RiemannHyp.pdf [11] www.research.att.com/njas/sequences/ [12] G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypoth`ese de Riemann. J. Math. Pures Appl. (9) 63 (1984), 187–213. [13] J. B. Rosser and L. Schoenfeld, Approximate formulas for some functions of prime numbers. Illinois J. Math. 6 (1962), 64–94.