Riemann Hypothesis from the Dedekind Psi Function Michel Planat

Home , Chebyshev function, Divisor function, Primorial

Riemann hypothesis from the Dedekind psi function Michel Planat

To cite this version:

Michel Planat. Riemann hypothesis from the Dedekind psi function. 2010. hal-00526454v1

HAL Id: hal-00526454 https://hal.archives-ouvertes.fr/hal-00526454v1 Preprint submitted on 14 Oct 2010 (v1), last revised 22 Oct 2010 (v2)

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Riemann hypothesis from the Dedekind psi function

Michel Planat Institut FEMTO-ST, CNRS, 32 Avenue de l’Observatoire, F-25044 Besan¸con, France.

Abstract. Let P be the set of all primes and ψ(n) = n Qn∈P,p|n(1+1/p) be the Dedekind psi function. We prove that, under the assumption that the n-th prime pn is always larger than the ﬁrst Chebyshev function θ(pn), the Riemann hypothesis is γ satisﬁed if and only if f(n) = ψ(n)/n − e log log n < 0 for all integers n>n0 = 30 (D), where γ ≈ 0.577 is Euler’s constant. This inequality is equivalent to Robin’s inequality that is recovered from (D) by replacing ψ(n) with the sum of divisor function σ(n) ≥ ψ(n) and the lower bound by n0 = 5040. For a square free number, both arithmetical functions σ and ψ are the same. We also prove that any exception to (D) may only occur at a positive integer n satisfying ψ(m)/m < ψ(n)/n, for any m

PACS numbers: 11A41, 11N37, 11M32

1. Introduction

−s 1 Riemann zeta function ζ(s)= n = − (where the product is taken over Pn>1 Qp∈P 1−p s the set P of all primes) converges for R(s) > 1. It has a analytic continuation to the complex plane with a simple pole at s = 1. The Riemann hypothesis (RH) states that 1 non-real zeros all lie on the critical line R(s) = 2 . RH has equivalent formulations, many of them having to do with the distribution of prime numbers [1, 2]. Let d(n) be the divisor function. There exists a remarkable parallel between the

error term ∆(x) = Pn≤x d(n) − x(log x +2γ − 1) in the summatory function of d(n) 1 (Dirichlet’s divisor problem) and the corresponding mean-square estimates |ζ( 2 )+ it| of ζ(s) on the critical line, see [3] for a review. This may explain Ramanujan’s interest for highly composite numbers [4]. A highly composite number is a positive integer n such that for any integer m < n, d(m) < d(n), i.e. it has more divisors than any positive integer smaller than itself. This landmark work eventually led to Robin’s formulation of RH in terms of the sum of divisor function σ(n) [5, 6, 7]. More precisely, σ(n) RH is true iﬀ g(n)= − eγ log log n< 0 for any n> 5040. (1) n 2

The numbers that do not satisfy (1) are in the set A = {2, 3, 4, 5, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 36, 48, 60, 72, 84, 120, 180, 240, 360, 720, 840, 2520, 5040}. If RH is false, the smallest value of n> 5040 that violates the inequality should be a superabundant number [8, 9, 10], i.e. a positive number n satisfying σ(m) σ(n) < for any m < n. (2) m n No counterexample has been found so far. In the present paper, Robin’s criterion is refined by replacing the sum of divisor 1 function by the Dedekind psi function ψ(n) = Qp∈P,p|n(1 + p ) ‡. Since ψ(n) ≤ σ(n), with equality when n is free of square, we establish the refined criterion ψ(n) RH is true iff f(n)= − eγ log log n< 0 for any n> 30. (3) n The numbers that do not satisfy (3) are in the set B = {2, 3, 4, 5, 6, 8, 10, 12, 18, 30}. A number that would possibly violate (3) should satisfy ψ(m) ψ(n) < for any m < n, (4) m n n and be a primorial number Nn = Qi=1 pi (the product of the first n primes) or one its multiples smaller than Nn+1 (Sloane sequence A060735). γ According to a Mertens theorem [2], one has lim ψ(Nn) = e log(p ) and we n→∞ Nn ζ(2) n show that none of the numbers larger than 30 in the sequence A060735 can violate (3). As a result, RH may only be true. In the next section, we provide the proof of (3) and compare it with the Robin’s criterion (1). Furthermore, we investigate the structure of numbers satisfying (4) and justify why they fail to provide counterexamples to RH.

Originally, Dedekind introduced ψ(n) as the index of the congruence subgroup Γ0(n) in the modular group (see [11], p. 79). In our recent work, the Dedekind psi function plays a role for understanding the commutation relations of quantum observables within the discrete Heisenberg/Pauli group [12]. In particular, it counts the cardinality of the

projective line P1(Zn) of the lattice Zn ×Zn. The relevance of the Dedekind psi function ψ(n) in the context of RH is novel. For other works aiming at a reﬁnement of Robin’s inequality, we mention [13], [7] and [14].

2. A proof of the reﬁned Robin’s inequality

Let us compute the sequence S of all positive numbers satisfying (4). For 2

210, 420, 630, 840, 1050, 1260, 1470, 1680, 1890, 2100, N5 = 2310, 4620, 6930, 9240}, which are the ﬁrst terms of Sloane sequence A060735, consisting of the primorials Nn and their multiples up to the next primorial Nn+1.

‡ The Dedekind function ψ(n) should not be confused with the second Chebyshev function ψT (n) = Ppk≤n log p. 3

It is straightforward to check that about half of the numbers in S are not superabundant (compare to Sloane sequence A004394). Based on calculations performed on the numbers in the ﬁnite sequence S, we are led to three properties that the inﬁnite sequence A060735 should satisfy

Proposition 1: For any l > 1 such that Nn < lNn < Nn+1 one has f(lNn) < f(Nn). ˜ ψ(n) Proof: One may use f(n)= n log log n instead of f(n). 2 One observes that lNn = p1p2 ··· l ··· pn for some pj = l. The corresponding Dedekind psi function is evaluated as

2 ψ(lNn)=(pj + pj) Y ψ(pi)= lψ(Nn). i6=j Then, with ψ(lNn) ψ(Nn) f˜(lNn)= = , lNn log log(lNn) Nn log log(lNn) one concludes that

f˜(lNn) log log Nn = < 1 n the required range 1

Proposition 2: Given l ≥ 1, for any m such that lNn

Proof: This proposition is proved by using inequality (4) at n =(l + 1)Nn

ψ(m) ψ[(l + 1)Nn] < for any m< (l + 1)Nn m (l + 1)Nn

and the relation ψ[(l + 1)Nn]=(l + 1)ψ(Nn). As a result

ψ(Nn) log log Nn f˜(m) < = f˜(Nn) < f˜(Nn) since m > Nn. Nn log log m log log m

Proposition 3: For any n > 1 one has (i) pn+1 > θ(pn) and as result (ii) n f(Nn+1) < f(Nn), where θ(pi)= Pi=1 log pi is the ﬁrst Chebyshev function at pi. This statement is reinforced by numerical calculations: an excerpt is in Table 1.

3 5 7 pn 10 10 10 10 θ(pn) 0.779 0.986 0.99905 0.999958 pn ˜ f(Nn+1) 0.987 0.9999980 0.99999999921 0.99999999999975 f˜(Nn)

f˜(Nn+1) Table 1. An excerpt of values of θ(pn)/pn and versus the number of primes f˜(Nn) in the primorial Nn. 4

Justiﬁcation n : At a primorial n = Nn, ψ(Nn) = Qi=1(1 + pi) so that ψ(Nn+1) = (1 + pn+1)ψ(pn). One would like to show that, for any pn > 2

1 f˜(N ) 1 log θ(p ) 1+ n+1 =(1+ ) n = pn+1 < 1 log pn+1 f˜(N ) pn+1 log θ(pn+1) 1 + log(1 + )/ log θ(p ) n θ(pn) n

This is equivalent to

log θ(p ) log p n < log(1 + n+1 ), pn+1 θ(pn) and this follows from the weaker inequality

pn+1 log pn+1 > θ(pn) log θ(pn),

that is satisﬁed if n> 2, pn+1 >pn > θ(pn). Thus (i) ⇒ (ii). Proving (i) is not an easy task. From the prime number theorem [1, 2], one knows that

θ(pn)= π(pn) log(pn) ∼ pn, where π(n) is the prime counting function. In addition, it has been proved that [16] (see also table 2 in [17])

θ(n) < n for 0 < n ≤ 1011, but no result is known to us concerning general values of n. The Riemann hypothesis Propositions 1 and 2 show that the worst case for the inequality (3), if not satisﬁed,

should occur at a primorial Nn. Proposition 3 stipulates that the function f(Nn) versus

Nn is a strictly decreasing function. Let us ﬁrst show that RH ⇒ (3). This is easy because if RH is true, then Robin’s inequality (1) is true, for any

n > 5040, including at primorials n = Nn [5]. Since Nn is free of square, one has

ψ(Nn)= σ(Nn) so that the refined inequality (3) is satisfied at n = Nn and, according to propositions 1 to 3, it is also satisfied at any n. The reverse implication (3) ⇒ RH is similar to that for Robin’s inequality (theorem 2 in [5]). By contradiction, if RH is false, there exists an infinity of numbers n such that γ ψ(n) g(n) > e and the bound on n is weakened as ψ(n) σ(n) 0.6482 for n ≥ 3, ≤ ≤ eγ log log n + . n n log log n We have proven that the modified Robin’s criterion is equivalent to RH. To prove that RH is true, it is sufficient to prove that no exception to this criterion can be found. 5

At large value of primorials Nn, we use Mertens theorem about the density of primes n 1 −γ ln pn (1 − ) ∼ e , or the equivalent relation § Qk=1 pk n γ 1 ψ(Nn) 1 1 e ≡ Y(1 + ) ∼ , (5) ln pn Nn ln pn pk ζ(2) k=1 and the lower bound given p. 206 of [5] 0.1235 for pn ≥ 20000, log log Nn > log pn − . (6) log pn Using (5) and (6), and with eγ (ζ(2)−1 − 1) ≈ −0.6983, one obtains a lower bound 0.1235 for pn ≥ 20000, f(n) < −0.6983 log pn + ∼ −6.90. (7) log pn

For values of 210 ≤ Nn ≤ 200000, computer calculations can be performed. The

calculation of f(Nn) = g(Nn) < 0 is fast using the multiplicative property of σ(n), i.e. n using σ(Nn) = Qi=1(1 + pi). One ﬁnd a decreasing function g(pn) that is negative if pn > 2 as expected (and in agreement with the exceptions in the set A).

pn 2 20 200 2000 20000 200000 f(Nn)= g(Nn) 0.96 −2.57 −4.87 -6.79 -8.60 -10.35

Table 2. The approximate values of the function f(Nn)= g(Nn) versus the number of 2748049 primes in the primorials Nn. The highest primorial in the table is N200000 ≈ 10 .

According to our calculations above and the bound (7), there is no exception to the refined Robin’s criterion. To summarize, if it can be proved that the n-th prime pn is always larger that the first Chebyshev function θ(pn), then the refined Robin’s criterion is equivalent to RH, and RH must be true.

Acknowledgements

The author thanks P. Sol´efor pointing out his paper [7], following the presentation of [12] at QuPa meeting in Paris on 09/23/2010. He also thanks Fabio Anselmi for his feedback on this subject.

Bibliography

[1] Edwards H M 1974 Riemann’s zeta function (Academic Press, New York). [2] Hardy G H and Wright E M 1979 An introduction to the theory of numbers, Fifth Edition (Oxford Univ. Press, Oxford). [3] Ivi´cA 1985 The Riemann zeta function. The theory of the Riemann zeta function with applications (Wiley, New York).

1 § A better approximation could be obtained from proposition 9 in [15], i.e. from Qp≤x(1 + p ) = eγ ζ(2) log x + O(1/ log x). 6

[4] Ramanujan S 1997 Highly composite numbers. Annotated and with a foreword by J L Nicolas and G Robin Ramanujan J. 1 119-153. [5] Robin G 1984 Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann J. Math. pures et appl. 63 187-213. [6] Nicolas J L 1983 Petites valeurs de la fonction d’Euler J. Numb. Th. 17 375-388. [7] Choie Y J, Lichiardopol N, Moree P and SoléP 2007 On Robin’s criterion for the Riemann hypothesis J. Theor. Nombres Bordeaux 19 351-366. [8] Aloglu L and Erdös 1944 On higly composite and similar numbers Trans. Am. Math. Soc. 56 448-469. [9] Akbary A and Friggstad Z 2009 Superabundant numbers and the Riemann hypothesis Amer. Math. Month. 116 273-275. [10] Erdös P and Nicolas 1975 J L Répartition des nombres superabondants Bull. Soc. math. France 103 65-90. [11] Schoeneberg B 1974 Elliptic modular functions (Springer Verlag, Berlin). [12] Planat M 2010 Pauli graphs when the Hilbert space dimension contains a square: why the Dedekind psi function? Preprint 1009.3858 [math-ph]. [13] Lagarias J C 2002 An elementary problem equivalent to Riemann hypothesis Amer. Math. Monthly 109 111-130. [14] Kupershmidt B A 2009 Remarks on Robin’s and Nicolas inequalities. Preprint 0903.1088 [math- ph]. [15] Carella N A 2010 A divisor function inequality. Preprint 0912.1866. [16] Schoenfeld L 1976 Sharper bounds for the Chebyshev functions θ(x) and ψ(x). II Math. Comp. 30 337-360. [17] Dusart P 1999 Sharper bounds for ψ, θ, π, pk. Rapport de recherche 1998-06 (Universitéde Limoges), http://www.unilim.fr/laco/rapports/1998/R1998 06.pdf.