arXiv:1703.02633v1 [math.NT] 7 Mar 2017 n a ics h rtmtcpoete fisvle,frexample for values, its of properties arithmetic the discuss can one ae n at and case) series seas 1 ]ad npriua,[,Scin25) hc pcaie at specializes which 2.5]), Section [8, particular, in and, 2] [1, also (see hc a esdpol’ mgnto ic h ie fElr oiet Notice Euler. of convergent times perfectly the a since is imagination (1) people’s teased has which nhs16 ae ndvretsre 5,L ue nrdcdadstu and introduced Euler L. [5], series divergent series on (hypergeometric) paper 1760 his In Introduction 1 abu nvriet OBx91,60 LNjee,TeNeth The Nijmegen, GL 6500 9010, Box PO Universiteit, Radboud rhmda valuations. Archimedean rtmtcadaayia usin eae oisvle i values its to related questions analytical and arithmetic h nvriyo ecsl,Clahn S 38 Australi 2308, NSW Callaghan, Newcastle, of University The 2 nttt o ahmtc,Atohsc n atcePhys Particle and Astrophysics Mathematics, for Institute ue’ atra eisadgoa relations global and series factorial Euler’s sn a´ prxmtost h series the to Pad´e approximations Using 1 z aeaika L30,904Olnyipso Finland yliopisto, Oulun 90014 3000, PL Matematiikka, = 3 colo ahmtcladPyia Sciences, Physical and Mathematical of School − aaiMatala-aho Tapani .Teirtoaiyo h atrspecialization, latter the of irrationality The 1. p ai eisi h disc the in series -adic E K W ( ac 2017 March 7 z = := ) := Abstract K X k 1 otemmr fMr Huttner Marc of memory the To p ∞ =0 X k ∞ 1 =0 n ai Zudilin Wadim and := ( − k X k 1) ( ! ∞ =0 E k − k k ( z z ! ! ) = ) , , k | z | P p ≤ k ∞ =0 o l primes all for 1 k both n ( ! − at , z 2,3 ) k eaddress we , z p z idteformal the died teWallis (the 1 = ai and -adic oWallis’ to 1 = a h series the hat (1947–2015) ics, erlands p a othat so , (3) (1) (2) for any prime p is a folklore conjecture [12, p. 17] (see [11], also for a link of the problem to a combinatorial conjecture of H. Wilf). Already the expectation K = 1 (so that | p|p Kp is a p-adic unit) for all primes p, which is an equivalent form of Kurepa’s conjecture [7] from 1971, remains open; see [4] for the latest achievements in this direction. In what follows, we refer to the series in (1) as Euler’s factorial series and label it Ep(z) when treat it as the function in the p-adic domain for a given prime p. Theorem 1. Given ξ Z 0 , let be a subset of prime numbers such that ∈ \{ } P n 2 2 lim sup c n! n! p =0, where c = c(ξ; ) := 4 ξ ξ p. (4) n | | P | | | | →∞ pY pY ∈P ∈P
Then either there exists a prime p for which Ep(ξ) is irrational, or there are two distinct primes p, q such that E∈( Pξ) = E (ξ) (while E (ξ), E (ξ) Q). ∈ P p 6 q p q ∈ Because n! =1/n! when the product is taken over all primes, condition (4) is p | |p clearly satisfiedQ for any subset whose complement in the set of all primes is finite. P This also suggests more exotic choices of . Furthermore, the conclusion of the theorem is contrasted with Euler’s sum P ∞ k k!= 1, · − Xk=0 n 1 which is valid in any p-adic valuation (and follows from − k k! = n! 1)—an k=0 · − example of what is called a global relation. P The idea behind the proof of Theorem 1 is to construct approximations pn/qn to the number ωp = Ep(ξ) in question, which do not depend on p and approximate the number considerably well for each p-adic valuation: q ω p = r for n =0, 1, 2,... ; n p − n n,p r 0 as n and there are infinitely many indices n for which r = 0 for | n,p|p → → ∞ n,p 6 at least one prime p . Assume that ωp = a/b, the same rational number, for all p . Then q a p∈b P Z 0 , so that 0 < q a p b 1, for infinitely many ∈ P n − n ∈ \{ } | n − n |p ≤ indices n and all primes p, hence
1= q a p b q a p b q a p b q a p b | n − n | | n − n |p ≤| n − n | | n − n |p Yp pY ∈P ( a + b ) max q , p br ≤ | | | | {| n| | n|} | n,p|p pY ∈P ( a + b ) max q , p r ≤ | | | | {| n| | n|} | n,p|p pY ∈P for those n. This means that the condition
lim sup max qn , pn rn,p p =0 (5) n {| | | |} | | →∞ pY ∈P contradicts the latter estimate, thus making the -global linear relation ωp = a/b impossible. P
2 The result in Theorem 1 can be put in a general context of global relations for Euler-type series; the corresponding settings can be found in the paper [3]. We do not pursue this route here as our principal motivation is a sufficiently elementary arithmetic treatment of an analytical function that have some historical value. The rational approximations to E(z) we construct in Section 2 are Pad´eapproximations; in spite of being known for centuries, implicitly from the continued fraction for E(z) given by Euler himself [5] and explicitly from the work of T. J. Stieltjes [13], these Pad´e approximations remain a useful source for arithmetic and analytical investigations. In Section 3 we revisit Euler’s summation of Wallis’ series (2) using the approximations; we also complement the derivation by providing ‘Archimedean analogue(s)’ for the divergent series K = E( 1) from (3) — the case when the classical strategy does not − work. Our way of constructing the Pad´eapproximations is inspired by a related Pad´e construction of M. Hata and M. Huttner in [6]. The construction was a particular favourite of Marc Huttner who, for the span of his mathematical life, remained a passionate advocate of interplay between Picard–Fuchs linear differential equations and Pad´eapproximations. We dedicate this work to his memory.
2 Hypergeometric series and Pad´eapproximations
Euler’s factorial series (1) is the particular a = 1 instance of the hypergeometric series
∞ k 2F0(a, 1 z)= (a) z . (6) | k Xk=0
Here and in what follows, we use the Pochhammer notation (a)k which is defined inductively by (a)0 = 1and(a)k+1 =(a+k)(a)k for k Z 0. Our Pad´eapproximations ≥ below are given more generally for the function (6). ∈
Theorem 2. For n, λ Z 0, take ∈ ≥ n i n i n ( 1) z − Bn,λ(z)= − . i (a) + Xi=0 i λ
Then degz Bn,λ = n and for a polynomial An,λ(z) of degree degz An,λ n + λ 1 we have ≤ − B (z) 2F0(a, 1 z) A (z)= L (z), (7) n,λ | − n,λ n,λ where ordz=0 Ln,λ(z)=2n + λ. Explicitly,
∞ n + k n + k + a + λ 1 L (z)=( 1)nn! z2n+λ k! − zk. (8) n,λ − k k Xk=0 Proof. Relation (7) means that there is a ‘gap’ of length n in the power series expansion
B (z) 2F0(a, 1 z)= A (z)+ L (z). n,λ | n,λ n,λ 3 n h Write Bn,λ(z)= h=0 bht and consider the series expansion of the product P ∞ l B (z) 2F0(a, 1 z)= r z , n,λ | l Xl=0 where rl = h+k=l bh(a)k; in particular, P n n i n (a)i+λ+m i n rl = ( 1) = ( 1) (a + i + λ)m − i (a) + − i Xi=0 i λ Xi=0 with m = l n λ for l > n + λ 1. To verify the desired ‘gap’ condition, − − −
rn+λ = rn+λ+1 = = rn+λ+n 1 =0, (9) ··· − we introduce the shift operators N = N and ∆ = ∆ = N id defined on functions a a − f(a) by Nf(a)= f(a +1) and ∆f(a)= f(a + 1) f(a). − It follows from
n n ∆ (a + λ)m =( 1) ( m)n(a + λ + n)m n for n Z 0 − − − ∈ ≥ that
n n r = ( 1)i N i(a + λ) = (id N)n(a + λ) l − i m − m Xi=0 n =( ∆) (a + λ)m =( m)n(a + λ + n)m n, − − − which, in turn, implies (9) because of the vanishing of ( m) for m =0, 1,...,n 1. − n − The explicit expression for rl just found also gives
n n + k n + k + a + λ 1 r2 + + =( n k) (a + λ + n) =( 1) n! k! − , n λ k − − n k − k k hence the closed form (8). We also have
n+λ 1 n+λ 1 n − − l l i n (a)i+l n An,λ(z)= rlz = z ( 1) − . − i (a)i+λ Xl=0 Xl=0 Xi=0 i n l ≥ − This concludes our proof of the theorem. From now on we choose a = 1, λ = 0, change z to z and renormalize the − corresponding Pad´eapproximations produced by Theorem 2 by multiplying them by
4 ( 1)nn!: − n n 2 n n n! n i n i Q (z) := ( 1) n! B 0( z)= z − = i! z , n − n, − i i! i Xi=0 Xi=0 n 1 n − n n l n i n n!(i + l n)! P (z) := ( 1) n! A 0( z)=( 1) ( z) ( 1) − − n − n, − − − − i i! Xl=0 i=Xn l − n 1 l 2 − n =( 1)n ( z)l ( 1)ii!(l i)! − − − − i Xl=0 Xi=0 and
2 ∞ n 2 2n k n + k k R (z) := ( 1) n! L 0( z)= n! z ( 1) k! z . n − n, − − k Xk=0 In this notation, the Pad´eapproximation formula (7) may be rewritten as
Q (z)E(z) P (z)= R (z) (10) n − n n for n Z 0. Observe that ∈ > 2 2n Q (z)P +1(z) Q +1(z)P (z)= n! z . (11) n n − n n Indeed, the standard Pad´ewalkabout proves the identity
Q (z)P +1(z) Q +1(z)P (z)= Q +1(z)R (z) Q (z)R +1(z), n n − n n n n − n n in which the degree of the left-hand side is at most 2n while the order of the right-hand side is at least 2n. Proof of Theorem 1. We may use the formal series identity (10) to get appropriate numerical approximations for any at z = ξ Z 0 . For n = 1, 2,... , take pn = P (ξ) Z, q = Q (ξ) Z and define ∈ \{ } n ∈ n n ∈ r = R (ξ)= q E (ξ) p n,p n n p − n for each prime p . Using elementary summation formulas and trivial estimates for ∈ P binomials we have
2 n n 2n q ξ nn! = ξ nn! < 4n ξ nn! , | n|≤| | i | | n | | Xi=0 n 1 2 n 2 − n 1 n n n n n p ξ − n i!(n 1 i)! ξ n! < 4 ξ n! | n|≤| | − − i ≤| | i | | Xi=0 Xi=0
5 and 2 ∞ n + k r = ξ 2n n! 2 ( 1)kk! ξk ξ 2n n! 2. | n,p|p | |p | |p − k ≤| |p | |p Xk=0 p
Therefore, condition (5) reads
n n 2n 2 lim sup 4 ξ n! ξ p n! p =0 n | | | | | | →∞ pY ∈P and because for at least one p we have either rn,p =0 or rn+1,p = 0 from (11), the theorem follows. ∈ P 6 6
3 Summation of divergent series
Fix a prime p. If we restrict, for simplicity, the sequence of indices n to the arithmetic progression n 0 mod p, then it is not hard to see from the calculation in Section 2 ≡ that the sequence of rational approximations pn/qn = Pn(ξ)/Qn(ξ) converges p-adically to E(ξ). Indeed,
p 1 2 n 2 − n n q =1+ i! ξi + i! ξi 1 mod p n i i ≡ Xi=1 Xi=p is a p-adic unit (all the binomials are divisible by p in the first sum, while i! is divisible by p in the second one), and
pn 1 E(ξ) = q − q E(ξ) p = r 0 as n . − q | n|p | n − n|p | n,p|p → →∞ n p
When ξ > 0, the same sequence of rational numbers pn/qn = Pn(ξ)/Qn(ξ) converges to the value at z = ξ of the integral
s ∞ e E˜(z)= − ds (12) Z0 1+ zs with respect to the Archimedean norm. The integral itself converges for any z ∈ C ( , 0], though its formal Taylor expansion at z = 0 (obtained by expanding 1/(1\ +−∞zs) into the power series under the integral sign) is precisely Euler’s factorial series E(z) as in (1). In particular, relation (10) remains valid with E(z) replaced by E˜(z) and Rn(z) by
n s ˜ 2n ∞ s e− Rn(z)= n! z n+1 ds for n =0, 1, 2,..., Z0 (1 + zs) so that p R (ξ) nn E˜(ξ) n = n 0 as n , − q Q (ξ) ≤ (n + 1)n+1 → →∞ n en
6 where the estimates Q (ξ) n! ξn and n ≥ n n 2n s ∞ s n n ˜ − Rn(ξ) n! ξ max n+1 e ds = n! ξ n+1 ≤ s>0 (1 + ξs) Z0 (n + 1) were used. This computation reveals us that
p ∞ ( x)k lim n = E˜(ξ)= xex γ + log x + − , n qn − k k! →∞ Xk=1 x=1/ξ · where γ =0.5772156649 ... is Euler’s constant. In particular,
k 1 ∞ ( 1) − W = E˜(1) = e γ + − =0.5963473623 ... − k k! Xk=1 · for Wallis’ series (2). The resulted quantity is known as the Euler–Gompertz constant [8]. The strategy in the previous paragraph does not apply to z = ξ < 0, somewhat already observed by Stieltjes in [13]. The analytical continuation of the the function E˜(z) depends on whether we perform the integration along the upper or lower banks of the ray [0, ) in (12); denote the corresponding values by E˜+(z) and E˜ (z), respec- ∞ s − tively. By considering the integration of e− /(1+zs) along the curvilinear triangle that √ 1θ consists of the segment [0, R] (along a particular bank), the arc [R,Re − ] followed √ 1θ by the segment [R e − , 0], where 0 <θ<π/2 for the upper bank and π/2 <θ< 0 − for the lower one, and then taking the limit as R (so that the integral along the arc tends to 0) we conclude that →∞
√ 1θ e − s k ∞ e− ∞ ( x) E˜ (z)= ds = xex γ + log x √ 1π + − , (13) ± Z0 1+ zs − | | ± − k k! Xk=1 x=1/z · with the choice of θ arbitrary in the interval 0 <θ<π/2 for E˜+(z) and in the interval π/2 <θ< 0 for E˜ (z). In particular, − − 1 ∞ 1 K = E˜ ( 1) = γ + √ 1π ± − e k k! ∓ − Xk=1 · =0.6971748832 ... √ 1 1.1557273497 ... ∓ − · for the series in (3).
4 Final remarks
In [14] we outline a different strategy of proving a result analogous to Theorem 1 on using the Hankel determinants generated by the tails of Euler’s factorial series (1). As the condition on a subset of primes in that result is spiritually similar to (4), P 7 we do not detail the derivation here. However we stress that a potential combination of the two methods, namely, using the Hankel determinants generated by the Pad´e approximations of Euler’s factorial series, may be a source of further novelties on the topic. A discussion on this type of construction in the Archimedean setting can be found in [15]. One consequence of the formula in (13), which uncovers a pair of complex conjugate values for E(ξ) when ξ < 0, is that the rational approximations pn/qn = Pn(ξ)/Qn(ξ) do not converge at all in such cases. Interestingly enough, the Hankel determinants ‘see’ those complex values (13) as experimentally observed in [14]. Finally, we would like to note that nothing is known about the irrationality and transcendence of the Archimedean valuations of Euler’s factorial series (1) at rational z = ξ (see the discussion in [8, Sections 3.15, 3.16]). This is in contrast with its q-analogue k ∞ zk (1 qi), − Xk=0 Yi=1 for which irrationality and linear independence results are known in Archimedean and non-Archimedean places alike — see [9]. Further details on a nice q-counterpart of the Pad´eapproximation analysis can be found in [10].
References
[1] E. J. Barbeau and P. J. Leah, Euler’s 1760 paper on divergent series, Historia Math. 3 (1976), no. 2, 141–160.
[2] E. J. Barbeau, Euler subdues a very obstreperous series, Amer. Math. Monthly 86 (1979), no. 5, 356–372.
[3] D. Bertrand, V. Chirskii and J. Yebbou, Effective estimates for global re- lations on Euler-type series, Ann. Fac. Sci. Toulouse Math. (6) 13 (2004), no. 2, 241–260.
[4] V. G. Chirskii, Arithmetic properties of Euler series, Vestnik Moskov. Univ. Ser. I Mat. Mekh. (2015), no. 1, 59–61; English transl., Moscow Univ. Math. Bull. 70 (2015), no. 1, 41–43.
[5] L. Euler, De seriebus divergentibus, Novi. Commentii Acad. Sci. Petrop. 5 (1760), 205–237; in: Opera Omnia, Series 1, Vol. 14, pp. 585–617.
[6] M. Hata and M. Huttner, Pad´eapproximation to the logarithmic derivative of the Gauss hypergeometric function, in: Analytic number theory (Beijing/Kyoto, 1999), Dev. Math. 6 (Kluwer Acad. Publ., Dordrecht, 2002), 157–172.
[7] D. Kurepa, On the left factorial function !n, Math. Balkanica 1 (1971), 147–153.
8 [8] J. C. Lagarias, Euler’s constant: Euler’s work and modern developments, Bull. Amer. Math. Soc. 50 (2013), no. 4, 527–628.
[9] T. Matala-aho, On q-analogues of divergent and exponential series, J. Math. Soc. Japan 61 (2009), no. 1, 291–313.
[10] T. Matala-aho, Type II Hermite–Pad´eapproximations of generalized hyperge- ometric series, Constr. Approx. 33 (2011), 289–312.
k [11] M. R. Murty and S. Sumner, On the p-adic series ∞ n n!, in: Number n=1 · theory, H. Kisilevsky and E. Z. Goren (eds.), CRM Proc.P Lecture Notes 36 (Amer. Math. Soc., Providence, RI, 2004), 219–227.
[12] W. H. Schikhof, Ultrametric calculus. An introduction to p-adic analysis, Cam- bridge Stud. Adv. Math. 4 (Cambridge Univ. Press, Cambridge, 1984).
[13] T.-J. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys. 8 (1894), no. 4, J1–J122.
[14] W. Zudilin, Arithmetic applications of Hankel determinants (joint with C. Krat- tenthaler, T. Matala-aho, V. Meril¨a, I. Rochev and K. V¨a¨an¨anen), in: Diophan- tische Approximationen (MFO, Oberwolfach, Germany, 22–28 April 2012), Ober- wolfach Reports (2012), no. 22, 1345–1347.
[15] W. Zudilin, A determinantal approach to irrationality, Constr. Approx. 45 (2017), no. 2, 301–310.
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