Summability methods aFrom Wikipedia, the free encyclopediaContents1 Abels summation formula 11.1 Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.1 EulerMascheroni constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2.2 Representation of Riemanns zeta function . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.3 Reciprocal of Riemann zeta function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Abels theorem 32.1 Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.3 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Outline of proof. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Related concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.7 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Abelian and tauberian theorems 63.1 Abelian theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.2 Tauberian theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.4 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 AbelPlana formula 84.1 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.2 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84.3 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 HardyLittlewood tauberian theorem 105.1 Statement of the theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.1.1 Series formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105.1.2 Integral formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11iii CONTENTS5.2 Karamatas proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.3.1 Non-positive coecients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.3.2 Littlewoods extension of Taubers theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 125.3.3 Prime number theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 Littlewoods Tauberian theorem 146.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.2 History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 Slowly varying function 167.1 Denition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 Wieners tauberian theorem 188.1 The condition in L1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.1.1 Tauberian reformulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188.1.2 Discrete version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.2 The condition in L2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198.5 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 WienerIkehara theorem 219.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.2 Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219.3 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229.4 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 239.4.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.4.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239.4.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Chapter 1Abels summation formulaAnother concept sometimes known by this name is summation by parts.In mathematics, Abels summation formula, introduced by Niels Henrik Abel, is intensively used in number theoryto compute series.1.1 IdentityLet anbe a sequence of real or complex numbers and (x) a function of class C1. Then1nxan(n) = A(x)(x) x1A(u)(u) duwhereA(x) :=1nxan .Indeed, this is integration by parts for a RiemannStieltjes integral.More generally, we havex1 . It may be used to derive Dirichlets theorem, that is, (s) has a simple pole withresidue 1 in s = 1.1.2.3 Reciprocal of Riemann zeta functionIf an= (n) is the Mbius function and (x) =1xs, then A(x) = M(x) =nx (n) is Mertens function and1(n)ns= s1M(u)u1+s du.This formula holds for (s) > 1 .1.3 See alsoSummation by parts1.4 ReferencesApostol, Tom(1976), Introduction to Analytic Number Theory, Undergraduate Texts in Mathematics, Springer-Verlag.Chapter 2Abels theoremThis article is about Abels theorem on power series. For Abels theorem on algebraic curves, see AbelJacobi map.For Abels theoremon the insolubility of the quintic equation, see AbelRuni theorem. For Abels theoremon lineardierential equations, see Abels identity. For Abels theorem on irreducible polynomials, see Abels irreducibilitytheorem.In mathematics, Abels theorem for power series relates a limit of a power series to the sum of its coecients. It isnamed after Norwegian mathematician Niels Henrik Abel.2.1 TheoremLet a = {ak: k 0} be any sequence of real or complex numbers and letGa(z) =k=0akzkbe the power series with coecients a. Suppose that the series k=0 akconverges. Thenlimz1Ga(z) =k=0ak, ()where the variable z is supposed to be real, or, more generally, to lie within any Stolz angle, that is, a region of theopen unit disk where|1 z| M(1 |z|)for some M. Without this restriction, the limit may fail to exist: for example, the power seriesn>0(z3n z23n)/nconverges to 0 at z=1, but is unbounded near any point of the form ei/3n, so the value at z=1 is not the limit as z tendsto 1 in the whole open disk.Note that Ga(z) is continuous on the real closed interval [0, t] for t < 1, by virtue of the uniform convergence of theseries on compact subsets of the disk of convergence.Abels theorem allows us to say more, namely that Ga(z) iscontinuous on [0, 1].34 CHAPTER 2. ABELS THEOREM2.2 RemarksAs an immediate consequence of this theorem, if z is any nonzero complex number for which the series k=0 akzkconverges, then it follows thatlimt1Ga(tz) =k=0akzkin which the limit is taken from below.The theorem can also be generalized to account for innite sums. Ifk=0ak= then the limit from below limz1 Ga(z) will tend to innity as well. However, if the series is only known to bedivergent, the theorem fails; take for example, the power series for11+z . The series is equal to 1 1 + 1 1 + at z= 1 , but 1/(1 + 1) = 1/2 .2.3 ApplicationsThe utility of Abels theorem is that it allows us to nd the limit of a power series as its argument (i.e.z ) approaches1 from below, even in cases where the radius of convergence,R , of the power series is equal to 1 and we cannotbe sure whether the limit should be nite or not. See e.g. the binomial series. Abels theorem allows us to evaluatemany series in closed form. For example, whenak=(1)k/(k+1) , we obtainGa(z) =ln(1+z)/z for0 0 we can take N large enough to make the initial segment of terms up to cNaverage to at most /2, while each term in the tail is bounded by /2 so that the average is also necessarily bounded.The name derives from Abels theorem on power series. In that case L is the radial limit (thought of within thecomplex unit disk), where we let r tend to the limit 1 from below along the real axis in the power series with termanznand set z = rei. That theorem has its main interest in the case that the power series has radius of convergence exactly1: if the radius of convergence is greater than one, the convergence of the power series is uniform for r in [0,1] sothat the sum is automatically continuous and it follows directly that the limit as r tends up to 1 is simply the sum ofthe an. When the radius is 1 the power series will have some singularity on |z| = 1; the assertion is that, nonetheless,if the sum of the an exists, it is equal to the limit over r. This therefore ts exactly into the abstract picture.3.2 Tauberian theoremsPartial converses to abelian theorems are called tauberian theorems. The original result of Tauber (1897) statedthat if we assume also63.3. REFERENCES 7an = o(1/n)(see Little o notation) and the radial limit exists, then the series obtained by setting z = 1 is actually convergent.This was strengthened by J. E. Littlewood: we need only assume O(1/n).A sweeping generalization is the HardyLittlewood tauberian theorem.In the abstract setting, therefore, an abelian theorem states that the domain of L contains convergent sequences, andits values there are equal to those of the Lim functional. A tauberian theorem states, under some growth condition,that the domain of L is exactly the convergent sequences and no more.If one thinks of L as some generalised type of weighted average, taken to the limit, a tauberian theorem allows oneto discard the weighting, under the correct hypotheses. There are many applications of this kind of result in numbertheory, in particular in handling Dirichlet series.The development of the eld of tauberian theorems received a fresh turn with Norbert Wiener's very general results,namely Wieners tauberian theorem and its large collection of corollaries. The central theorem can now be proved byBanach algebra methods, and contains much, though not all, of the previous theory.3.3 ReferencesKorevaar, Jacob (2004). Tauberian theory. A century of developments. Grundlehren der MathematischenWissenschaften 329. Springer-Verlag. ISBN 978-3-540-21058-0.Montgomery, Hugh L.; Vaughan, Robert C. (2007). Multiplicative number theory I. Classical theory. Cam-bridge tracts in advanced mathematics 97. Cambridge: Cambridge Univ. Press. pp. 147167. ISBN 0-521-84903-9.Tauber, A. (1897). Ein Satz aus der Theorie der unendlichen Reihen (A theorem from the theory of inniteseries)". Monatsh. F. Math. (in German) 8: 273277. doi:10.1007/BF01696278. JFM 28.0221.02.Wiener, N. (1932). Tauberian theorems. Annals of Mathematics33 (1): 1100. doi:10.2307/1968102.JSTOR 1968102.3.4 External linksHazewinkel, Michiel, ed. (2001), Tauberian theorems, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4Chapter 4AbelPlana formulaIn mathematics, the AbelPlana formula is a summation formula discovered independently by Niels Henrik Abel(1823) and Giovanni Antonio Amedeo Plana (1820). It states thatn=0f(n) =0f(x) dx +12f(0) + i0f(it) f(it)e2t 1dt.It holds for functions f that are holomorphic in the region Re(z) 0, and satisfy a suitable growth condition in thisregion; for example it is enough to assume that |f| is bounded by C/|z|1+ in this region for some constants C, > 0,though the formula also holds under much weaker bounds. (Olver 1997, p.290).An example is provided by the Hurwitz zeta function,(s, ) =n=01(n + )s=1ss 1+12s+ 20sin(s arctant)(2+ t2)s2dte2t 1.Abel also gave the following variation for alternating sums:n=0(1)nf(n) =12f(0) + i0f(it) f(it)2 sinh(t)dt.4.1 See alsoEulerMaclaurin summation formula4.2 ReferencesAbel, N.H. (1823), Solution de quelques problmes laide dintgrales dniesButzer, P. L.; Ferreira, P. J. S. G.; Schmeisser, G.; Stens, R. L. (2011), The summation formulae of Euler-Maclaurin, Abel-Plana, Poisson, and their interconnections with the approximate sampling formula of signalanalysis, Results in Mathematics 59 (3):359400, doi:10.1007/s00025-010-0083-8, ISSN 1422-6383, MR2793463Olver, Frank W. J. (1997) [1974], Asymptotics and special functions, AKP Classics, Wellesley, MA: AKPetersLtd., ISBN 978-1-56881-069-0, MR 1429619Plana, G.A.A. (1820), Sur une nouvelle expression analytique des nombres Bernoulliens, propre exprimeren termes nis la formule gnrale pour la sommation des suites, Mem. Accad. Sci. Torino 25: 40341884.3. EXTERNAL LINKS 94.3 External linksAnderson, David, Abel-Plana Formula, MathWorld.Chapter 5HardyLittlewood tauberian theoremIn mathematical analysis, the HardyLittlewood tauberian theorem is a tauberian theorem relating the asymptoticsof the partial sums of a series with the asymptotics of its Abel summation. In this form, the theorem asserts that if,as y 0, the non-negative sequence an is such that there is an asymptotic equivalencen=0aneny1ythen there is also an asymptotic equivalencenk=0ak nas n . The integral formulation of the theorem relates in an analogous manner the asymptotics of the cumulativedistribution function of a function with the asymptotics of its Laplace transform.The theorem was proved in 1914 by G. H. Hardy and J. E. Littlewood.[1]:226 In 1930 Jovan Karamata gave a new andmuch simpler proof.[1]:2265.1 Statement of the theorem5.1.1 Series formulationThis formulation is from Titchmarsh.[1]:226 Suppose an 0 for all n, and as x 1 we haven=0anxn11 x.Then as n goes to we havenk=0ak n.The theorem is sometimes quoted in equivalent forms, where instead of requiring an 0, we require an = O(1), orwe require an K for some constant K.[2]:155 The theorem is sometimes quoted in another equivalent formulation(through the change of variable x = 1/ey).[2]:155 If, as y 0,105.2. KARAMATAS PROOF 11n=0aneny1ythennk=0ak n.5.1.2 Integral formulationThe following more general formulation is from Feller.[3]:445 Consider a real-valued function F : [0,) R ofbounded variation.[4] The LaplaceStieltjes transform of F is dened by the Stieltjes integral(s) =0estdF(t).The theorem relates the asymptotics of with those of F in the following way. If is a non-negative real number,then the following are equivalent(s) Cs, as s 0F(t) C( + 1)t, as t .Here denotes the Gamma function. One obtains the theorem for series as a special case by taking = 1 and F(t) tobe a piecewise constant function with value nk=0 ak between t=n and t=n+1.A slight improvement is possible. A function L(x) is slowly varying at innity ifL(tx)L(x) 1, x for every positive t. Let L be a function slowly varying at innity and a non-negative real number. Then the followingare equivalent(s) sL(s1), as s 0F(t) 1( + 1)tL(t), as t .5.2 Karamatas proofKaramata (1930) found a short proof of the theorem by considering the functions g such thatlimx1(1 x)anxng(xn) =10g(t)dtAn easy calculation shows that all monomials g(x)=xkhave this property, and therefore so do all polynomials g. Thiscan be extended to a function g with simple (step) discontinuities by approximating it by polynomials from above andbelow (using the Weierstrass approximation theorem and a little extra fudging) and using the fact that the coecientsan are positive.In particular the function given by g(t)=1/t if 1/e