On a series of Ramanujan

Transcription

1 On a series of Ramanujan Olivier Oloa To cite this version: Olivier Oloa. On a series of Ramanujan. Gems in Experimental Mathematics, pp.35-3,, <.9/conm/57/48>. <hal-55866> HAL Id: hal Submitted on 8 Jul 7 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 ON A SERIES OF RAMANUJAN Olivier Oloa Abstract. We uncover a remarkable evaluation of a family of infinite series involving the logarithmic derivative of the Gamma function. We are then led to a new evaluation of an integral involving Riemann s zeta function on the critical line.. Introduction Recently, B. C. Berndt and A. Dixit [5] offered a clear proof of an interesting identity coming from some manuscripts of Ramanujan involving an infinite series with the logarithmic derivative of the Gamma function and an integral with the Riemann zeta function. They proved Theorem.. If α and β are positive numbers such that αβ, then α {γ logπα α β + ψnα lognα + } nα { γ logπβ β π 3/ + ξ + it ψnβ lognβ + } nβ + it 4 cos t log α + t dt,. where γ denotes Euler s constant, ψ denotes the logarithmic derivative of the function, called the Digamma function, ψx : x x γ k + x, Re x >,. k + k and ξx is defined by ξs : ss π s s ζs, with Riemann s zeta function ζ. In the present note we uncover a new evaluation of a family of infinite series including the one involved in Ramanujan s Theorem. Our main result is Professor of Mathematics, IUT of Velizy-Rambouillet, University of Versailles, FRANCE. olivier.oloa@wanadoo.fr

3 O. OLOA Theorem.. If α and s are complex numbers such that Re α > and Reα + s >, then ψαn + s logαn + s + αn + s log + + / ψ / logπ α u /α u + u du..3 α u The preceding result yields different consequences. First, the right hand side of.3 shows clearly that the infinite series on the left hand side is expressible in finite terms of standard functions, whenever α and s are positive rational numbers. Next, combining Theorem. and Theorem. gives new informations for the non elementary integrals involving Riemann s ζ function. Hence, when α tends to +, one may deduce in particular ξ + it + it moreover, one has the asymptotic expansion, π 3/ ξ + it + it 4 4 cos t log α + t dt π3/ log α,.4 α cos t log α + t dt log α logπ γ + π α α 7 α α π4 8 α 3 α + O α 5..5 α Observe that the above expansion may be read in terms of Fourier cosine integral and thus may give, via the inverse transform, a possible path to estimate ξ + it + it 4 + t,.6 or equivalently ζ + it. In section we give a proof of our main result Theorem., then we display certain closed forms in section 3 and we establish the asymptotic expansion.5 in section 4.

4 ON A SERIES OF RAMANUJAN 3. The Proof In this section we establish our main result, Theorem.. Recall Binet s formula [3], p. 48, Re z >, log z z log z z + logπ + x + e zx e x dx,. x which, upon making x log v, can be written as log z z log z z + logπ log v + v v z dv.. log v If one differentiates. with respect to z, one may obtain ψz log z + z log v + v v z dv,.3 or equivalently, Gauss formula [3], p. 49 ψz log z log v + v z dv..4 v Let α > and s >. From.3, we deduce ψαn + s logαn + s + αn + s log v + v v αn+s dv, v α+s log v + v dv,.5 vα where we have interchanged the integral and sum, which may be justified by considering the finite sum and estimating the remainder. Substitute u v α in the latter integral to obtain log u + α u /α u du..6 α u We then split.6 in two integrals log u + u u u du + α u /α u + u du..7 α u Thus to prove Theorem. it is sufficient to evaluate log u + u u du..8 u One may check by a direct calculation that log u + u u u d { u u du log u + log u + u } + u u log u log u + u..9 u

5 4 O. OLOA Hence the integral in.8 is the sum of three integrals. The first, I u d du and, using.4, I u d du { u The second, applying., I } du log u + u log u + u u + s α log u + u du,. u { u log u + } du u + s ψ + log +.. α log u + u u log u du log / log + + logπ.. The third, using.4, I 3 log u + u du ψ + log +..3 u Consequently, log u + u u u du I + I + I 3 log / ψ + / + logπ..4 Using.5,.6,.7,.4, and analytic continuation, gives Theorem.. 3. Closed Forms Theorem. yields different closed forms. For example, the substitution u v in the following integrand leads us to integrate a rational function and produces 4 u /4 u 3 u /6 u du yielding, with.3, 3 6 log ψ4n + /3 log4n + /3 + 4n + /3 log 7/6 3 ψ7/6 logπ 3 6 log π π π 8, 3. 3π

6 In the same manner, from u / u + u / 4 u du one deduces, with.3, ON A SERIES OF RAMANUJAN 5 5 log ψn + /5 logn + /5 + n + /5 log / 3 5 ψ/ 5 logπ log One may observe the following particular family of unexpected closed forms. Corollary 3.. If s is a complex number such that Re s >, then ψn + s logn + s + n + s log s + log +3π , 3.3 3π log s + ψs + + s + logπ. 3.5 Proof. Put α in.3. Particular cases are, ψn log n + γ n logπ +, 3.6 ψn + / logn + / + γ + log, 3.7 n + / ψn / logn / + log, 3.8 n / ψn + /3 logn + /3 + n + /3 log /3 + 4 log γ logπ π One may differentiate 3.5 several times with respect to s, obtaining Proposition 3.. If s is a complex number such that Re s > and m is a natural number with m then, ψ m n + s mψ m s + sψ m s +. 3.

7 6 O. OLOA For example, putting m 3 and s in 3. gives n ζ4 k 4 ζ3. 3. where we have used k ψ m s + m m! Another interesting particular case of Theorem. is given by. 3. n + s m+ Corollary 3.3. If α is a complex number such that Re α >, then ψαn logαn + γ αn logπ + Proof. Put s in.3. α u /α u + α du u Asymptotic Expansion Theorem., via Corollary 3.3, gives a tractable way to obtain an asymptotic expansion for the infinite series ψαn logαn +. αn Inserting the asymptotic expansion of the previous series in Theorem. allows us to estimate, when α is great, the non elementary integrals involving Riemann s ζ function. Hence, we find Theorem 4.. Let α tend to +, then π 3/ ξ + it + it cos t log α 4 + t dt log α logπ γ + π α α 7 α α π4 8 α 3 α + O α α Proof. Let < u < and let α tend to +. We readily have α u /α u + α u log u + u u log u u α + log3 u 7 u α 4 + O α 6, 4.

8 from which we deduce α u /α u + du α u log u + u du u α ON A SERIES OF RAMANUJAN 7 log u u du+ log 3 u 7α 4 u du+ O α 6 Now, using.4 with s, log u + u du u γ logπ and considering the well-known results log u du π u 6, 4.5 we obtain α u /α u + α log 3 u du π4 u 5, 4.6 du u γ logπ + + π 7 α π4 8 α 4 + O α Finally inserting the expansion 4.7 in Corollary 3.3 and combining with Theorem. yields Theorem 4.. References [] M. Abramowitz and I.A. Stegun, eds., Handbook of Mathematical Functions, Dover, New York, 965. [] G.E. Andrews and B.C. Berndt, Ramanujan s Lost Notebook, Part IV, Springer, New York, to appear. [3] B.C. Berndt, Ramanujan s quarterly reports, Bull. London Math. Soc , [4] B.C. Berndt, Ramanujan s Notebooks, Part I, Springer Verlag, New York, 985. [5] B.C. Berndt and A. Dixit A transformation formula involving the Gamma and Riemann zeta functions in Ramanujan s Lost Notebook, to appear, [6] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol., New York: Krieger, 98. [7] I.S. Gradshteyn and I.M. Ryzhik, eds., Table of Integrals, Series, and Products, 5th ed., Academic Press, San Diego, 994. [8] A.P. Guinand, Some formulae for the Riemann zeta-function, J. London Math. Soc. 947, 4 8. [9] A.P. Guinand, A note on the logarithmic derivative of the Gamma function, Edinburgh Math. Notes 38 95, 4. [] S. Ramanujan, New expressions for Riemann s functions ξs and Ξt, Quart. J. Math , [] S. Ramanujan, Collected Papers, Cambridge University Press, Cambridge, 97; reprinted by Chelsea, New York, 96; reprinted by the American Mathematical Society, Providence, RI,. [] S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 988. [3] E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, Chemin Orme Aigu, ABLIS, 7866, FRANCE address: olivier.oloa@wanadoo.fr