GENERALIZED STIELTJES CONSTANTS AND INTEGRALS INVOLVING THE LOG LOG FUNCTION: KUMMER S THEOREM IN ACTION OMRAN KOUBA. 1. Introduction and notation

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1 Journal of Classical Analysis Volume 9, Number 16, doi:1715/jca-9-9 GENERALIZED STIELTJES CONSTANTS AND INTEGRALS INVOLVING THE LOG LOG FUNCTION: KUMMER S THEOREM IN ACTION OMRAN KOUBA Abstract In this note, we recall Kummer s Fourier series expansion of the 1-periodic function that coincides with the logarithm of the Gamma function on the unit interval,1, and we use it to find closed forms for some numerical series related to the generalized Stieltjes constants, and some integrals involving the function x lnln1/x 1 Introduction and notation The aim of this paper is to present an alternative proof of the reflection principle of the first order generalized Stieltjes constants, and to give an alternative approach to the evaluation of some integrals involving the function x ln ln1/x The basic tool for this investigation is a result of Kummer recalled below Theorem 1 The first order generalized Stieltjes constant γ 1 a is defined for a,1 by γ 1 a= lim n From this, it is easy to show that n = n γa γ1 a= lim n = n lna + 1 a + ln n + a ln a + def = a + ln a + n n Z a + n, where the primed sum denotes the principal value as shown above For integers p and with < p < the difference γp/ γ1 p/ can be expressed as follows γp/ γ1 p/= lne γ cot p 1 + sin jp j lnγ The formula is attributed to Almvist and Meurman who obtained it by calculating the derivative of the functional euation for the Hurwitz zeta function ζs,v with respect to s at rational v, see[] However, it was recently discovered that an euivalent form of this formula was already obtained by Carl Malmsten in 1846 see [5] An elementary proof of this formula will be presented in Proposition In a recent series of articles [], [9], [1], [11], [14], the authors proved some formulas from the Table of integrals, Series, and Products, of Gradshteyn and Ryzhi Mathematics subject classification 1: B15 Keywords and phrases: Gamma function, log-log integrals, Fourier series, numerical series c D l,zagreb Paper JCA

2 8 O KOUBA [7] Further, the monographs [1, 1] are devoted to providing proofs for the formulas in [7] In fact, we are particularly interested in integrals involving the function x ln ln1/x Indeed, entries 45 of [7] contain the following evaluations: lnln1/x 1 + x = Γ/4 ln Γ1/4 lnln1/x 1 + x + x = Γ/ ln Γ1/ lnln1/x 1 + xcost + x = sint ln t/ Γ 1 + t Γ 1 t These integrals can be traced bac to [6] The first of them was the object of a detailed investigation in [14], where the author says that his approach can be adapted to prove also the second one A general approach that yields the first two integrals, and much more evaluations, can also be found in [] This line of investigation was completed by adapting the methods of [14] to obtain general results that include all the above mentioned integrals in [11] Our aim is to present an alternative approach to the evaluation of these integrals Our starting point will be Kummer s Fourier expansion of LogΓ, Theorem1, where Γ is the well-nown Eulerian gamma function This result is attributed to Kummer in 1847, a more accessible reference is [4, Section 17]: THEOREM 1 Kummer, [8] For < x < 1, ln Γx = lnsinx 1 +γ + ln x + 1 where γ is the Euler-Mascheroni constant ln sinx, The reflection formula for the first order generalized Stieltjes constants As we explained in the introduction, this formula relates the first order generalized Stieltjes constant γ 1 a to its reflected value γ 1 1 a for rational a The presented proof is different from that of Almvist and Meurman, and has the advantage of being elementary in the sense that it does not mae use of the functional euation of the Hurwitz zeta function PROPOSITION For positive integers p and with p <, we have ln n + p p n Z n + p = lne γ 1 jp cot + sin lnγ where the primed sum denotes the principal value, defined as follows: n a n = lim a n Z n = n j

3 KUMMER S THEOREM IN ACTION 81 Proof The statement of Theorem 1 is written as ln sinx= ln + lnsinx+lneγ x 1 + lnγx 1 Now, consider a positive integer with For j {1,,, 1} we have ln j sin = ln + j lnsin j + lne γ Multiply both sides of bysin and add the resulting eualities for j = 1,, 1, to obtain where ln A p,= ln B p, + C p, + lnγ j 1 jp,where p is some integer from {1,, 1}, + lne γ 1 D p, + sin 1 A p, = 1 B p, = 1 C p, = D p, = 1 jp jp j sin sin jp sin jp j sin lnsin j 1 jp sin These sums are now simplified Let ω = exp i j lnγ,, and use 1 j= ωnj = χ n where χ n=1ifn mod and χ n= otherwise The imaginary part of the identity gives B p, = 4 Also, A p, = 1 1 cos = 1 R 1 j= jp jp + cos ω p j 1 j= ω p+ j = χ p χ p +

4 8 O KOUBA That is A p, = { if p mod, if p mod 5 On the other hand, the change of summation index j j in the formula for C p, shows that 1 C p, = sin p jp lnsin j = C p, Thus, Finally, use 4 to obtain D p, = 1 C p, = 6 1 j sin jp Now for, < θ <,wehave cos jθ= e ijθ = eiθ e i1 θ e iθ 1 sin 1θ = = sinθcotθ cosθ sin θ Taing the derivative with respect to θ and substituting θ = p/ we get D p, = 1 p cot 7 Replacing 4,5,6 and7in we obtain ln + p ln + p + p/ + 1 p/ = = lne γ cot 1 + sin jp p The final step is to use the well-nown cotangent partial fraction expansion: = 1 + p/ 1 = lim + 1 p/ N N = N = p + p/ p/ = cot j lnγ p/ = N p p/ p/ p 9 Thus, subtracting ln times 9 from 8 we obtain the desired conclusion

5 KUMMER S THEOREM IN ACTION 8 EXAMPLES Taing p = 1and {,4} we obtain ln n + 1 n Z n + 1 = Γ 1 ln e γ ln n Γ n Z n + 1 = ln 4 4 e γ The evaluation of some integrals involving the log-log function In this section we use Theorem 1, to evaluate some difficult integrals PROPOSITION For < x < 1, we have: ln ln1/u u cosxu + 1 du = sinx 1 xln+ln And, taing the limit as x tend to 1/, we obtain 1 ln ln1/u u + 1 du = ln + Γ 1/ Γ1/ = ln γ Γ1 x Γx Proof Indeed, subtracting the corresponding Kummer s Formulas, for lnγx and lnγ1 x we see that, for < x < 1wehave or euivalently, Γx ln =γ + ln1 x+ Γ1 x ln x Γx 1 x Γ1 x = γ 1 x+ ln sinx, 1 ln sinx 11 Now, using the fact that for Rs > and 1wehave Γs s = ts 1 e t dt,we conclude that for s >, and < x < 1, we have e ix s = 1 Γs t s 1 e t e ix dt = 1 e t+ix Γs 1 e t+ix ts 1 dt Restricting our attention to the imaginary parts we get sinx s = sinx Γs t s 1 e t + e t dt 1 cosx

6 84 O KOUBA Now, taing the derivative with respect to s at s = 1 we obtain, for < x < 1, the following: ln sinx= Γ 1 sinx Γ 1 e t + e t cosx dt sinx lnt Γ1 e t + e t dt 1 cosx Taing into account the facts Γ 1= γ, Γ1=1, and sinx 1 e t + e t cosx dt = x, for < x < 1 we conclude that ln sinx= γ 1 x sinx lnt e t + e t dt 14 cosx The change of variables t = ln1/u yields: ln sinx 1 ln ln1/u sinx+γ1 x= u du 15 cosxu + 1 Finally, combining 11 and15 we obtain the desired result Concerning the limit as x tend to 1/, we use the well-nown fact that Γ 1/ = ψ1/= γ ln,see[1, Γ1/ 6] EXAMPLES Taing x = 1/, x = 1/4 andx = 1/6 we obtain ln ln1/u 1 u du = + u ln Γ1 16 ln ln1/u 1 1 u du = ln 4 Γ ln ln1/u 1 1 u du = u + 1 ln 5 Γ6 = 6 1 ln 7 8 Γ1 18 where we used freely the duplication, and the reflection formulas for the gamma function [1, 6117 and 6118] In particular, we used Γ 1 6 = / Γ 1 that follows readily from these formulas The second degree polynomial in the integrand s denominator in Proposition has negative discriminant In the next proposition the corresponding denominator has real roots outside the interval [,1] This case seems to be new to the best nowledge of the author PROPOSITION 4 Let A Γ : R R be the function defined by A Γ y= ln y + sinhy ln ln1/u u + coshyu + 1 du

7 KUMMER S THEOREM IN ACTION 85 Then, for y R we have 1 + iy Γ = coshy/ eia Γy Proof Let us rephrase Proposition, bytaingx = t+1 in order to give more symmetric aspect to the formula there: ln ln1/u t 1,1, u + costu + 1 du = Γ 1+t lnt + ln sint Γ 1 t, or euivalently, for 1 < t < 1, we have exp lnt + sint 1+t ln ln1/u u + costu + 1 du Γ = Γ 1 t Using analytic continuation we deduce that, for 1 < Rz < 1wehavealso exp lnz + sinz ln ln1/u u + coszu + 1 du = In particular, setting z = iy with y R, we obtain 1+iy e iaγy Γ = Γ 1 iy But, by Euler s reflection formula [1, 6117]we nowthat 1 + iy 1 + iy 1 + iy 1 + iy 1 iy Γ = Γ Γ = Γ Γ therefore, the suare of the continuous function: coshy/ 1 + iy y Γ e ia Γy Γ 1+z Γ 1 z = coshy/, is eual to 1 for every y R, hence, it must be constant and conseuently identical to 1 which is its value for y = COROLLARY 1 Let the principal determination of the argument of a nonzero complex number z be denoted by Arg, and let α be defined by the formula { } 1 + iy α = inf y > :Γ = coshy/ Then, for every y α,α we have ln ln1/u u + coshyu + 1 du = ln y sinhy iy sinhy ArgΓ Moreover, using Mathematica [15] we readily obtain α

8 86 O KOUBA Proof The definition of α implies that 1 + iy y α,α, Γ C \,] {} Thus, the function y ArgΓ 1+iy A Γy is continuous on α,α, taes its values in Z, and is eual to for y = Therefore, A Γ y=argγ 1+iy, forevery y α,α, which is the desired conclusion EXAMPLES ln ln1/u ln u du = ln + 4u ArgΓ ln ln1/u ln u du = lnφ+ ArgΓ + u i ln 1 + i lnφ 1 + where φ = is the golden ratio More generally, for < < coshα , the following holds ln ln1/u ln u du = + u lnφ ArgΓ + i lnφ + + with φ = It is worth mentioning that Mathematica [15] gives the results of examples 16, 17and18, but it fails to give the results of the previous examples However, numerical uadrature confirms the results In our final proposition we consider the evaluation of another log-log integral This integral was given in [] as a corollary of a more difficult evaluation Our approach is straightforward and simpler PROPOSITION 5 [] For any complex number z with Rz >, we have Fz def t z 1 lnln1/t = 1 + t z dt = ln z Logz where Log is the principal branch of the logarithm Proof We start by evaluating F1 Note that So, F1= lnln1/t dt = 1 + t e x lnxdx 1 + e x n F1 1 1 e x lnxdx lnx 1 + e x e nx dx

9 KUMMER S THEOREM IN ACTION 87 Because x lnx 1+e x is integrable on,+, we conclude using Lebesgue s dominated convergence theorem that lnx lim n 1 + e x e nx dx = Thus, F1= 1 1 e x lnxdx A simple change of variables shows that e x lnxdx = 1 e u lnu lndu = γ ln since lnue u du = Γ 1= γ It follows that F1= γ ln = γ ln + Now, note that ln + 1 ln = ln ln ln = ln = ln ln + O ln + O ln 1 ln This proves that the series ln + 1 ln ln is convergent Conseuently, if we define G n = n ln then there is a real number l such that G n = 1 ln n+l+o1 But n 1 ln n n ln ln = =lnh n + G n G n =lnlnn + γ+ 1 ln n ln n + o1 = 1 ln + γ ln + o1, where we used H n = n 1/ = lnn + γ + o1,see[1, 41] Now, let n tend to + to obtain 1 ln = 1 ln + γ ln Combining this with we conclude that F1= 1 ln

10 88 O KOUBA Next, for z,+ the change of variables t z = u shows that Fz= 1 lnln1/u lnz z 1 + u and the desired conclusion follows by analytic continuation du = F1 lnzln z z = lnz ln, z Acnowledgement The author would lie to than the anonymous reviewers for their comments that greatly improved the manuscript REFERENCES [1] M ABRAMOWITZ AND I A STEGAN Eds, it Handboo of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, Dover Boos on Mathematics, Dover Publication, Inc, New Yor, 197 [] V ADAMCHIK, A class of Logarithmic Integrals, Proceedings of the 1997 International Symposium on Symbolic and Algebraic Computation, ACM, Academic Press, 1, 1 8 [] M ALBANO, T AMDEBERHAN, E BEYERSTEDT AND V H MOLL, The integrals in Gradshteyn and Ryzhi Part 19: The error function, Scientia, 1, 11, 5 4 [4] G E ANDREWS, R ASKEY AND R ROY, Special functions, Encyclopedia of Mathematics and Its Applications, 71, Cambridge University Press, 1999 [5] I V BLAGOUCHINE, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory Elsevier, 148, 57 59, 15 [6] D BIERENS DE HAAN, Nouvelles tables d intégrales définies, Amsterdam, 1867, reprint G E Stechert & Co, New Yor, 199 [7] I S GRADSHTEYN AND I M RYZHIK, Table of Integrals, Series, and Products, 8th edition, D Zwillinger, and V Moll, eds Academic Press, Elsevier Inc, 15 [8] E KUMMER, Beitrag zur Theorie der Function Γx, J Reine Ang Math, 5, 1847, 1 4 [9] V H MOLL, The integrals in Gradshteyn and Ryzhi Part 4: The gamma function, Scientia, 15, 7, 7 46 [1] V H MOLL, The integrals in Gradshteyn and Ryzhi Part 6: The beta function, Scientia, 16, 8, 9 4 [11] L A MEDINA AND V H MOLL, A Class of Logarithmic Integrals, Ramanujan Journal,, 9, [1] V H MOLL, Special Integrals of Gradshteyn and Ryzhi: the Proofs Volume I, CRC Press, Taylor & Francis Group, LLC, 15 [1] V H MOLL, Special Integrals of Gradshteyn and Ryzhi: the Proofs Volume II, CRC Press, Taylor & Francis Group, LLC, 16 [14] I VARDI, Integrals, an Introduction to Analytic Number Theory, The American Mathematical Monthly, 95, no 4, 1988, 8 15 [15] Wolfram Research, Inc Mathematica, Version 11, Wolfram Research, Inc Champaign, Illinois, 16 Received December, 15 Omran Kouba Department of Mathematics Higher Institute for Applied Sciences and Technology P O Box 198, Damascus, Syria omran ouba@hiastedusy Journal of Classical Analysis wwwele-mathcom jca@ele-mathcom