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An Introduction to the Stieltjes Constants

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An Introduction to the Stieltjes Constants Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers Volume II(b) Donal F. Connon 18 February 2008 Abstract In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function, including the following: ∞ n 11⎛⎞n kp+1 γ p ()uuk=−⎜⎟(1)log( − + ) ∑∑k pn++11nk==00⎝⎠ x (1)− p+1 γςς()udu=− [(1)pp++ (0,)x (1) (0)] ∫ p p +1 1 ⎛⎞11 2 1⎡ ⎛⎞1⎤ γγ11⎜⎟=−[2 15log 2 − 6 γπγπ log 2] −++−Γ⎢ 4log 2 3log 4log ⎜⎟⎥ ⎝⎠42 2⎣ ⎝⎠4⎦ ∞ ⎡⎤1 1 log log(1/t ) −=dtγγ() u+ 2 γ () u+ γ log u+ log2 u ∫ ⎢⎥u 01 1 ⎣⎦1− tt log(1/ t ) t ∞ n 11⎛⎞n k 221212 ∑∑⎜⎟(−+=−−++ 1)kk log ( 1)γ1 log(2πγ ) πlog (2 π ) nk==00n +12⎝⎠k 242 ∞ n ks∞ −−+1(1)yu n ⎛⎞n xxtuextxt⎛ ⎞ tdusy⎜⎟ s ==2 Φ⎜,,+ 1 ⎟ ∑∑k (ky+−Γ ) (1 t ) ( s ) ∫ ⎡⎤−u (1− t ) (1 − t ) nk==11⎝⎠ 0 11−+()xe t ⎝ ⎠ ⎣⎦ where ()u is the Stieltjes constant and is the Hurwitz-Lerch zeta function. γ p Φ(,,zsu ) Whilst this paper is mainly expository, some of the formulae reported in it are believed to be new, and the paper may also be of interest specifically due to the fact that most of the various identities have been derived by elementary methods. CONTENTS OF VOLUMES I TO VI: Volume/page SECTION: 1. Introduction I/11 2. An integral involving cot x I/17 The Riemann-Lebesgue lemma I/21 3. Results obtained from the basic identities I/24 Some stuff on Stirling numbers of the first kind I/69 Euler, Landen and Spence polylogarithm identities I/86 An application of the binomial theorem I/117 Summary of harmonic number series identities I/154 4. Elementary proofs of the Flajolet and Sedgewick identities II(a)/5 Some identities derived from the Hasse/Sondow equations II(a)/13 A connection with the gamma, beta and psi functions II(a)/26 An application of the digamma function to the derivation of Euler sums II(a)/31 Gauss hypergeometric summation II(a)/48 Stirling numbers revisited II(a)/51 Logarithmic series for the digamma and gamma functions II(a)/64 An alternative proof of Alexeiewsky’s theorem II(a)/74 A different view of the crime scene II(a)/91 An easy proof of Lerch’s identity II(a)/109 Stirling’s approximation for logΓ (u ) II(a)/114 The Gosper/Vardi functional equation II(a)/125 A logarithmic series for log A II(a)/137 2 Asymptotic formula for logGu (+ 1) II(a)/139 Gosper’s integral II(a)/144 The vanishing integral II(a)/147 Another trip to the land of G II(a)/165 x Evaluation of ∫ππuudun cot II(a)/169 0 An observation by Glasser II(a)/188 An introduction to Stieltjes constants II(b)/5 A possible connection with the Fresnel integral II(b)/16 Evaluation of some Stieltjes constants II(b)/21 A connection with logarithmic integrals II(b)/67 A hitchhiker’s guide to the Riemann hypothesis II(b)/84 A multitude of identities and theorems II(b)/101 Various identities involving polylogarithms III/5 Sondow’s formula for γ III/42 Evaluation of various logarithmic integrals III/61 Some integrals involving polylogarithms III/66 A little bit of log(π / 2) III/88 Alternative derivations of the Glaisher-Kinkelin constants III/108 Some identities involving harmonic numbers IV/5 Some integrals involving logΓ (x ) IV/77 Another determination of logG (1/ 2) IV/88 An application of Kummer’s Fourier series for logΓ (1+ x ) IV/92 Some Fourier series connections IV/109 3 Further appearances of the Riemann functional equation IV/124 More identities involving harmonic numbers and polylogarithms IV/133 5. An application of the Bernoulli polynomials V/5 6. Trigonometric integral identities involving: - Riemann zeta function V/7 - Barnes double gamma function V/31 - Sine and cosine integrals V/45 - Yet another derivation of Gosper’s integral V/108 7. Some applications of the Riemann-Lebesgue lemma V/141 8. Some miscellaneous results V/145 APPENDICES (Volume VI): A. Some properties of the Bernoulli numbers and the Bernoulli polynomials B. A well-known integral C. Euler’s reflection formula for the gamma function and related matters π 2 ∞ 1 D. A very elementary proof of = ∑ 2 8(21)n=0 n + E. Some aspects of Euler’s constant γ and the gamma function F. Elementary aspects of Riemann’s functional equation for the zeta function ACKNOWLEDGEMENTS REFERENCES 4 AN INTRODUCTION TO THE STIELTJES CONSTANTS We recall Hasse’s formula for the Hurwitz zeta function from (4.3.106a) that for Re (s )≠ 1 ∞ 1(1)n ⎛⎞n − k (1)(,)ssu−= ς ∑∑⎜⎟ s−1 nk==00nuk++1()⎝⎠k and note that for s > 1 ∞∞11n ⎛⎞n (1)− k (1)s −= ∑∑∑s ⎜⎟ s−1 nnk===000()nu+++ n 1⎝⎠k () uk ∞ 1(1)n ⎛⎞n − k Hence, in passing, we see that is positive for s > 1 and is ∑∑⎜⎟ s−1 nk==00nuk++1()⎝⎠k monotonically decreasing with u (which is not intuitively obvious). We have the limit ∞∞n 11⎛⎞n k (4.3.200) lim(ssu−= 1)ςδ ( , ) ( −== 1)n,0 1 s→1 ∑∑⎜⎟ ∑ nk==00nn++11⎝⎠k n = 0 This limit is independent of u and we therefore have ∂∂ lim[(ssu−= 1)ςς ( , )] lim [( ssu −= 1) ( , )] 0 ∂∂uuss→→11 We also note that ∂∂∞ 1 (4.3.200a) (,su ) s( s 1,) u ςς==∑ s −+ ∂∂+uunun=0 () and hence as before we have ∂ lim [(1)(,)]ssusssu−=−−+=ςς lim[( 1)( 1,)]0 ss→→11∂u As a result of (4.3.200) we may apply L’Hôpital’s rule to the limit ⎡⎤⎡⎤(s−− 1)(,)1ςς su (s − 1)(,)′ su +ς (,) su lim ⎢⎥⎢⎥= lim ss→→11⎣⎦⎣⎦s −11 5 From (4.3.107a) we see that ∞ 1 n ⎛⎞n log(uk+ ) (4.3.201) (1)(,)(,)ssusu−+=−−ςς′ (1)k ∑∑⎜⎟ s−1 nk==00nu++1()⎝⎠k k and hence we have ∞ 1 n ⎛⎞n lim([]ssusu−+=−−+ 1)(,ςς′ ) (, ) (1)log(k uk ) s→1 ∑∑⎜⎟ nk==00n +1 ⎝⎠k Referring back to (4.3.74) in Volume II(a) we may then deduce that (4.3.202) lim[ (ssusuu−+=− 1)ςς′ ( , ) ( , )] ψ ( ) s→1 and therefore conclude that ⎡⎤⎡⎤(1)(,)1ssu−−ς 1 (4.3.203) lim ⎢⎥⎢⎥=−=−limςψ (su , ) () u ss→→11⎣⎦⎣⎦ss−−11 The above limit is derived in a more complex manner in [126, p.91] and [135, p.271] using Hermite’s integral formula for the Hurwitz zeta function. A further derivation is contained in Volume VI. With u =1 we immediately see that ⎡⎤⎡⎤11 (4.3.204) lim⎢⎥⎢⎥ς (ss ,1)−= limςψγ ( ) −=−= (1) ss→→11⎣⎦⎣⎦ss−−11 which is derived in an alternative way in (4.4.99m) in Volume III. The above limits then imply that (4.3.204a) lim[ςς (su , )−=−− ( s )] γψ ( u ) s→1 In [135, p.266] we see that ⎡⎤ς (,su ) (4.3.204b) lim⎢⎥=− 1 s→1 ⎣⎦Γ−(1s ) This is easily demonstrated as follows. We have ς (,su ) (1− s )(,ς su ) = Γ−(1sss ) (1 −Γ− ) (1 ) 6 and hence we see that ⎡⎤⎡⎤(1−−ssu )ςς ( , ) (1 ssu ) ( , ) lim⎢⎥⎢⎥= lim=− 1 ss→→11⎣⎦⎣⎦(1−Γ−ss ) (1 ) Γ−(2 s ) L’Hôpital’s rule then gives us ⎡⎤ς ′(,su ) (4.3.205) lim⎢⎥= 1 s→1 ⎣⎦Γ−′(1s ) Differentiating (4.3.202) with respect to u gives us d ⎡⎤1 lim⎢⎥ςψ (su , )−=−′ ( u ) du s→1 ⎣⎦s −1 and hence d limςψ (su , )=− ′ ( u ) du s→1 We have by interchanging the derivative and the limit operations d ∂ limςς (su , )= lim ( su , ) duss→→11∂ u =−+lim[ssς ( 1, u )] =−ς (2, u ) s→1 giving the well-known result ς (2,uu )=ψ ′ ( ) This may also be derived by differentiating (4.3.200a). From (4.3.203) we see that ⎡⎤1 lim⎢⎥ςψ (su , )−+ ( u ) = 0 s→1 ⎣⎦s −1 and we have the Laurent expansion of the Hurwitz zeta function ς (su , ) about s =1 ∞ p 1(1)− p (4.3.206) ςγ(,su )=+∑ p ()(u s − 1) sp−1!p=0 where γ p (u ) are known as the generalised Stieltjes constants. We have (4.3.206a) γ 0 ()uu=−ψ () and γ 0 (1)= −=ψγ (1) . 7 We have for p ≥ 0 p pp⎡ () (1)!− p ⎤ (4.3.207) γςp ()usu=− (1)lim⎢ (,)− p+1 ⎥ s→1 ⎣ (1)s − ⎦ and, in particular, we have ⎡⎤1 (4.3.208) γς0 ()usuu=−=− lim(,)⎢⎥ ψ () s→1 ⎣⎦s −1 ⎡ 1 ⎤ ′ γς1()usu=− lim⎢ (,) + 2 ⎥ s→1 ⎣ (1)s − ⎦ Applying L’Hôpital’s rule again gives us ⎡⎤1 ςψ(,su )−+ () u ⎢⎥⎡⎤1 s −1 ′ (4.3.208a) lim⎢⎥=+=− lim⎢⎥ςγ (su , )2 1 ( u ) ss→→11⎢⎥ss−−1(1)⎣⎦ ⎣⎦ Using (4.3.202) we see that lim[ (ssusuu−++= 1)ςςψ′ ( , ) ( , ) ( )] 0 s→1 and applying L’Hôpital’s rule gives us ⎡⎤(ssusuu−++ 1)(,ςςψ′′ ) (, ) ()⎡⎤ ( ssusu −+ 1)(, ς′ ) 2 ς′ (, ) lim⎢⎥= lim ⎢⎥ ss→→11⎣⎦s −11⎣⎦ From (4.3.201) we see that ∞ n 2 ∂+1l⎛⎞n k og()ku [(ssusu 1)′ ( , ) ( , )] ( 1) −+=ςς ∑∑⎜⎟ − s−1 ∂++snuknk==001()⎝⎠k and thus we have ∞ 1 n ⎛⎞n (4.3.209) lim([]ssusu−+= 1)(,ςς′′ ) 2 ′ (, ) ( − 1)log(k 2 ku + ) s→1 ∑∑⎜⎟ nk==00n +1 ⎝⎠k Accordingly we see that 8 ⎡⎤(1)(,)(,)()ssusuu−++ςςψ′ ∞ 1n ⎛⎞n lim= (−+ 1)k log2 (ku ) s→1 ⎢⎥∑∑⎜⎟ ⎣⎦sn−+11nk==00⎝⎠k and the limit may be written as ⎡⎤1(,)()1ςψsu+ u ′ lim⎢⎥ς (su , ) ++22 − s→1 ⎣⎦(1)sss−−− 1 (1) ⎡⎤⎡⎤1(,)()1ςψsu+ u ′ =++lim⎢⎥⎢⎥ς (su , )22 lim − ss→→11⎣⎦⎣⎦(1)sss−−− 1 (1) =−2()γ1 u where we have used (4.3.207) and (4.3.208a).
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