Erratum and Addendum to: Rediscovery of Malmsten’s integrals, their evaluation by contour integration methods and some related results [Ramanujan J. (2014), 35:21–110]
1 Erratum to: Ramanujan J (2014) 35:21–110 DOI 10.1007/s1113901395285
2 Addendum to Section 2.2
The historical analysis of functional Eqs. (20)–(22) on pp. 35–37 is far from exhaustive. In order to give a larger vision of this subject, several complimentary remarks may be needed.
First, on p. 37, lines 1–5, the text “By the way, the above reflection formula (21) for L(s) was also obtained by Oscar Schlömilch; in 1849 he presented it as an exercise for students [55], and then, in 1858, he published the proof [56]. Yet, it should be recalled that an analog of formula (20) for the alternating...” may be replaced by the following one: “By the way, between 1849 and 1858, the above reflection formula for L(s) was also obtained by several other mathematicians, including Oscar Schlömilch [55, 56], Gotthold Eisenstein [73, 84], and Thomas Clausen [75].^{1} Yet, it should be noted that formula (20) itself was rigorously proved by Kinkelin a year before Riemann [79, p. 100], [78], and its analog for the alternating...”
“An alternative historical analysis of functional Eqs. (20)–(21) in the context of contributions of various authors may be found in [85], [31, p. 23], [84], [82, p. 4], [78, p. 193], [74, pp. 326–328], [83, p. 298], [73]. Note, however, that Butzer et al.’s statement [74, p. 328] “Malmstén included the functional equation without proof” is rather incorrect. Thus, André Weil [84, p. 8] points out that “Malmstén included the proof in a long paper written in May 1846”. Moreover, our investigations show that this proof was not only included in his paper [41] written in 1846, but also was present in an earlier work [40] published in 1842. By the way, Malmsten remarked that reflection formulas of such kind were first announced by Euler in 1749, the fact which was not mentioned by Schlömilch [55, 56], nor by Clausen [75], nor by Kinkelin [79], nor by Riemann [54].”
3 Addendum to Section 4.1.2, Exercise no. 18
4 Addendum to Section 4.2, Exercise no. 29
5 Addendum to Section 4.5, Exercise no. 62b
6 Some minor corrections and additions

p. 42, line 20: “has no branch points.” should read “has no branch points except at poles of \(\varGamma (z)\).”

p. 42, line 27: “points at all, which allows” should read “points at all in the right half–plane, which allows”.

p. 66, in Nota Bene of exercise no. 19: “derived by Malmsten in [41, unnumbered” should read “derived by Malmsten in [40, p. 24, Eq. (37)], [41, unnumbered”.

p. 68, first line: “no. 21e” should read “no. 21d”.

p. 73, last line, “\(\mathrm{Re}r\,<2\pi \)” should read “\(\mathrm{Im}r\,<2\pi \)”.

p. 82, line 7, “no. 39c is given” should read “no. 39e is given”.

p. 83, exercise no. 40: formula given in exercise no. 40b, as well as formula (55), were also obtained by Nørlund in [81, p. 107].

p. 97, footnote 40 “formula (c) was” should read “formula (b.2) was”.^{3}

p. 100, exercise no. 64: closedform expressions equivalent to those we gave for Open image in new window , Open image in new window , Open image in new window and Open image in new window were also obtained by Connon in [76, pp. 1, 50, 53, 54–55], [77, pp. 17–18].
7 References
 72.
Blagouchine, Ia.V.: A theorem for the closed–form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, J. Number Theory 148, 537–592 and 151, 276–277, arXiv:1401.3724 (2015)
 73.
Bombieri, E.: The classical theory of zeta and Lfunctions. Milan J. Math. 78(1), 11–59 (2010)
 74.
Butzer, P.M., Jansche, S.: A direct approach to the Mellin transform. J. Fourier Anal. Appl. 3(4), 325–376 (1997)
 75.
Clausen, T.: Beweis des von Schlömilch Archiv Bd. XII. No. XXXV. aufgestellten Lehssatzes; — über die Ableitung des Differentials von \(\log \Gamma x\); und — über eine allgemeine Aufgabe über die Funktionen von Abel, Grunert Archiv der Mathematik und Physik, vol. XXX, pp. 166–170 (1858)
 76.
Connon, D.F.: Some series and integrals involving the Riemann zeta function, binomial coefficients and the harmonic numbers. vol. II(b), arXiv:0710.4024 (2007)
 77.
Connon, D.F.: The difference between two Stieltjes constants, arXiv:0906.0277 (2009)
 78.
Dutka, J.: On the summation of some divergent series of Euler and the zeta functions. Arch. Hist. Exact Sci. 50(2), 187–200 (1996)
 79.
Kinkelin, H.: Ueber einige unendliche Reihen. Mitteilungen der Naturforschenden Gesellschaft in Bern, no. 419–420, 89–104 (1858)
 80.
Linnik, Y.V.: On Hilbert’s eighth problem. In: Alexandrov, P.S. (ed.) Hilbert’s Problems (in Russian), pp. 128–130. Nauka, Moscow (1969)
 81.
Nörlund, N.E.: Vorlesungen über Differenzenrechnung. Springer, Berlin (1924)
 82.
Patterson, S.J.: An Introduction to the Theory of the Riemann Zetafunction. Cambridge Studies in Advanced Mathematics, vol. 14. Cambridge University Press, Cambridge (1988)
 83.
Srinivasan, G.K.: The gamma function: an eclectic tour. Am. Math. Mon. 114(4), 297–315 (2007)
 84.
Weil, A.: Prehistory of the zetafunction. In: Aubert, K., Bombieri, E., Goldfeld, D. (eds.) Number Theory, Trace Formulas and Discrete Groups, pp. 1–9. Academic Press, London (1989)
 85.
Wieleitner, H.: Geschichte der Mathematik (in 2 vols.), Berlin (1922–1923)
Footnotes
 1.
In 1849, Schlömilch presented the theorem as an exercise for students [55]. In 1858, Clausen [75] published the proof to this exercise. The same year, Schlömilch published his own proof [56]. Eisenstein did not publish the proof, but left some drafts dating back to 1849, see e.g. [73, 84].
 2.
Estimation of \(\bigl \zeta (1+i\alpha )\bigr \) was found to be connected with \(\mathrm{Re}\rho \), where \(\rho \) are the zeros of \(\zeta (s)\) in the critical strip \(0\leqslant \mathrm{Re}{s}\leqslant 1\), see e.g. [80, p. 128].
 3.
It may also noted that in a later work [72, pp. 542–543], we showed that (b.1), which is a shifted version of (b.2), was already known to Malmsten in 1846.