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I'm reading Donal F. Connon, Some integrals involving the Stieltjes constants.

It gives a definition of the generalized Stieltjes constants $\gamma_n(u)$ as coefficients in the Laurent series expansion of the Hurwitz zeta function (formula $(2.3)$ on page $7$): $$\zeta(s,u)=\sum_{n=0}^\infty\frac1{(n+u)^s}=\frac1{s-1}+\sum_{n=0}^\infty\frac{(-1)^n}{n!}\gamma_n(u)\,(s-1)^n$$

It also gives the following integral representation (formula $(2.4)$ on page $7$): $$\int_0^1\left[\frac1{1-y}+\frac1{\log y}\right]y^{u-1}\log\left|\log y\right|dy=-2\gamma_1(u)-\gamma\,\gamma_0(u)-\log^2u-\gamma\log u$$ Recall that $\gamma_0(u)$ in the second term is just a negation of the digamma function: $\gamma_0(u)=-\psi(u)$.

The given integral representation does not check numerically for me, e.g. for $u=1$ the integral on the left is $$-0.26036207832404194945778976048034290752168080191529...$$ but the numeric value of the right hand side is $$-0.18754623284036522459720338460544158838394446358095...$$ Further numeric calculations suggest that the right hand side should instead be $$-\gamma_1(u)-\gamma\,\gamma_0(u)-\frac{\log^2u}2-\gamma\log u$$

Is it an error in the paper, or I just misunderstand something?

Vladimir Reshetnikov
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    I contacted the author and he was very kind to review my comments and agreed that there is indeed an error in the paper. Actually, he states in the section $2$ of the paper "We now boldly assume that $(1.6)$ is also valid in the case where $r$ is a continuous variable" and later, in the comment after the formula $(\text{B}.30)$, mentions that it "unfortunately differs from $(2.5)$". It looks like this unproved assumption was actually wrong and was the basis of some erroneous formulae in the paper. – Vladimir Reshetnikov Oct 25 '14 at 21:13
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    So was the error resolved? – MathGod Oct 09 '16 at 12:17
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    The author acknowledged the error, but I do not know if a corrected version has been published anywhere. – Vladimir Reshetnikov Oct 09 '16 at 19:32

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