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Bernoulli umbra is defined in classical umbral calculus as in Taylor - Difference equations via the classical umbral calculus.

Yu - Bernoulli Operator and Riemann's Zeta Function shows that $\operatorname{eval}\log (B+1)=-\gamma$ (the later paper uses the symbol $B$ for what is defined as $B+1$ in the first paper).

Since we can define Bernoulli umbra as a formal Laurent series in Levi-Civita field as

$$B=\varepsilon^{-1} -\frac{1}{2}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}+\dotsb$$

and

$$B+1=\varepsilon^{-1} +\frac{1}{2}-\frac{\varepsilon }{24}+\frac{3 \varepsilon ^3}{640}-\frac{1525 \varepsilon ^5}{580608}+\dotsb$$

(the numerators of the terms are given in https://oeis.org/A118050 and the denominators are in https://oeis.org/A118051), the question becomes simple: what's the intuition behind the fact that $\operatorname{st}\log (B+1)=-\gamma$ (in some closure of Levi-Civita field under logarithm operation)?

Here $\operatorname{st}$ means the $0$-th coefficient of the power series and is equivalent to $\operatorname{eval}$ from the first linked paper.

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    $\begingroup$ Could you make the question more self-contained? What is the standardisation operator on the Levi-Civita field? $\endgroup$ Feb 5, 2022 at 12:56
  • $\begingroup$ @MarkWildon the coefficient of the 0-th power of the indeterminant. That is of $\varepsilon^0$. $\endgroup$
    – Anixx
    Feb 5, 2022 at 12:57
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    $\begingroup$ "in some closure of Levi-Civita field under logarithm operation" exactly which closure? And how is $\mathrm{st}$ defined on this closure? $\endgroup$
    – Wojowu
    Feb 5, 2022 at 16:53

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