May 30, 2026
We introduce a family of constants \[C_m := \lim_{n\to\infty} \left( \sum_{k=1}^n \operatorname{Li}_m\!\left(\frac{1}{k}\right) - \log n \right),\] which may be regarded as polylogarithmic analogues of Euler’s constant. We study their basic properties and derive representations in terms of iterated logarithmic integral structures associated with the gamma function. We further introduce associated polylogarithmic zeta potentials and polylogarithmic gamma functions, establish differential relations and integral representations, and describe logarithmic branch asymptotics near the singular points. As an application, we relate the constants \(C_m\) to special values of certain Dirichlet series involving the Riemann zeta function.
Among the classical constants appearing in analytic number theory, Euler’s constant \[\gamma = \lim_{n\to\infty} \left( \sum_{k=1}^n \frac{1}{k} - \log n \right)\] occupies a distinguished position through its close connections with the gamma function, zeta functions, and logarithmic phenomena. For historical background and modern developments concerning Euler’s constant, see Lagarias [1]. Motivated by the classical definition of \(\gamma\), for integers \(m\geqslant 2\), we introduce the constants \[C_m := \lim_{n\to\infty} \left( \sum_{k=1}^n \operatorname{Li}_m\!\left(\frac{1}{k}\right) - \log n \right),\] where \[\operatorname{Li}_m(z) = \sum_{n=1}^{\infty} \frac{z^n}{n^m}\] denotes the polylogarithm. The constants \(C_m\) may naturally be viewed as polylogarithmic analogues of Euler’s constant. Indeed, the logarithmic divergence of the partial sums is analogous to the classical harmonic case, while the polylogarithmic structure introduces new connections with zeta values, iterated integrals, and logarithmic gamma-type functions.
One of the fundamental identities satisfied by \(C_m\) is \[\sum_{k=2}^{\infty} \frac{\zeta(k)-1}{k^m} =1-\gamma-\zeta(m)+C_m,\] which connects the constants \(C_m\) with Dirichlet series involving the Riemann zeta function \(\zeta(s)\). Related Dirichlet series were studied in [2]. This identity may be viewed as a direct extension of Euler’s classical identity \[\sum_{k=2}^{\infty} \frac{\zeta(k)-1}{k} = 1-\gamma.\]
Motivated by this identity, we introduce two associated function families: polylogarithmic zeta potentials and polylogarithmic gamma functions. Although the latter are not Barnes multiple gamma functions in the usual sense, they are motivated by a related philosophy: higher logarithmic structures arise through recursive procedures associated with the gamma function. \[\begin{array}{ccc} \boldsymbol{Barnes hierarchy} & & \boldsymbol{Present hierarchy} \\[0.3em] \text{difference equations} & \leadsto & \text{integral recursion} \\[0.3em] \text{factorial recursion} & \leadsto & \text{polylogarithmic recursion} \\[0.3em] \text{multiple gamma functions} & \leadsto & \text{iterated logarithmic gamma functions} \end{array}\] The present framework is distinguished by its recursive logarithmic-gamma realization, its associated logarithmic branch structures, and its connection with polylogarithmic analogues of Euler’s constant.
The paper is organized as follows. In Section 2, we introduce the polylogarithmic zeta potentials \[\mathcal{Z}_s(z) := \sum_{n=1}^{\infty} \left\{ \operatorname{Li}_s\!\left(\frac{z}{n}\right) - \frac{z}{n} \right\},\] which serve as generating functions for the constants introduced above. We study their logarithmic gamma structure and analytic continuation, together with the logarithmic branch singularities arising at \(z=1,2,3,\dots\). In Section 3, we introduce the polylogarithmic gamma functions \(G_s(z)\), which satisfy \[G_1(z) = e^{-\gamma z}\Gamma(1-z).\] We derive iterated integral representations, differential relations, and special-value formulas relating \(G_s(z)\) to the constants \(C_m\). We also discuss their relation to classical logarithmic gamma integrals, including connections with the Kinkelin function and Barnes-type multiple gamma functions. In Section 4, we investigate arithmetic properties of the constants \(C_m\), including asymptotic and summation formulas, and their relation to Dirichlet series involving the Riemann zeta function.
We begin by introducing a family of generating functions built from polylogarithms and zeta values.
Definition 1 (Polylogarithmic zeta potentials). For \(s\in\mathbb{C}\) and \(|z|<1\), define \[\mathcal{Z}_s(z) := \sum_{n=1}^{\infty} \left\{ \operatorname{Li}_s\!\left(\frac{z}{n}\right) - \frac{z}{n} \right\}.\]
The terminology “potential” is used heuristically, emphasizing the role of \(\mathcal{Z}_s(z)\) as a generating analytic object encoding zeta values through polylogarithmic layers. Indeed, the generating structure becomes apparent upon expanding the polylogarithm term-by-term: \[\operatorname{Li}_s\!\left(\frac{z}{n}\right) - \frac{z}{n} = \sum_{r=2}^{\infty} \frac{z^r}{r^s n^r}.\] Summing over \(n\), and interchanging the order of summation in the region of absolute convergence, we obtain \[\mathcal{Z}_s(z) = \sum_{r=2}^{\infty} \frac{\zeta(r)}{r^s}z^r.\] Hence, for each fixed \(s\in\mathbb{C}\), the function \(\mathcal{Z}_s(z)\) is analytic for \(|z|<1\). Moreover, since \(\zeta(r)\to1\) exponentially fast as \(r\to\infty\), while \(r^{-s}\) contributes only polynomial growth, the associated power series has radius of convergence equal to \(1\) by the Cauchy–Hadamard formula.
Evaluating the polylogarithmic zeta potentials at \(z=1\), we obtain the associated arithmetic constants.
Definition 2 (Polylogarithmic Euler constants). For \(\Re(s)>1\), define \[C_s := \gamma+\mathcal{Z}_s(1) = \gamma+ \sum_{r=2}^{\infty} \frac{\zeta(r)}{r^s}.\] For integers \(m\geqslant 2\), this recovers the constants defined in (1.1). Equivalently, \[C_s = \sum_{n=1}^{\infty}\frac{a_n}{n^s}, \qquad a_n= \begin{cases} \gamma & (n=1),\\ \zeta(n) & (n\geqslant 2). \end{cases}\]
Thus the constants \(C_s\) form a zeta-weighted Dirichlet series. The following proposition describes the meromorphic continuation of \(\mathcal{Z}_s(1)\).
Proposition 1. The function \(\mathcal{Z}_s(1)\), and therefore \(C_s\), extends meromorphically to the whole complex \(s\)-plane with a simple pole at \(s=1\).
Proof. Separating the constant term in \(\zeta(r)\), we obtain \[\mathcal{Z}_s(1) = \zeta(s)-1 + \sum_{r=2}^{\infty} \frac{\zeta(r)-1}{r^s}.\] Since \[\zeta(r)-1 = O(2^{-r}),\] the latter series defines an entire function of \(s\). The conclusion follows. ◻
By analytic continuation, \(C_s\) will also denote the resulting meromorphic function.
The special value \(s=1\) plays a distinguished role. Indeed, the classical expansion [3] \[\log\Gamma(1-z) = \gamma z + \sum_{r=2}^{\infty} \frac{\zeta(r)}{r}z^r, \qquad |z|<1,\] yields \[\mathcal{Z}_1(z) = \log\Gamma(1-z)-\gamma z, \qquad |z|<1.\]
Define the integral operator \[(\mathcal{I} f)(z) := \int_0^z \frac{f(w)}{w}\,dw.\] For integers \(m\geqslant 2\), the functions \(\mathcal{Z}_m(z)\) satisfy the recursive integral relation \[\mathcal{Z}_m(z) = (\mathcal{I}^{m-1}\mathcal{Z}_1)(z).\]
Proposition 2. The function \(\mathcal{Z}_1(z)\) admits an analytic continuation to \[\Omega:=\mathbb{C}\setminus[1,\infty),\] where \(\log\Gamma(1-z)\) is defined using the principal branch of the logarithm. For every integer \(m\geqslant 2\), the function \(\mathcal{Z}_m(z)\) extends holomorphically to \(\Omega\) and admits the finite value \[\mathcal{Z}_m(1)=C_m-\gamma.\] In particular, for every integer \(m\geqslant 2\), the iterated integral structure regularizes the logarithmic singularity at \(z=1\), so that \(\mathcal{Z}_m(z)\) extends holomorphically across \(z=1\). No additional branch points are created, and the branch structure is inherited from \(\log\Gamma(1-z)\).
Proof. Since \(\Gamma(1-z)\) has poles only at \(z=1,2,3,\dots\) and has no zeros, the function \(\Gamma(1-z)\) is holomorphic and nonvanishing on \[\Omega:=\mathbb{C}\setminus[1,\infty).\] The slit \([1,\infty)\) is introduced to avoid the poles and fix a single-valued branch of the logarithm. Since \(\Omega\) is simply connected, there exists a holomorphic branch of \[\log\Gamma(1-z)\] on \(\Omega\), obtained by analytic continuation from the principal branch near \(z=0\). By the expansion (2.4) of \(\log\Gamma(1-z)\) at \(z=0\), we obtain \[\mathcal{Z}_1(z)=O(z^2) \qquad (z\to0).\] Hence \(\mathcal{Z}_1(z)/z\) extends holomorphically to \(z=0\). More generally, if \(f\) is holomorphic on \(\Omega\) and satisfies \[f(z)=O(z^2) \qquad (z\to0),\] then \(f(z)/z\) is holomorphic on \(\Omega\). Since \(\Omega\) is simply connected, the integral \[(\mathcal{I} f)(z) = \int_0^z \frac{f(w)}{w}\,dw\] is path-independent and defines a holomorphic function on \(\Omega\). Moreover, since \(f(w)/w = O(w)\), \[(\mathcal{I} f)(z)=O(z^2) \qquad (z\to0).\] Thus the assertion follows by induction. For \(m\geqslant 2\), the series (2.2) converges at \(z=1\), and therefore \[\mathcal{Z}_m(1) = \sum_{r=2}^{\infty} \frac{\zeta(r)}{r^m} = C_m-\gamma,\] which agrees with the definition (2.3) of \(C_m\). ◻
The relation between \(\mathcal{Z}_1(z)\) and the logarithmic gamma function provides the motivation for the polylogarithmic gamma hierarchy introduced in Section 3.
The preceding discussion describes the local analytic behaviour of \(\mathcal{Z}_s(z)\) near \(z=0\), together with its analytic continuation and logarithmic gamma structure. We now discuss, from a heuristic analytic viewpoint, the expected singularity structure of \(\mathcal{Z}_s(z)\).
From the defining series (2.1), we formally expect that the singularities of \(\mathcal{Z}_s(z)\) are inherited from the branch structure of the individual polylogarithm terms. Upon analytic continuation in the variable \(z\), this suggests the appearance of singularities at the positive integers \[z=1,2,3,\ldots.\] These singularities are not poles in the usual sense, but branch singularities inherited from the polylogarithmic structure. Indeed, the term corresponding to \(n=k\) contains \[\operatorname{Li}_s\!\left(\frac{z}{k}\right),\] which possesses the classical branch point of the polylogarithm at \(1\). Consequently, the singularity at \(z=k\) is expected to exhibit a polylogarithmic branch behaviour.
More precisely, for non-integral \(s\), the local singular behaviour near \(z=k\) is governed by \[\Gamma(1-s) \left( -\log\frac{z}{k} \right)^{s-1},\] in accordance with the expansion of the polylogarithm near \(1\) (cf. [4]; see also [5])). The case \(s=1\) is exceptional, since \[\operatorname{Li}_1(z) = -\log(1-z),\] so that the singularity reduces to a pure logarithmic branch point. For integral values \(s\geqslant 2\), the fractional logarithmic branch structure reduces to ordinary logarithmic branch singularities. Thus, from a global viewpoint, the sequence \[z=1,2,3,\ldots\] may be regarded heuristically as locations of polylogarithmic branch points. This viewpoint motivates the use of the slit domain \[\Omega=\mathbb{C}\setminus[1,\infty),\] which avoids the branch singularities arising from the polylogarithmic structure.
The logarithmic singularity of \(\mathcal{Z}_1(z)\) at \(z=1\) is inherited by the functions \(\mathcal{Z}_m(z)\) through the iterated integral construction, with the singular behaviour becoming progressively weaker as \(m\) increases. More precisely, iterated integration transforms the logarithmic divergence of \(\mathcal{Z}_1(z)\) into weaker logarithmic branch terms with progressively higher vanishing order.
As \(z\to1^{-}\), the classical pole behaviour \[\Gamma(1-z)\sim \frac{1}{1-z}\] implies \[\log\Gamma(1-z) = -\log(1-z)+O(1).\] Combining this with (2.5), we obtain \[\mathcal{Z}_1(z) = -\log(1-z)+O(1) \qquad (z\to1^-).\] Moreover, since \[\mathcal{Z}_m = \mathcal{I}^{m-1}\mathcal{Z}_1,\] repeated integration progressively weakens the logarithmic singularity at \(z=1\). Formally, successive integrations of the leading logarithmic term \[-\log(1-z)\] produce factors of \(1-z\) together with milder logarithmic terms, thereby softening the singular behaviour through the iterated integral hierarchy.
For example, since \[\mathcal{Z}_2(z) = \int_0^z \frac{\mathcal{Z}_1(w)}{w}\,dw,\] we have \[\mathcal{Z}_2(1)-\mathcal{Z}_2(1-\varepsilon) = \int_{1-\varepsilon}^{1} \frac{\mathcal{Z}_1(w)}{w}\,dw.\] Using (2.7), we obtain \[\mathcal{Z}_2(1)-\mathcal{Z}_2(1-\varepsilon) = \int_{1-\varepsilon}^{1} \frac{-\log(1-w)+O(1)}{w}\,dw.\] Putting \(t=1-w\), this becomes \[\int_0^\varepsilon \frac{-\log t+O(1)}{1-t}\,dt = \int_0^\varepsilon \left( -\log t+O(1) \right)\,dt = \varepsilon\log\frac{1}{\varepsilon} + O(\varepsilon).\] Since \[\mathcal{Z}_2(1)=C_2-\gamma,\] we obtain \[\mathcal{Z}_2(1-\varepsilon) = C_2-\gamma - \varepsilon\log\frac{1}{\varepsilon} + O(\varepsilon), \qquad (\varepsilon\to0^+).\] Thus, iterated integration weakens the logarithmic divergence. Although the divergent logarithmic singularity occurs only for \(m=1\), logarithmic branch behaviour persists for all \(m\geqslant 2\).
In view of (2.4) and (2.6), it is natural to construct a hierarchy generated from the logarithmic gamma function through repeated application of the logarithmic integral operator. Motivated by this structure, we introduce the following hierarchy of polylogarithmic gamma functions.
Definition 3 (Polylogarithmic gamma functions). For \(s\in\mathbb{C}\), define \[G_s(z) := \exp\left( \sum_{r=2}^{\infty} \frac{\zeta(r)}{r^s}z^r \right), \qquad |z|<1.\]
Remark 3. For each fixed \(z\) with \(|z|<1\), the defining series converges absolutely and defines an entire function of \(s\). By definition (3.1), \[\log G_m(z)=\mathcal{Z}_m(z)\] together with Proposition 2.2 yields an analytic continuation of \(G_m(z)\) to the domain \(\Omega\).
The terminology “polylogarithmic gamma functions” reflects the identity \[G_1(z) = e^{-\gamma z}\Gamma(1-z),\] together with the recursive structure induced by repeated application of the logarithmic integral operator \[(\mathcal{I} f)(z) = \int_0^z \frac{f(w)}{w}\,dw.\] This is analogous to the classical hierarchy of polylogarithms, whose adjacent orders are related through the Euler differential operator \[\left( z\frac{d}{dz} \right) \operatorname{Li}_{s+1}(z) = \operatorname{Li}_s(z).\] Since \[\log G_s(z) = \mathcal{Z}_s(z)\] by definition (3.1), the family satisfies the recursive relation \[\log G_{s+1}(z) = \mathcal{I} (\log G_s)(z).\]
Thus, the functions \(G_s(z)\) may be regarded as a polylogarithmic hierarchy generated from the logarithmic gamma function. For integer values \(m\geqslant 2\), the functions \(G_m(z)\) are obtained from the base function \(G_1(z)\) through repeated application of the operator \(\mathcal{I}\). Indeed, \[\log {G}_m(z) = \mathcal{I}^{m-1} (\log {G}_1)(z).\] For example, \[\log {G}_2(z) = \int_0^z \frac{ \log\Gamma(1-w)-\gamma w }{w} \,dw,\] while \[\log {G}_3(z) = \int_0^z \frac{1}{u} \left( \int_0^u \frac{ \log\Gamma(1-w)-\gamma w }{w} \,dw \right)\,du.\] In view of its logarithmic gamma integral structure, the function \(G_2(z)\) may be regarded as a polylogarithmic analogue of the Barnes \(G\)-function. Further comparisons with the classical Barnes hierarchy will be given in Section 3.3.
The iterated logarithmic integral representation admits a natural extension to complex parameters \(s\) with \(\Re(s)>1\).
Theorem 4 (Integral representation of \(G_s(z)\)). For \(\Re(s)>1\) and \(z\in\Omega\), \[\log G_s(z) = \frac{1}{\Gamma(s-1)} \int_0^z \frac{ \log\Gamma(1-w)-\gamma w }{w} \left( \log\frac{z}{w} \right)^{s-2} \,dw,\] where the integral is taken along the line segment joining \(0\) and \(z\).
Remark 5. By Definition 3.1, \(G_s(z)\) is entire in \(s\) for each fixed \(z\) with \(|z|<1\). The condition \(\Re(s)>1\) is required for the integral representation, which provides an analytic continuation of \(G_s(z)\) to \(\Omega\). In particular, the value at \(z=1\) is well defined for \(\Re(s)>1\), and is obtained by taking the radial limit \(z\to1^{-}\).
Proof. Temporarily, we assume \(|z|<1\). By (2.4) \[\frac{ \log\Gamma(1-w)-\gamma w }{w} = \sum_{r=2}^{\infty} \frac{\zeta(r)}{r}w^{r-1}.\] Since the series converges absolutely for \(|w|<1\) and the kernel \[\left(\log\frac{z}{w}\right)^{s-2}\] is integrable along the line segment for \(\Re(s)>1\), term-by-term integration is justified. Substituting the expansion above into the right-hand side of the asserted integral formula, we get \[\frac{1}{\Gamma(s-1)} \sum_{r=2}^{\infty} \frac{\zeta(r)}{r} \int_0^z w^{r-1} \left( \log\frac{z}{w} \right)^{s-2} \,dw.\] Along the line segment joining \(0\) and \(z\), let \[w=ze^{-u}.\] Then \[dw=-ze^{-u}\,du,\] and hence \[\int_0^z w^{r-1} \left( \log\frac{z}{w} \right)^{s-2} \,dw = z^r \int_0^\infty e^{-ru}u^{s-2}\,du.\] Since \(\Re(s)>1\), the last integral equals \[\frac{\Gamma(s-1)}{r^{\,s-1}}.\] Therefore, we recover the following series: \[\sum_{r=2}^{\infty} \frac{\zeta(r)}{r} \frac{z^r}{r^{\,s-1}} = \sum_{r=2}^{\infty} \frac{\zeta(r)}{r^s}z^r = \log G_s(z).\] This proves the identity for \(|z|<1\).
It remains to show that the integral extends to a well-defined holomorphic function on \(\Omega\). Near \(w=0\), \[\frac{\log\Gamma(1-w)-\gamma w}{w} = O(w).\] Hence the integrand is locally integrable at \(w=0\) for every \(\Re(s)>1\). If \(z\neq1\), the integrand is analytic along the line segment joining \(0\) and \(z\). When \(z=1\), using \[\log\Gamma(1-w) = -\log(1-w)+O(1) \qquad (w\to1),\] and \[\log\frac{1}{w} \sim 1-w,\] we obtain \[\frac{\log\Gamma(1-w)-\gamma w}{w} \left( \log\frac{1}{w} \right)^{s-2} = O\!\left( -\log(1-w) (1-w)^{\Re(s)-2} \right),\] which is integrable for \(\Re(s)>1\).
Therefore the integral is well defined for all \(z\in\Omega\). Moreover, the convergence is locally uniform on compact subsets of \(\Omega\), so the integral defines a holomorphic function on \(\Omega\). Since it agrees with \(\log G_s(z)\) for \(|z|<1\), the identity theorem yields the asserted continuation. ◻
Thus, the iterated logarithmic integral representation admits a natural extension to complex order parameters \(s\). Moreover, the recursive structure yields a differential relation between adjacent orders.
Proposition 6. For \(|z|<1\) and \(s\in\mathbb{C}\), \[z\frac{d}{dz} \log G_{s+1}(z) = \log G_s(z).\]
Proof. By definition, \[\log G_{s+1}(z) = \sum_{r=2}^\infty \frac{\zeta(r)}{r^{s+1}}z^r.\] Differentiating term-by-term, we obtain \[z\frac{d}{dz}\log G_{s+1}(z) = \sum_{r=2}^\infty \frac{\zeta(r)}{r^s}z^r = \log G_s(z),\] which proves the assertion. ◻
By Proposition 2.3, the polylogarithmic constants \(C_s\) extend meromorphically to the whole complex \(s\)-plane with a simple pole at \(s=1\). Thus, evaluating at \(z=1\), the polylogarithmic gamma functions recover the arithmetic constants introduced earlier.
Theorem 7 (Integral representation of the polylogarithmic constants). The identity \[C_s=\log G_s(1)+\gamma\] yields a meromorphic continuation of \(G_s(1)\) to \(s\in\mathbb{C}\setminus\{1\}\). For \(\Re(s)>1\), \[C_s = \frac{1}{\Gamma(s-1)} \int_0^1 \frac{\log\Gamma(1-t)}{t} (-\log t)^{s-2} \,dt .\]
Proof. By Definition 2.2 and the defining series for \(G_s(z)\), evaluating at \(z=1\) gives \[C_s=\log G_s(1)+\gamma.\] By Theorem 3.3 and Remark 3.4, the radial limit \(z\to1^{-}\) exists and yields \[\log G_s(1) = \frac{1}{\Gamma(s-1)} \int_0^1 \frac{ \log\Gamma(1-t)-\gamma t }{t} (-\log t)^{s-2} \,dt .\] Hence \[\begin{align} \log G_s(1) &= \frac{1}{\Gamma(s-1)} \int_0^1 \frac{\log\Gamma(1-t)}{t} (-\log t)^{s-2} \,dt \\ &\quad - \frac{\gamma}{\Gamma(s-1)} \int_0^1 (-\log t)^{s-2} \,dt . \end{align}\] Using \[\int_0^1 (-\log t)^{s-2} \,dt = \Gamma(s-1),\] we obtain \[\log G_s(1) = \frac{1}{\Gamma(s-1)} \int_0^1 \frac{\log\Gamma(1-t)}{t} (-\log t)^{s-2} \,dt -\gamma .\] Combining this with the identity \(C_s=\log G_s(1)+\gamma\), we obtain the asserted integral representation. ◻
The present construction may be compared with logarithmic gamma integrals appearing in the theory of the Kinkelin function and Barnes-type multiple gamma functions (cf. [6]–[10]). Recall that the Kinkelin function, which is closely related to the Barnes \(G\)-function, admits the logarithmic integral representation \[\log G(x) := \int_0^x \log\Gamma(t)\,dt + \frac{x(x-1)}{2} - \frac{x}{2}\log(2\pi),\] whereas the present function satisfies \[\log G_2(x) = \int_0^x \frac{ \log\Gamma(1-t)-\gamma t }{t} \,dt .\] Thus, both constructions arise from logarithmic gamma integrals, although the present construction contains a singular logarithmic kernel reflecting the underlying polylogarithmic structure. This analogy may be summarized schematically as \[\log G(x) \qquad\leadsto\qquad \log G_2(x),\] \[\int_0^x \log\Gamma(t)\,dt \qquad\leadsto\qquad \int_0^x \frac{\log\Gamma(1-t)}{t}\,dt.\] At the level of special values, Raabe’s formula (cf. [11]), \[\int_0^1 \log\Gamma(x)\,dx = \frac{1}{2}\log(2\pi)\] corresponds to \[\int_0^1 \frac{\log\Gamma(1-x)}{x}\,dx = C_2.\] Symbolically, \[\frac{1}{2}\log(2\pi) \qquad\leadsto\qquad C_2.\] In this sense, the constants \(C_s\) and the functions \(G_s(z)\) may be viewed as polylogarithmic analogues of classical logarithmic gamma constructions. The analogy with the Kinkelin function should be understood at the level of logarithmic gamma integrals rather than factorial-type difference equations.
Evaluating at \(z=-1\) gives rise to an alternating analogue of the constants \(C_m\). Indeed, for \(\Re(s)>1\), \[\log G_s(-1) = \sum_{r=2}^{\infty} \frac{(-1)^r\zeta(r)}{r^s}.\] Rewriting the series, we obtain \[\log G_s(-1) = 1-\eta(s) + \sum_{r=2}^{\infty} \frac{(-1)^r(\zeta(r)-1)}{r^s},\] where \(\eta(s)\) denotes the Dirichlet eta function. Since \(\eta(s)\) is entire and \(\zeta(r)-1=O(2^{-r})\), the right-hand side defines an entire function of \(s\). Thus \(\log G_s(-1)\) extends holomorphically to the whole complex \(s\)-plane.
For integers \(m\geqslant 2\), define alternating analogues of the constants \(C_m\) \[\widehat C_m := - \lim_{n\to\infty} \left( \sum_{k=1}^n \operatorname{Li}_m\!\left(-\frac{1}{k}\right) + \log n \right).\] Expanding the polylogarithm, we obtain \[\operatorname{Li}_m\!\left(-\frac{1}{k}\right) = -\frac{1}{k} + \sum_{r=2}^{\infty} \frac{(-1)^r}{r^m k^r},\] and hence \[\widehat C_m = \gamma - \sum_{r=2}^{\infty} \frac{(-1)^r\zeta(r)}{r^m}.\] Consequently, \[\log G_m(-1) = \gamma-\widehat C_m.\] Thus the values \(G_m(-1)\) may be interpreted as alternating counterparts of the values \(G_m(1)\). In particular, since \[G_1(z) = e^{-\gamma z}\Gamma(1-z),\] we recover the classical identity (cf. [3]): \[\log G_1(-1) = \sum_{r=2}^{\infty} \frac{(-1)^r \zeta(r)}{r} = \gamma.\] Equivalently, \[G_1(-1)=e^\gamma .\]
Although explicit evaluations of the individual constants \(C_m\) appear difficult in general, the family \(\{C_m\}_{m\geqslant 2}\) exhibits several remarkable collective structures. These include asymptotic properties, summation identities, and connections with shifted Stieltjes-type constants.
We begin by investigating the asymptotic behaviour of \(C_m\) as \(m\to\infty\). The identity in Theorem 3.6, \[C_m-\gamma=\log G_m(1),\] shows that the asymptotic behaviour of \(C_m-\gamma\) is equivalent to that of \(\log G_m(1)\).
Proposition 8 (Asymptotic and summation formulas for \(C_m\)). The polylogarithmic constants \(C_m\) satisfy the asymptotic relation \[C_m-\gamma \;\sim\; \frac{\zeta(2)}{2^{m}} \qquad (m\to\infty).\] Moreover, \[\begin{align} &\sum_{m=2}^{\infty}(C_m-\gamma) \;=\; \gamma+\gamma_1(0)-\gamma_1(1), \\ &\sum_{m=2}^{\infty}(-1)^m(C_m-\gamma) \;=\; \log\sqrt{2\pi}-\frac{\gamma}{2}. \end{align}\] Here \[\gamma_1(a) := \lim_{n\to\infty} \left( \sum_{k=1}^{n}\frac{\log k}{k+a} -\frac{1}{2}\log^2 n \right)\] denotes a shifted analogue of the first Stieltjes constant. In particular, \(\gamma_1(0)\) coincides with the classical first Stieltjes constant \(\gamma_1\).
Remark 9. The summation identities above may also be obtained as special cases of Theorem 2.5 in [2]. For completeness, we provide a direct proof in the present setting.
Proof. By the series representation (2.3) of \(C_m\), we have \[C_m-\gamma = \sum_{r=2}^{\infty} \frac{\zeta(r)}{r^m}.\] Hence \[C_m-\gamma = \frac{\zeta(2)}{2^m} + \sum_{r=3}^{\infty} \frac{\zeta(r)}{r^m} = \zeta(2)\cdot 2^{-m} + O(3^{-m}),\] which establishes the asymptotic formula.
We now prove the summation identities. By absolute convergence, we may interchange the order of summation and obtain \[\sum_{m=2}^{\infty}(C_m-\gamma) = \sum_{r=2}^{\infty} \zeta(r) \sum_{m=2}^{\infty}\frac{1}{r^m} = \sum_{r=2}^{\infty} \frac{\zeta(r)}{r(r-1)}.\] Using the Dirichlet series expansion \(\zeta(r)\), we obtain \[\sum_{r=2}^{\infty} \frac{\zeta(r)}{r(r-1)} = \sum_{k=1}^{\infty} \sum_{r=2}^{\infty} \frac{1}{r(r-1)k^r}.\] For \(0<x\leqslant 1\), we have \[\sum_{r=2}^{\infty} \frac{x^r}{r(r-1)} = x+(1-x)\log(1-x),\] where the value at \(x=1\) is understood as a limit. Therefore \[\sum_{m=2}^{\infty}(C_m-\gamma) = 1+ \sum_{k=2}^{\infty} \left\{ \frac{1}{k} + \frac{k-1}{k}\log\left(1-\frac{1}{k}\right) \right\}.\] Taking partial sums up to \(N\), the right-hand side is \[H_N + \sum_{k=1}^{N-1} \frac{\log k}{k(k+1)} - \frac{N-1}{N}\log N.\] Letting \(N\to\infty\), we obtain \[\sum_{m=2}^{\infty}(C_m-\gamma) = \gamma + \sum_{k=1}^{\infty} \frac{\log k}{k(k+1)}.\] On the other hand, from the definition of \(\gamma_1(a)\), \[\gamma_1(0)-\gamma_1(1) = \sum_{k=1}^{\infty} \left( \frac{\log k}{k} - \frac{\log k}{k+1} \right) = \sum_{k=1}^{\infty} \frac{\log k}{k(k+1)}.\] Thus \[\sum_{m=2}^{\infty}(C_m-\gamma) = \gamma+\gamma_1(0)-\gamma_1(1).\]
It remains to prove the alternating identity. Again by absolute convergence, \[\sum_{m=2}^{\infty}(-1)^m(C_m-\gamma) = \sum_{r=2}^{\infty} \zeta(r) \sum_{m=2}^{\infty} \left(-\frac{1}{r}\right)^m = \sum_{r=2}^{\infty} \frac{\zeta(r)}{r(r+1)}.\] Using the Dirichlet series expansion \(\zeta(r)\), we obtain \[\sum_{r=2}^{\infty} \frac{\zeta(r)}{r(r+1)} = \sum_{k=1}^{\infty} \sum_{r=2}^{\infty} \frac{1}{r(r+1)k^r}.\] For \(0<x\leqslant 1\), one has \[\sum_{r=2}^{\infty} \frac{x^r}{r(r+1)} = \frac{x}{2} + \frac{1-x}{x}\log(1-x) + 1-x,\] again with the value at \(x=1\) understood as a limit. It follows that \[\sum_{m=2}^{\infty}(-1)^m(C_m-\gamma) = \frac{1}{2} + \sum_{k=2}^{\infty} \left\{ \frac{1}{2k} + (k-1)\log\left(1-\frac{1}{k}\right) + 1-\frac{1}{k} \right\}.\] Taking partial sums and simplifying, we find \[\sum_{r=2}^{\infty} \frac{\zeta(r)}{r(r+1)} = \lim_{N\to\infty} \left\{ N - \frac{1}{2} H_N + \log N! - N\log N \right\}.\] By Stirling’s formula, \[\log N! = N\log N-N+\frac{1}{2}\log N+\log\sqrt{2\pi}+o(1),\] and since \(H_N=\log N+\gamma+o(1)\), this gives \[\sum_{m=2}^{\infty}(-1)^m(C_m-\gamma) = \log\sqrt{2\pi}-\frac{\gamma}{2}.\] This completes the proof. ◻
We recall the Dirichlet series studied in [2]: \[D(s_1,s_2;\lambda) = \sum_{k=1}^{\infty} \frac{\zeta(s_1+k)-1}{(k+\lambda)^{s_2}}.\] This series may be regarded as a generating function for shifted zeta values. The classical identity \[D(1,1;1)=1-\gamma\] goes back to Euler [12]. More recently, special values \(D(1,1;\mu)\) for \(\mu=2,3\) were evaluated by Srivastava [13] and Choi–Srivastava [14], in terms of the Stirling constant and the Glaisher–Kinkelin constant: \[\begin{align} D(1,1;2) &= \frac{3}{2} - \frac{\gamma}{2} - \log A_0, \\ D(1,1;3) &= \frac{11}{6} - \frac{\gamma}{3} - \log A_0 - 2\log A_1. \end{align}\] Here \[A_0=\sqrt{2\pi}\] denotes the Stirling constant, while \(A_1\) is the Glaisher–Kinkelin constant. More generally, the constants \(A_j\) (\(j\geqslant 2\)) are known as the Bendersky constants (cf. [15]). In [2], the following general formula was obtained: \[D(1,1;\mu) = H_\mu - \frac{\gamma}{\mu} - \sum_{j=0}^{\mu-2} \binom{\mu-1}{j}\log A_j.\] We now present several examples for higher values \(m\geqslant 2\).
Example 1. For \(m\geqslant 2\), we have \[\begin{align} D(1,m;1) &= (C_m-\gamma)-(\zeta(m)-1), \\ D(2,m;2) &= (C_m-\gamma)-(\zeta(m)-1) -\frac{\zeta(2)-1}{2^m}. \end{align}\]
Example 2. Let \[I_0 := \int_0^1 \log\Gamma(x)\log(1-x)\,dx, \qquad I_1 := \int_0^1 x\log\Gamma(x)\log(1-x)\,dx.\] Then \[\begin{align} D(1,2;2) ={}& I_0 + \log A_0 - \frac{\gamma}{4} + \frac{5}{4} - \zeta(2), \\ D(1,2;3) ={}& 2I_0 - 2I_1 + \frac{1}{2}\log A_0 + \log A_1 - \frac{\gamma}{9} + \frac{49}{36} - \zeta(2). \end{align}\]
The latter identities may be derived from classical generating function techniques associated with the digamma function. These examples are related to logarithmic gamma moments. For \(m\geqslant 0\), let \[M_m := \int_0^1 x^m\log\Gamma(x)\,dx.\] The moments \(M_m\) are naturally connected with the constants \(C_m\).
Example 3. For the logarithmic gamma moments \(M_m\), the constant \(C_2\) admits the representation \[C_2 = \sum_{m=0}^{\infty} M_m.\]
In this paper, we introduced polylogarithmic analogues of Euler’s constant together with a family of polylogarithmic gamma functions. The resulting family admits recursive integral structures closely related to logarithmic gamma integrals appearing in the theory of the Barnes \(G\)-function and related logarithmic gamma constructions.
The present framework is also related to Dirichlet series involving shifted zeta values, which arise naturally in the study of generalized Euler-type constants. At the same time, the associated integral representations give rise to new logarithmic gamma moments, suggesting further analogies with Barnes-type logarithmic gamma constructions.
Several problems remain open. Among them are the detailed study of the branch singularity structure of the functions \(\mathcal{Z}_s(z)\), and the investigation of further polylogarithmic operator hierarchies generated from logarithmic special functions. These questions suggest that the present constructions may represent only the first layer of a broader theory linking polylogarithms, zeta values, and logarithmic gamma hierarchies.
Finally, we note an operator-theoretic interpretation of the polylogarithmic gamma family constructed in this paper. Since \[(\mathcal{I} z^r)(z) = \frac{z^r}{r},\] the zeta potentials satisfy the formal relation \[\mathcal{Z}_s = \mathcal{I}^s \mathcal{Z}_0.\] Recalling (2.5), \[\mathcal{Z}_1(z) = \log\Gamma(1-z)-\gamma z,\] we obtain \[\log G_s(z) = \mathcal{Z}_s(z) = \mathcal{I}^{\,s-1}\mathcal{Z}_1(z),\] where \(\mathcal{I}^{\,s-1}\) is understood through the complex-order integral representation established in Section 3. Together with the identity in Theorem 3.6 \[C_s-\gamma = \mathcal{Z}_s(1),\] these relations yield the following formal scheme: \[\mathcal{Z}_0 \xrightarrow{\;\mathcal{I}\;} \mathcal{Z}_1 = \log\Gamma(1-z)-\gamma z \xrightarrow{\;\mathcal{I}^{\,s-1}\;} \log G_s = \mathcal{Z}_s \xrightarrow{\;z=1\;} C_s-\gamma.\] This viewpoint suggests that the hierarchy is generated by repeated application of the inverse of the Euler differential operator \[\mathcal{I} = \left( z\frac{d}{dz} \right)^{-1}.\] It is also natural to consider the hierarchy at non-positive indices. Indeed, \[\log G_0(z) = -z\psi(1-z)-\gamma z,\] showing that the level \(s=0\) is naturally expressed in terms of the digamma function \(\psi(z)\). More generally, one may expect the functions \(G_{-m}\) to be related to higher polygamma functions. This operator-theoretic viewpoint deserves further investigation.