Shellability of the higher pinched Veronese posets

The pinched Veronese poset \({\mathcal {V}}^{\bullet }_n\) is the poset with ground set consisting of all nonnegative integer vectors of length \(n\) such that the sum of their coordinates is divisible by \(n\) with exception of the vector \((1,\ldots ,1)\) . For two vectors \(\mathbf {a}\) and \(\mathbf {b}\) in \({\mathcal {V}}^{\bullet }_n\) , we have \(\mathbf {a}\preceq \mathbf {b}\) if and only if \(\mathbf {b}- \mathbf {a}\) belongs to the ground set of \({\mathcal {V}}^{\bullet }_n\) . We show that every interval in \({\mathcal {V}}^{\bullet }_n\) is shellable for \(n \ge 4\) . In order to obtain the result, we develop a new method for showing that a poset is shellable. This method differs from classical lexicographic shellability. Shellability of intervals in \({\mathcal {V}}^{\bullet }_n\) has consequences in commutative algebra. As a corollary, we obtain a combinatorial proof of the fact that the pinched Veronese ring is Koszul for \(n \ge 4\) . (This also follows from a result by Conca, Herzog, Trung, and Valla.)     

Randomized large distortion dimension reduction

Consider a random matrix \(H:{\mathbb {R}}^{n}\longrightarrow {\mathbb {R}}^{m}\) . Let \(D\ge 2\) and let \(\{W_l\}_{l=1}^{p}\) be a set of \(k\) -dimensional affine subspaces of \({\mathbb {R}}^{n}\) . We ask what is the probability that for all \(1\le l\le p\) and \(x,y\in W_l\) , $$\begin{aligned} \Vert x-y\Vert _2\le \Vert Hx-Hy\Vert _2\le D\Vert x-y\Vert _2. \end{aligned}$$ We show that for \(m=O\big (k+\frac{\ln {p}}{\ln {D}}\big )\) and a variety of different classes of random matrices \(H\) , which include the class of Gaussian matrices, existence is assured and the probability is very high. The estimate on \(m\) is tight in terms of \(k,p,D\) .     

Distributions and the Global Operator of Slice Monogenic Functions

In this paper we consider functions \(f\) defined on an open set \(U\) of the Euclidean space \(\mathbb{R }^{n+1}\) and with values in the Clifford Algebra \(\mathbb{R }_n\) . Slice monogenic functions \(f: U \subseteq \mathbb{R }^{n+1} \rightarrow \mathbb{R }_n\) belong to the kernel of the global differential operator with non constant coefficients given by \( \mathcal{G }=|{\underline{x}}|^2\frac{\partial }{\partial x_0} \ + \ {\underline{x}} \ \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}. \) Since the operator \(\mathcal{G }\) is not elliptic and there is a degeneracy in \( {\underline{x}}=0\) , its kernel contains also less smooth functions that have to be interpreted as distributions. We study the distributional solutions of the differential equation \(\mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})\) and some of its variations. In particular, we focus our attention on the solutions of the differential equation \( ({\underline{x}}\frac{\partial }{\partial x_0} \ - E)F(x_0,{\underline{x}})=G(x_0,{\underline{x}}), \) where \(E= \sum _{j=1}^n x_j\frac{\partial }{\partial x_j}\) is the Euler operator, from which we deduce properties of the solutions of the equation \( \mathcal{G }F(x_0,{\underline{x}})=G(x_0,{\underline{x}})\) .     

Estimates for vector valued Dirichlet polynomials

We estimate the \(\ell _1\) -norm \(\sum _{n=1}^N \Vert a_n\Vert \) of finite Dirichlet polynomials \(\sum _{n=1}^N a_n n^{-s},\,s \in {\mathbb {C}}\) with coefficients \(a_n\) in a Banach space. Our estimates quantify several recent results on Bohr’s strips of uniform but non absolute convergence of Dirichlet series in Banach spaces.     

The \bar{\partial }-Equation on an Annulus Between Two Strictly q-Convex Domains with Smooth Boundaries

In this paper, we study the global boundary regularity of the \(\bar{\partial }\) - equation on an annulus domain \(\Omega \) between two strictly \(q\) -convex domains with smooth boundaries in \(\mathbb{C }^n\) for some bidegree. To this finish, we first show that the \(\bar{\partial }\) -operator has closed range on \(L^{2}_{r, s}(\Omega )\) and the \(\bar{\partial }\) -Neumann operator exists and is compact on \(L^{2}_{r,s}(\Omega )\) for all \(r\ge 0\) , \(q\le s\le n-q- 1\) . We also prove that the \(\bar{\partial }\) -Neumann operator and the Bergman projection operator are continuous on the Sobolev space \(W^{k}_{r,s}(\Omega )\) , \(k\ge 0\) , \(r\ge 0\) , and \(q\le s\le n-q-1\) . Consequently, the \(L^{2}\) -existence theorem for the \(\bar{\partial }\) -equation on such domain is established. As an application, we obtain a global solution for the \(\bar{\partial }\) equation with Hölder and \(L^p\) -estimates on strictly \(q\) -concave domain with smooth \(\mathcal C ^2\) boundary in \(\mathbb{C }^n\) , by using the local solutions and applying the pushing out method of Kerzman (Commun Pure Appl Math 24:301–380, 1971).     

The topology of parabolic character varieties of free groups

Let $G$ be a complex affine algebraic reductive group, and let $K\,\subset \, G$ be a maximal compact subgroup. Fix h $\,:=\,(h_{1}\,,\ldots \,,h_{m})\,\in \, K^{m}$ . For $n\, \ge \, 0$ , let $\mathsf X _{\mathbf{{h}},n}^{G}$ (respectively, $\mathsf X _{\mathbf{{h}},n}^{K}$ ) be the space of equivalence classes of representations of the free group on $m+n$ generators in $G$ (respectively, $K$ ) such that for each $1\le i\le m$ , the image of the $i$ -th free generator is conjugate to $h_{i}$ . These spaces are parabolic analogues of character varieties of free groups. We prove that $\mathsf X _{\mathbf{{h}},n}^{K}$ is a strong deformation retraction of $\mathsf X _{\mathbf{{h}},n}^{G}$ . In particular, $\mathsf X _{\mathbf{{h}},n}^{G}$ and $\mathsf X _{\mathbf{{h}},n}^{K}$ are homotopy equivalent. We also describe explicit examples relating $\mathsf X _{\mathbf{{h}},n}^{G}$ to relative character varieties.     

Semiclassical limits of ground state solutions to Schrödinger systems

This paper is concerned with the existence and concentration properties of the ground state solutions to the following coupled Schrödinger systems $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)G_{v}(z)~\hbox { in }\ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)G_{u}(z)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ and $$\begin{aligned} \left\{ \begin{array}{l} -\varepsilon ^2\varDelta u+u+V(x)v=W(x)(G_{v}(z)+|z|^{2^*-2}v)~\hbox {in } \ {\mathbb {R}}^N,\\ -\varepsilon ^2\varDelta v+v+V(x)u=W(x)(G_{u}(z)+|z|^{2^*-2}u)~\hbox {in } \ {\mathbb {R}}^N,\\ u(x)\rightarrow 0\ \hbox {and }v(x)\rightarrow 0\ \hbox {as } \ |x|\rightarrow \infty , \end{array} \right. \end{aligned}$$ where \(z=(u,v)\in {\mathbb {R}}^2\) , \(G\) is a power type nonlinearity, having superquadratic growth at both \(0\) and infinity but subcritical, \(V\) can be sign-changing and \(\inf W>0\) . We prove the existence, exponential decay, \(H^2\) -convergence and concentration phenomena of the ground state solutions for small \(\varepsilon >0\) .     

On a subsemigroup of the universal covering of Lie semigroups

Let \(G\) be a connected Lie group and \(S\) a generating Lie semigroup. An important fact is that generating Lie semigroups admit simply connected covering semigroups. Denote by \(\widetilde{S}\) the simply connected universal covering semigroup of \(S\) . In connection with the problem of identifying the semigroup \(\Gamma (S)\) of monotonic homotopy with a certain subsemigroup of the simply connected covering semigroup \(\widetilde{S}\) we consider in this paper the following subsemigroup $$\begin{aligned} \widetilde{S}_{L}=\overline{\left\langle \mathrm {Exp}(\mathbb {L} (S))\right\rangle } \subset \widetilde{S}, \end{aligned}$$ where \(\mathrm {Exp}:\mathbb {L}(S)\rightarrow S\) is the lifting to \( \widetilde{S}\) of the exponential mapping \(\exp :\mathbb {L}(S)\rightarrow S\) . We prove that \(\widetilde{S}_{L}\) is also simply connected under the assumption that the Lie semigroup \(S\) is right reversible. We further comment how this result should be related to the identification problem mentioned above.     

Nonlinear Schrödinger equations near an infinite well potential

The paper deals with standing wave solutions of the dimensionless nonlinear Schrödinger equation where the potential \(V_\lambda :\mathbb {R}^N\rightarrow \mathbb {R}\) is close to an infinite well potential \(V_\infty :\mathbb {R}^N\rightarrow \mathbb {R}\) , i. e. \(V_\infty =\infty \) on an exterior domain \(\mathbb {R}^N\setminus \Omega \) , \(V_\infty |_\Omega \in L^\infty (\Omega )\) , and \(V_\lambda \rightarrow V_\infty \) as \(\lambda \rightarrow \infty \) in a sense to be made precise. The nonlinearity may be of Gross–Pitaevskii type. A standing wave solution of \((NLS_\lambda )\) with \(\lambda =\infty \) vanishes on \(\mathbb {R}^N\setminus \Omega \) and satisfies Dirichlet boundary conditions, hence it solves We investigate when a standing wave solution \(\Phi _\infty \) of the infinite well potential \((NLS_\infty )\) gives rise to nearby solutions \(\Phi _\lambda \) of the finite well potential \((NLS_\lambda )\) with \(\lambda \gg 1\) large. Considering \((NLS_\infty )\) as a singular limit of \((NLS_\lambda )\) we prove a kind of singular continuation type results.     

Occupation Times of Refracted Lévy Processes

A refracted Lévy process is a Lévy process whose dynamics change by subtracting off a fixed linear drift (of suitable size) whenever the aggregate process is above a pre-specified level. More precisely, whenever it exists, a refracted Lévy process is described by the unique strong solution to the stochastic differential equation $$\begin{aligned} {\mathrm{d}}U_t=-\delta \mathbf 1 _{\{U_t>b\}}{\mathrm{d}}t +{\mathrm{d}}X_t,\quad t\ge 0 \end{aligned}$$ where \(X=(X_t, t\ge 0)\) is a Lévy process with law \(\mathbb{P }\) and \(b,\delta \in \mathbb{R }\) such that the resulting process \(U\) may visit the half line \((b,\infty )\) with positive probability. In this paper, we consider the case that \(X\) is spectrally negative and establish a number of identities for the following functionals $$\begin{aligned} \int \limits _0^\infty \mathbf 1 _{\{U_t where \(\kappa ^+_c=\inf \{t\ge 0: U_t> c\}\) and \(\kappa ^-_a=\inf \{t\ge 0: U_t< a\}\) for \(a . Our identities extend recent results of Landriault et al. (Stoch Process Appl 121:2629–2641, 2011) and bear relevance to Parisian-type financial instruments and insurance scenarios.     

Simply transitive geodesic ball packings to \mathbf{S}^2\!\times \!\mathbf{R} space groups generated by glide reflections

The \(\mathbf{S}^2\!\times \!\mathbf{R}\) geometry can be derived by the direct product of the spherical plane \(\mathbf{S}^2\) and the real line \(\mathbf{R}\) . In (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 42:235–250, 2001), Farkas has classified and given the complete list of the space groups of \(\mathbf{S}^2\!\times \!\mathbf{R}\) . The \(\mathbf{S}^2\!\times \!\mathbf{R}\) manifolds were classified by Molnár and Farkas in [2] by similarity and diffeomorphism. In Szirmai (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 52(2):413–430, 2011), we have studied the geodesic balls and their volumes in \(\mathbf{S}^2\!\times \!\mathbf{R}\) space; moreover, we have introduced the notion of geodesic ball packing and its density and have determined the densest geodesic ball packing for generalized Coxeter space groups of \(\mathbf{S}^2\!\times \!\mathbf{R}\) . In this paper, we study the locally optimal ball packings to the \(\mathbf{S}^2\!\times \!\mathbf{R}\) space groups having Coxeter point groups, and at least one of the generators is a glide reflection. We determine the densest simply transitive geodesic ball arrangements for the above space groups; moreover, we compute their optimal densities and radii. The density of the densest packing is \(\approx 0.80407553\) , may be surprising enough in comparison with the Euclidean result \(\frac{\pi }{\sqrt{18}}\approx 0.74048\) . Molnár has shown in (Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry) 38(2):261–288, 1997) that the homogeneous 3-spaces have a unified interpretation in the real projective 3-sphere \(\mathcal PS ^3(\mathbf{V}^4,\varvec{V}_4,\mathbb R )\) . In our work, we shall use this projective model of \(\mathbf{S}^2\!\times \!\mathbf{R}\) geometry.     

Logarithmic Mean Oscillation on the Polydisc,Multi-Parameter Paraproducts and Iterated Commutators

We introduce another notion of bounded logarithmic mean oscillation in the \(N\) -torus and give an equivalent definition in terms of boundedness of multi-parameter paraproducts from the dyadic little \(\mathrm {BMO}\) , \(\mathrm {bmo}^d(\mathbb {T}^N)\) to the dyadic product \(\mathrm {BMO}\) space, \(\mathrm {BMO}^d(\mathbb {T}^N)\) . We also obtain a sufficient condition for the boundedness of the iterated commutators from the subspace of \(\mathrm {bmo}(\mathbb {R}^N)\) consisting of functions with support in \([0,1]^N\) to \(\mathrm {BMO}(\mathbb {R}^N)\) .     

Euclidean hypersurfaces with genuine deformations in codimension two

We classify hypersurfaces of rank two of Euclidean space ${\mathbb{R}^{n+1}}$ that admit genuine isometric deformations in ${\mathbb{R}^{n+2}}$ . That an isometric immersion ${\hat{f}\colon M^n \to \mathbb{R}^{n+2}}$ is a genuine isometric deformation of a hypersurface ${f\colon M^n\to\mathbb{R}^{n+1}}$ means that ${\hat f}$ is nowhere a composition ${\hat f=\hat F\circ f}$ , where ${\hat{F} \colon V\subset \mathbb{R}^{n+1} \to\mathbb{R}^{n+2}}$ is an isometric immersion of an open subset V containing the hypersurface.     

Approximating minimum-cost edge-covers of crossing biset-families

Part of this paper appeared in the preliminary version [16]. An ordered pair ? = (S, S ₊) of subsets of a groundset V is called a biset if S ? S+; (V S ⁺;V S) is the co-biset of ?. Two bisets \(\hat X,\hat Y\) intersect if ^X X ∩ Y ≠ \(\not 0\) and cross if both X ∩ Y \(\not 0\) and X ⁺ ∪ Y ⁺ ≠= V. The intersection and the union of two bisets \(\hat X,\hat Y\) are defined by \(\hat X \cap \hat Y = (X \cap Y,X^ + \cap Y^ + )\) and \(\hat X \cup \hat Y = (X \cup Y,X^ + \cup Y^ + )\) . A biset-family \(\mathcal{F}\) is crossing (intersecting) if \(\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}\) for any \(\hat X,\hat Y \in \mathcal{F}\) that cross (intersect). A directed edge covers a biset ? if it goes from S to V S ⁺. We consider the problem of covering a crossing biset-family \(\mathcal{F}\) by a minimum-cost set of directed edges. While for intersecting \(\mathcal{F}\) , a standard primal-dual algorithm computes an optimal solution, the approximability of the case of crossing \(\mathcal{F}\) is not yet understood, as it includes several NP-hard problems, for which a poly-logarithmic approximation was discovered only recently or is not known. Let us say that a biset-family \(\mathcal{F}\) is k-regular if \(\hat X \cap \hat Y,\hat X \cup \hat Y \in \mathcal{F}\) for any \(\hat X,\hat Y \in \mathcal{F}\) with |V (X∪Y)≥k+1 that intersect. In this paper we obtain an O(log |V|)-approximation algorithm for arbitrary crossing \(\mathcal{F}\) if in addition both \(\mathcal{F}\) and the family of co-bisets of \(\mathcal{F}\) are k-regular, our ratios are: \(O\left( {\log \frac{{|V|}} {{|V| - k}}} \right) \) if |S ⁺ \ S| = k for all \(\hat S \in \mathcal{F}\) , and \(O\left( {\frac{{|V|}} {{|V| - k}}\log \frac{{|V|}} {{|V| - k}}} \right) \) if |S ⁺ \ S| = k for all \(\hat S \in \mathcal{F}\) . Using these generic algorithms, we derive for some network design problems the following approximation ratios: \(O\left( {\log k \cdot \log \tfrac{n} {{n - k}}} \right) \) for k-Connected Subgraph, and O(logk) \(\min \{ \tfrac{n} {{n - k}}\log \tfrac{n} {{n - k}},\log k\} \) for Subset k-Connected Subgraph when all edges with positive cost have their endnodes in the subset.     

The Fourier transform of multiradial functions

We obtain an exact formula for the Fourier transform of multiradial functions, i.e., functions of the form \(\varPhi (x)=\phi (|x_1|, \dots , |x_m|), x_i\in \mathbf R^{n_i}\) , in terms of the Fourier transform of the function \(\phi \) on \(\mathbf R^{r_1}\times \cdots \times \mathbf R^{r_m}\) , where \(r_i\) is either \(1\) or \(2\) .     

Stability of the Derivative of a Canonical Product

With each sequence \(\alpha =(\alpha _n)_{n\in \mathbb{N }}\) of pairwise distinct and non-zero points which are such that the canonical product $$\begin{aligned} P_\alpha (z) := \lim _{r\rightarrow \infty }\prod _{|\alpha _n|\le r}\left( 1-\frac{z}{\alpha _n}\right) \end{aligned}$$ converges, the sequence $$\begin{aligned} \alpha ^{\prime } := \bigl (P_\alpha ^{\prime }(\alpha _n)\bigr )_{n\in \mathbb{N }} \end{aligned}$$ is associated. We give conditions on the difference \(\beta -\alpha \) of two sequences which ensure that \(\beta ^{\prime }\) and \(\alpha ^{\prime }\) are comparable in the sense that $$\begin{aligned} \exists \,c,C>0:\quad c|\alpha ^{\prime }_n| \le |\beta ^{\prime }_n| \le C|\alpha ^{\prime }_n|, \quad n\in \mathbb{N }. \end{aligned}$$ The values \(\alpha ^{\prime }_n\) play an important role in various contexts. As a selection of applications we present: an inverse spectral problem, a class of entire functions and a continuation problem.     

Error bounds for Gauss-type quadratures with Bernstein–Szegő weights

The paper is concerned with the derivation of error bounds for Gauss-type quadratures with Bernstein?Szeg? weights, $${\int\limits_{-1}^{1}}f(t)w(t)\, dt=G_{n}[f]+R_{n}(f),\quad G_{n}[f]=\sum\limits_{\nu=1}^{n}\lambda_{\nu} f(\tau_{\nu}) \quad(n\in\textbf{N}),$$ where f is an analytic function inside an elliptical contour \(\mathcal{E}_{\rho}\) with foci at \(\mp 1\) and sum of semi-axes \(\rho > 1\) , and w is a nonnegative and integrable weight function of Bernstein?Szeg? type. The derivation of effective bounds on \(|R_{n}(f)|\) is possible if good estimates of \(\max_{z\in\mathcal{E}_{\rho}}|K_{n}(z)|\) are available, especially if one knows the location of the extremal point \(\eta\in\mathcal{E}_{\rho}\) at which \(|K_{n}|\) attains its maximum. In such a case, instead of looking for upper bounds on \(\max_{z\in\mathcal{E}_{\rho}}|K_{n}(z)|\) , one can simply try to calculate \(|K_{n}(\eta,w)|\) . In the case under consideration, i.e. when $$w(t)= \frac{(1-t^{2})^{-1/2}}{\beta(\beta-2\alpha)\,t^{2} +2\delta(\beta-\alpha)\,t+\alpha^{2}+\delta^{2}},\quad t\in(-1,1),$$ for some \(\alpha,\beta,\delta\) , which satisfy \(0<\alpha<\beta,\ \beta\ne 2\alpha,\vert\delta\vert<\beta-\alpha\) , the location on the elliptical contours where the modulus of the kernel attains its maximum value is investigated. This leads to effective bounds on \(|R_{n}(f)|\) . The quality of the derived bounds is analyzed by a comparison with other error bounds proposed in the literature for the same class of integrands.     

Donsker-type theorems for nonparametric maximum likelihood estimators

Let ${\mathcal{P}}$ be a nonparametric probability model consisting of smooth probability densities and let ${\hat{p}_{n}}$ be the corresponding maximum likelihood estimator based on n independent observations each distributed according to the law ${\mathbb{P}}$ . With $\hat{\mathbb{P}}_{n}$ denoting the measure induced by the density ${\hat{p}_{n}}$ , define the stochastic process ${\hat{\nu}}_{n}: f\longmapsto \sqrt{n} \int fd({\hat{\mathbb{P}}}_{n} -\mathbb{P})$ where f ranges over some function class ${\mathcal{F}}$ . We give a general condition for Donsker classes ${\mathcal{F}}$ implying that the stochastic process $\hat{\nu}_{n}$ is asymptotically equivalent to the empirical process in the space ${\ell ^{\infty }(\mathcal{F})}$ of bounded functions on ${ \mathcal{F}}$ . This implies in particular that $\hat{\nu}_{n}$ converges in law in ${\ell ^{\infty }(\mathcal{F})}$ to a mean zero Gaussian process. We verify the general condition for a large family of Donsker classes ${\mathcal{ F}}$ . We give a number of applications: convergence of the probability measure ${\hat{\mathbb{P}}_{n}}$ to ${\mathbb{P}}$ at rate ${\sqrt{n}}$ in certain metrics metrizing the topology of weak(-star) convergence; a unified treatment of convergence rates of the MLE in a continuous scale of Sobolev-norms; ${\sqrt{n}}$ -efficient estimation of nonlinear functionals defined on ${\mathcal{P}}$ ; limit theorems at rate ${\sqrt{n}}$ for the maximum likelihood estimator of the convolution product ${\mathbb{P\ast P}}$ .     

On the rank of incidence matrices in projective Hjelmslev spaces

Let \(R\) be a finite chain ring with \(|R|=q^m\) , \(R/{{\mathrm{Rad}}}R\cong \mathbb {F}_q\) , and let \(\Omega ={{\mathrm{PHG}}}({}_RR^n)\) . Let \(\tau =(\tau _1,\ldots ,\tau _n)\) be an integer sequence satisfying \(m=\tau _1\ge \tau _2\ge \cdots \ge \tau _n\ge 0\) . We consider the incidence matrix of all shape \(\varvec{m}^s=(\underbrace{m,\ldots ,m}_s)\) versus all shape \(\tau \) subspaces of \(\Omega \) with \(\varvec{m}^s\preceq \tau \preceq \varvec{m}^{n-s}\) . We prove that the rank of \(M_{\varvec{m}^s,\tau }(\Omega )\) over \(\mathbb {Q}\) is equal to the number of shape \(\varvec{m}^s\) subspaces. This is a partial analog of Kantor’s result about the rank of the incidence matrix of all \(s\) dimensional versus all \(t\) dimensional subspaces of \({{\mathrm{PG}}}(n,q)\) . We construct an example for shapes \(\sigma \) and \(\tau \) for which the rank of \(M_{\sigma ,\tau }(\Omega )\) is not maximal.     

Asymptotics of Landau Constants with Optimal Error Bounds

We study the asymptotic expansion for the Landau constants \(G_n\) , $$\begin{aligned} \pi G_n\sim \ln N + \gamma +4\ln 2 + \sum _{s=1}^\infty \frac{\beta _{2s}}{ N^{2s}},\quad n\rightarrow \infty , \end{aligned}$$ where \(N=n+3/4, \gamma =0.5772\ldots \) is Euler’s constant, and \((-1)^{s+1}\beta _{2s}\) are positive rational numbers, given explicitly in an iterative manner. We show that the error due to truncation is bounded in absolute value by, and of the same sign as, the first neglected term for all nonnegative \(n\) . Consequently, we obtain optimal sharp bounds up to arbitrary orders of the form $$\begin{aligned} \ln N+\gamma +4\ln 2+\sum _{s=1}^{2m}\frac{\beta _{2s}}{N^{2s}}< \pi G_n < \ln N+\gamma +4\ln 2+\sum _{s=1}^{2k-1}\frac{\beta _{2s}}{N^{2s}} \end{aligned}$$ for all \(n=0,1,2,\ldots , m=1,2,\ldots \) , and \(k=1,2,\ldots \) . The results are proved by approximating the coefficients \(\beta _{2s}\) with the Gauss hypergeometric functions involved and by using the second-order difference equation satisfied by \(G_n\) , as well as an integral representation of the constants \(\rho _k=(-1)^{k+1}\beta _{2k}/(2k-1)!\) .