We establish the following new Stirling-type approximation formulas for the factorial function n! ≈ √ 2πnne−n √ n+ 1 6 + 1 72n − 31 6480n2 − 139 155520n3 + 9871 6531840n4 and n! ≈ √ 2πnne−n 4 √ n2 + n 3 + 1 18 − 2 405n − 31 9720n2 . Our estimations give much more accurate values for the factorial function than some previously published strong formulas. We also derive new sequences converging to...

We prove that the constant δ studied by Masser, Gramain, and Weber, satisfies 1.819776 < δ < 1.819833, and disprove a conjecture of Gra-main. This constant is a two-dimensional analogue of the Euler-Mascheroni constant; it is obtained by computing the radius r k of the smallest disk of the plane containing k Gaussian integers. While we have used the original algorithm for smaller values of k, t...

We construct sequences of finite sums [Formula: see text] and [Formula: see text] converging increasingly and decreasingly, respectively, to the Euler-Mascheroni constant γ at the geometric rate 1/2. Such sequences are easy to compute and satisfy complete monotonicity-type properties. As a consequence, we obtain an infinite product representation for [Formula: see text] converging in a monotone...

We study the nonlinear O(N) sigma model on S 2 with the gravitational coupling term, by evaluating the effective potential in the large-N limit. It is shown that there is a critical curvature R c of S 2 for any positive gravitational coupling constant ξ, and the dynamical mass generation takes place only when R < R c. The critical curvature is analytically found as a function of ξ (> 0), which ...

We show some consequences of Caro and Tuza’s [J. Graph Theory 15 (1991), 99–107] lower bound on the independence number of aK-uniform hypergraph H. This bound has the form CK · ∑n i=1(di+1) −1/(K−1), where CK is a constant depending only on K, and d1, . . . , dn are the degrees of the vertices in H. We improve on the best known bounds for CK : in particular, we prove that C3 ≥ √π/2 and that CK ...

Robin's criterion states that the Riemann Hypothesis (RH) is true if and only if Robin's inequality σ(n) := d|n d < e γ n log log n is satisfied for n ≥ 5041, where γ denotes the Euler(-Mascheroni) constant. We show by elementary methods that if n ≥ 37 does not satisfy Robin's criterion it must be even and is neither squarefree nor squarefull. Using a bound of Rosser and Schoenfeld we show, mor...