Marek Wolf
In the first part we present results of four ``experimental'' determinations of the Euler-Mascheroni constant $γ$. Next we give new formulas expressing the $γ$ constant in terms of the Ramanujan-Soldner constant $μ$. Employing the cosine integral we obtain the infinity of formulas for $γ$.
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By integrating a series provided by Knopp, a series representation of the Euler-Mascheroni constant arises. The infinite sum representation of γ is determined through Fourier series (sawtooth wave).
On Dirichlet Series Involving $ζ(s)$ and Extensions of the Euler-Mascheroni Constant
本文研究了涉及ζ(s)的狄利克雷级数及其扩展的欧拉-马歇罗尼常数，通过积分表示和特殊值表达，为评估相关级数提供了统一框架。
In this paper, we present two new generalizations of the Euler-Mascheroni constant arising from the Dirichlet series associated to the hyperharmonic numbers. We also give some inequalities related to upper and lower estimates, and evaluation formulas.
Approximation Of Logarithm, Factorial And Euler Mascheroni Constant Using Odd Harmonic Series
本文通过奇数调和级数推导出对数、阶乘和欧拉-马歇罗尼常数的近似公式
Extreme values of Euler-Kronecker constants of abelian cubic fields
本文在广义黎曼猜想下，研究三次阿贝尔域和二次域的Euler-Kronecker常数的极值，给出了相应结果。
We introduce a one-parameter family of series associated to the Riemann $ζ$-function and prove that the values of the elements of this family at integers are linearly independent over the rationals for almost all values of the parameter, where almost all is with respect to any sufficiently nice measure. We also give similar results for the Euler--Mascheroni constant, for $\sum_{n=1}^\infty \frac{1}{n^n}$ and for $\sum_{n=1}^\infty \frac{1}{n! +1}$. Finally, specialising the criteria used, we give some new criteria for the irrationality of $ζ(k)$, the Euler--Mascheroni constant and the latter t…
Divisors in windows and the Euler--Mascheroni constant
该数学研究推导了区间内除数个数的平均值公式，涉及欧拉-马歇罗尼常数等，结果在Lean 4中得到形式化验证。
On the Uniqueness of Ein(1) among Linear Combinations of the Euler-Mascheroni and Euler-Gompertz Constants
该研究探讨Euler-Mascheroni常数与Euler-Gompertz常数线性组合的超越性与收敛性，证明$α=1/e$是使对应Borel可和发散级数线性组合后常规收敛的唯一系数，并将该唯一性推广到Gumbel分布矩定义的广义线性组合序列。
As a natural generalization of the Euler-Mascheroni constant $γ$, Y. Ihara introduced the Euler-Kronecker constant $γ_K$ attached to any number field $K$. In this paper, we prove that a certain bound on $γ_K$ in a tower of number fields $\mathcal{K}$ implies the generalized Brauer-Siegel conjecture for $\mathcal{K}$ as formulated by Tsfasman and Vlǎduţ. Moreover, we use known bounds on $γ_K$ for cyclotomic fields to obtain a finer estimate for the number of zeros of the Dedekind zeta-function $ζ_K(s)$ in the critical strip.
On finite analogues of Euler's constant
本文在有限多重zeta值的框架下引入欧拉常数的有限模拟，从正则化ζ(1)及级数表达式定义候选值，并证明其差异属于由1和对数有限模拟值张成的有理向量空间。
We present linear forms with integer coefficients containing the Euler-Mascheroni and Euler-Gompertz constants. The forms are defined by four-terms recurrence relations. Asymptotics of the forms and their coefficients are obtained.
Quadratic Euler-Kronecker constants in positive characteristic
本文通过Dirichlet $L$-函数对数导数研究有理全局函数域几何二次扩张上Euler-Kronecker常数的分布，利用概率模型证明其极限分布与偏差定理，并得出带CM的秩2 Drinfeld模高度的新分布结果。
In this article, we derive an expression for the complex magnitude of the Dirichlet beta function $β(s)$ represented as a Euler prime product and compare with similar results for the Riemann zeta function. We also obtain formulas for $β(s)$ valid for an even and odd $k$th positive integer argument and present a set of generated formulas for $β(k)$ up to $11$th order, including Catalan's constant and compute these formulas numerically. Additionally, we derive a second expression for the complex magnitude of $β(s)$ valid in the critical strip from which we obtain a formula for the Euler-Maschero…
We present logarithmic series for u, ln u and the Euler-Mascheroni constant gamma. It was indicated by J. Sondow that Theorem 4 and all proofs are new. All proofs are elementary. We present some conjectures.
For any natural number $n$, let $X'_n$ be the set of primitive Dirichlet characters modulo $n$. We show that if the Riemann hypothesis is true, then the inequality $|X'_{2n_k}|\le C_2 e^{-γ} ϕ(2n_k)/\log\log(2n_k)$ holds for all $k\ge 1$, where $n_k$ is the product of the first $k$ primes, $γ$ is the Euler-Mascheroni constant, $C_2$ is the twin prime constant, and $ϕ(n)$ is the Euler function. On the other hand, if the Riemann hypothesis is false, then there are infinitely many $k$ for which the same inequality holds and infinitely many $k$ for which it fails to hold.
We investigate a recursively generated sequence of random variables that begins with an Exponential random variable with parameter (i.e., inverse-mean) 1, and continues with additional Exponentials, each of whose random parameter possesses the distribution of the prior term in the sequence. Although such sequences enjoy some (limited) applicability as models of parameter uncertainty, our present interest is primarily theoretical. Specifically, we observe that the implied sequence of distribution functions manifests a surprising connection to two well-known mathematical quantities first studied…
An Infinite Family of Apéry-like Recurrences for Euler's Constant
针对欧拉常数无理性未证、类阿佩里近似稀少的问题，研究提出含Meijer G函数的恒等式，经推广后基于Conservative Matrix Field构造生成无穷多个类阿佩里近似。
The Euler-Mascheroni constant $γ=0.5772\dots\!$ is the $K=\mathbb{Q}$ example of an Euler-Kronecker constant $γ_K$ of a number field $K.$ In this note we consider the size of the $γ_q=γ_{K_q}$ for cyclotomic fields $K_q:=\mathbb{Q}(ζ_q).$ Assuming the Elliott-Halberstam Conjecture (EH), we prove uniformly in $Q$ that $$\frac{1}{Q}\sum_{Q<q\le 2Q} \left |γ_q - \log q\right |= o(\log Q).$$ In other words, under EH the $γ_q / \log q$ in these ranges converge to the one point distribution at $1$. This theorem refines and extends a previous result of Ford, Luca, and Moree for prime $q.$
Conjunctions of Three "Euler Constants" in Poisson-Related Expressions
本文研究了欧拉数、欧拉-马歇罗尼常数和欧拉-贡珀茨常数在泊松相关表达式中的两次联合出现，分别涉及Hardy方程和递归条件指数分布生成的概率序列，并推广初步观察以提供更全面的结果，包括Hardy方程的扩展。
Stirling-Ramanujan constants are exponential periods
研究斯特林 - 拉马努金常数，通过找到其在\(\mathbb{Q}(t,e^{-t})\)域中作为指数周期的自然积分表示，揭示其超代数性质，给出新积分公式，还可推广证明其他常数是该域上的指数周期。
Expression of Farhi's integral in terms of known mathematical constants
本文证明了 Farhi 积分 $\eta=2\int_0^1\log\Gamma(x)\sin(2\pi x)dx$ 等于 $(\gamma+\log(2\pi))/\pi$，其中 $\gamma$ 为欧拉-马歇罗尼常数，并给出两种证明方法。
We obtain two sequences of rational numbers which converge to the Euler-Gompertz constant. Denote by <f(x)> the integral of f(x)e^{-x} from 0 to infinity. Recall that the Euler-Gompertz constant δis <ln(x+1)>. Main idea. Let P_n(x) be a polynomial with integer coefficients. It is easy to prove that <P_n(x)ln(x+1)>=a_n+<ln(x+1)>b_n$ for some integers a_n, b_n. Hence if <P_n(x)ln(x+1)>/b_n converges to zero, a_n/b_n converges to - δ. Main Theorem. Let u be positive real. There exists polynomials P_n(x)(they are explicitly given in the paper) such that <P_n(x)ln(xu+1)> tends to u as n tends to in…
On the Algebraic Independence of a Set of Generalized Constants
针对欧拉-马歇罗尼常数等广义序列对，证明集合{γ^(n)+δ^(n)/e}_{n≥0}的代数无关性，推导得相关数的超越性及多组序列对的析取超越性，解决广义常数的代数性质问题。
This appendix to the beautiful paper of Ihara puts it in the context of infinite global fields of our papers. We study the behaviour of Euler--Kronecker constant $γ\_{K}$ when the discriminant (respectively, the genus) tends to infinity. Results of our paper easily give us good lower bounds on the ratio ${γ\_{K}/\log\sqrt{| d\_{K}|}}$. In particular, for number fields, under the generalized Riemann hypothesis we prove $$\liminf{γ\_{K}\over\log\sqrt{| d\_{K}|}}\ge -0.26049...$$ Then we produce examples of class field towers, showing that $$\liminf{γ\_{K}\over\log\sqrt{| d\_{K}|}}\le -0.17849...…
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