Related papers: A curious formula related to the Euler Gamma funct…

Let $Z(t)=\chi^{-1/2}(1/2+it)\zeta(1/2+it)=e^{i\theta(t)}\zeta(1/2+it)$ be Hardy's function and $g(n)$ be the $n$-th Gram points defined by $\theta(g(n))=\pi n$. Titchmarsh proved that $\sum_{n \leq N} Z(g(2n)) =2N+O(N^{3/4}\log^{3/4}N) $…

In the first part we present results of four ``experimental'' determinations of the Euler-Mascheroni constant $\gamma$. Next we give new formulas expressing the $\gamma$ constant in terms of the Ramanujan-Soldner constant $\mu$. Employing…

Let $\Gamma_+$ be a normal subgroup of index $2n$ of a group $\Gamma$ and $\gamma_i \in \Gamma \setminus \Gamma_+$ be involutions. We first prove that if $\Gamma = \Gamma_+ \rtimes (\mathbb{Z}_2(\gamma_1) \times \cdots \times…

We give all possible holomorphic Eisenstein series on $\Gamma_0(p)$, of rational weights greater than $2$, and with multiplier systems the same as certain rational-weight eta-quotients at all cusps. We prove they are modular forms and give…

In this paper, we focus on the explicit expression of an extended version of Riemann zeta function. We use two different methods, Mellin inversion formula and Cauchy's residue theorem, to calculate a Mellin-Barnes type integral of the…

The classical Mertens' formula states that $ \prod_{p\le N}\big(1-\frac1p)^{-1}\sim e^\gamma\log N, $ where the product is over all primes $p$ less than or equal to $N$, and $\gamma$ is the Euler-Mascheroni constant. By the Euler product…

Let $\Gamma$ denote the modular group $SL(2,\Bbb Z)$ and $C_n(\Gamma)$ the number of congruence subgroups of $\Gamma$ of index at most $n$. We prove that $\lim\limits_{n\to \infty} \frac{\log C_n(\Gamma)}{(\log n)^2/\log\log n} =…

We prove a short general theorem which immediately implies some classical results of Hasse, Guillera and Sondow, Paolo Amore, and also Alzer and Richards. At the end we obtain a new representation for the Euler constant gamma. The theorem…

Let $\psi(x)$ be the di-gamma function, the logarithmic derivative of the classical Euler's gamma function $\Gamma(x)$. In the paper, the author shows that the completely monotonic degree of the function $[\psi'(x)]^2+\psi''(x)$ is $4$,…

In this paper new series for the first and second Stieltjes constants (also known as generalized Euler's constant), as well as for some closely related constants are obtained. These series contain rational terms only and involve the…

Using a variational approach, two new series representations for the incomplete Gamma function are derived: the first is an asymptotic series, which contains and improves over the standard asymptotic expansion; the second is a uniformly…

Let $ x\geq 1 $ be a large number, let $ [x]=x-\{x\} $ be the largest integer function, and let $ \varphi(n)$ be the Euler totient function. The asymptotic formula for the new finite sum over the primes $ \sum_{p\leq…

We present a large number of analytic evaluations of Euler sums, namely sums such as \begin{align} M(m,n_0,n_1,n_2, \ldots, n_t) &= \sum_{k=1}^\infty \frac{H(k)^m}{k^{n_0} (k+1)^{n_1} (k+2)^{n_2} \cdots (k+t)^{n_t}}, \nonumber \end{align}…

Dating back to Euler, in classical analysis and number theory, the Hurwitz zeta function $$ \zeta(z,q)=\sum_{n=0}^{\infty}\frac{1}{(n+q)^{z}}, $$ the Riemann zeta function $\zeta(z)$, the generalized Stieltjes constants $\gamma_k(q)$, the…

We show how the asymptotic expansion for the gamma function $\Gamma(x)$, similar to that obtained by Boyd [Proc. Roy. Soc. London A447 (1994) 609--630], can be obtained by using a form of Lagrange's inversion theorem with a remainder. A…

In this paper, we develop Windschitl's approximation formula for the gamma function to two asymptotic expansions by using a little known power series. In particular, for $n\in \mathbb{N}$ with $n\geq 4$, we have \begin{equation*} \Gamma…

We call $n$ a cyclic number if every group of order $n$ is cyclic. It is implicit in work of Dickson, and explicit in work of Szele, that $n$ is cyclic precisely when $\gcd(n,\phi(n))=1$. With $C(x)$ denoting the count of cyclic $n\le x$,…

In analogy with values of the classical Euler Gamma-function at rational numbers and the Riemann zeta-function at positive integers, we consider Thakur's geometric Gamma-function evaluated at rational arguments and Carlitz zeta-values at…

In this paper, we established a sharp version of the difference analogue of the celebrated H\"{o}lder's theorem concerning the differential independence of the Euler gamma function $\Gamma$. More precisely, if $P$ is a polynomial of $n+1$…

The multiple gamma function $\Gamma_n$, defined by a recurrence-functional equation as a generalization of the Euler gamma function, was originally introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the pioneer work of…