相关论文: Some integrals of the Dedekind $\eta$ function
In this note, we recall Kummer's Fourier series expansion of the 1-periodic function that coincides with the logarithm of the Gamma function on the unit interval $(0,1)$, and we use it to find closed forms for some numerical series related…
We show for all $1/2 \le \sigma \le 1$ and $t\ge 3$ that $\zeta(\sigma+it)| \le 76.2 t^{4.45 (1-\sigma)^{3/2}}$, where $\zeta$ is the Riemann zeta function. This significantly improves the previous bounds, where $4.45$ is replaced by…
For any $\sigma$ with $0\leq \sigma\leq 1$ and any $T>10$ sufficiently large, let $N_{\zeta}(\sigma,K,T)$ be the number of zeros $\rho=\beta+i\gamma$ of $\zeta_{K}(s)$ with $|\gamma|\leq T$ and $\beta\geq \sigma$ and the zero being counted…
The Stieltjes constants $\gamma_n(K)$ of a number field $K$ are the coefficients of the Laurent expansion of the Dedekind zeta function $\zeta_K(s)$ at its pole $s=1$. In this paper, we establish a similar expression of $\gamma_n(K)$ as…
An easy generalization of Beukers' integrals allows us to conjecture a double integral formula involving the zeta and the gamma functions. A special case of this formula is Sondow's double integral formula for Euler's constant gamma.
In 2021, Hu and Kim defined a new type of gamma function $\widetilde{\Gamma}(x)$ from the alternating Hurwitz zeta function $\zeta_{E}(z,x)$, and obtained some of its properties. In this paper, we shall further investigate the function…
Recently, the degenerate gamma functions are introduced as a degenerate version of the usual gamma function by Kim-Kim. In this paper, we investigate several properties of them. Namely, we obtain an analytic continuation as a meromorphic…
In this paper we provide a new series representation for the values of Riemann zeta function at integer arguments, namely: $ \zeta(m)=\sum_{n=1}^{\infty}\frac{m(-1)^{n-1}\Gamma(1-\omega_{m}n)...\Gamma(1-\omega_{m}^{m-1}n)}{n!n^m}$, where…
The Stieltjes coefficients $\gamma_k(a)$ arise in the expansion of the Hurwitz zeta function $\zeta(s,a)$ about its single simple pole at $s=1$ and are of fundamental and long-standing importance in analytic number theory and other…
We shall make use of the method of partial fractions to generalize some of Ramanujan's infinite series identities, including Ramanujan's famous formula for $\zeta(2n+1)$, and we shall also give a generalization of the transformation formula…
Let $a\in (0, \infty)$, $\gamma(a)$ be the Generalized Euler-Mascheroni Constant, and let \begin{align*} &x_n=\frac1a+\frac{1}{a+1}+\cdots+\frac{1}{a+n-1}-\ln\frac{a+n}{a},\\…
Chen's iterated integrals may be generalized by interpolation of functions of the positive integer number of times which particular forms are iterated in integrals along specific paths, to certain complex values. These generalized iterated…
A recently published result states inequalities of the harmonic mean of the digamma function. In this work, we prove among others results that for all positive real numbers $x\neq 1$, $$-\gamma<-\gamma…
In [6], Kohnen proves that if $\Gamma=\Gamma_0(N)$ where $N$ is a square-free integer, then any modular function of weight $0$ for $\Gamma$ having a divisor supported at the cusps is an $\eta$-product. Under the condition of having rational…
The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, $ \int x^\alpha e^{\eta x^\beta}dx, \int x^\alpha \cosh\left(\eta x^\beta\right)dx, \int x^\alpha \sinh\left(\eta x^\beta\right)dx, \int…
One-dimensional and two-dimensional integrals containing $E_b(-u)$ and $E_{\alpha ,\beta }\left(\delta x^{\gamma }\right)$ are considered. $E_b(-u)$ is the Mittag-Leffler function and the integral is taken over the rectangle $0 \leq x <…
Dedekind sums, arithmetic correlation sums that arose in Dedekind's study of the modular transformation of the logarithm of the eta-function, are surprisingly ubiquitous. Their arithmetic properties attracted the attention of number…
The Dedekind eta function $\eta(\tau)$ is defined by \[\eta(\tau)=e^{\pi i\tau/12}\prod_{n=1}^{\infty}\left(1-e^{2\pi i n\tau}\right),\quad\text{when}\;\text{Im}\,\tau>0.\] It plays an important role in number theory, especially in the…
In this paper we present a new formula relating Stieltjes numbers $\gamma _{n}$ and Laurent coefficinets $\eta_{n}$ of logarithmic derivative of the Riemann's zeta function. Using it we derive an explicit formula for the oscillating part of…
Logarithmic integrals revisited. We consider integrals of the form $\int_0^1 \ln{\ln{(\frac{1}{x})}}R{(x)}{\rm d}x$ again, where $R{(x)}$ is a rational function, and we will explain a way to obtain their values.