For $s \in {\mathbb{C}}$ and $0 < a <1$, let $ζ(s,a)$ and ${\rm{Li}}_s (e^{2πia})$ be the Hurwitz and periodic zeta functions, repectively. For $0 < a \le 1/2$, put $Z(s,a) := ζ(s,a) + ζ(s,1-a)$, $P(s,a) := {\rm{Li}}_s (e^{2πia}) + {\rm{Li}}_s (e^{2πi(1-a)})$, $Y(s,a) := ζ(s,a) - ζ(s,1-a)$ and $O(s,a):= -i \bigl( {\rm{Li}}_s (e^{2πia}) - {\rm{Li}}_s (e^{2πi(1-a)}) \bigr)$.
Let $n \ge 0$ be an integer and $b := r/q$, where $q>r>0$ are coprime integers. In this paper, we prove that the values $Z(-n,b)$, $π^{-2n-2} P(2n+2,b)$, $Y(-n,b)$ and $π^{-2n-1} O(2n+1,b)$ are rational numbers, in addition, $π^{-2n-2} Z(2n+2,b)$, $P(-n,b)$, $π^{-2n-1} Y(2n+1,b)$ and $O(-n,b)$ are polynomials of $\cos (2π/q)$ and $\sin (2π/q)$ with rational coefficients. Furthermore, we show that $Z(-n,a)$, $π^{-2n-2} P(2n+2,a)$, $Y(-n,a)$ and $π^{-2n-1} O(2n+1,a)$ are polynomials of $0<a<1$ with rational coefficient, in addition, $π^{-2n-2} Z(2n+2,a)$, $P(-n,a)$, $π^{-2n-1} Y(2n+1,a)$ and $O(-n,a)$ are rational functions of $\exp (2 πia)$ with rational coefficients. Note that the rational numbers, polynomials and rational functions mentioned above are given explicitly.
Moreover, we show that $P(s,a) \equiv 0$ for all $ 0 < a < 1/2$ if and only if $s$ is a negative even integer. We also prove similar assertions for $Z(s,a)$, $Y(s,a)$, $O(s,a)$ and so on. In addition, we prove that the function $Z(s,|a|)$ appears as the spectral density of some stationary self-similar Gaussian distributions.
Inspired by a famous formula of Ramanujan for odd zeta values, we prove an analogous formula involving the Hurwitz zeta function. We introduce a new integral kernel related to the Hurwitz zeta function, generalizing the integral kernel associated to Ramanujan's identity. We also derive several infinite families of identities analogous to Ramanujan's formula.
We give a result on the asymptotic behavior of the Hurwitz-Lerch multiple zeta functions near non-positive integer points by using the Apostol-Bernoulli polynomials. From this result, we can evaluate limit values at non-positive integer points.
Maximum of Exponential Random Variables, Hurwitz's Zeta Function, and the Partition Function
在优惠券收集者问题的上下文中，一个自然的问题是独立几何分布随机变量（具有不同参数）的最大值的行为。Brennan等人（British J. of Math. & CS. 8（2015），330-336）。在这里，我们提供明确的渐近表达式的最大值的时刻，以及指数随机变量与相应的参数的最大值。我们还处理的概率，每个变量是最大的。计算导致表达式涉及赫维茨的zeta函数在某些特殊点。我们在这里明确地找到了函数在这些点上的值。此外，我们处理的最大值的分布函数与分拆函数的生成函数密切相关。因此，我们的结果（和证明）依赖于经典的结果有关的分拆函数。
In this note we prove that for all $a \in \mathbb{N}$, $x \in \mathbb{R}_+ \cup \{0\}$, and $s \in \mathbb{C}$ with $\Re(s) > a + 2$, the (alternating) weighted series of the Hurwitz zeta function, $$ \sum_{k \geq 1} (\pm 1)^k (k + x)^aζ(s,k + x), $$ resolves into a finite combination of Hurwitz (Lerch) zeta functions. This applies in Marichal and Zenaïdi's theory on analogues of the Bohr-Mollerup theorem for higher-order convex functions.
Semigroup's series for negative degrees of the gaps values in numerical semigroups generated by two integers and identities for the Hurwitz zeta functions
The definition for the $p$-adic Hurwitz-type Euler zeta functions has been given by using the fermionic $p$-adic integral on $\mathbb Z_p$. By computing the values of this kind of $p$-adic zeta function at negative integers, we show that it interpolates the Euler polynomials $p$-adically. Many properties are provided for the $p$-adic Hurwitz-type Euler zeta functions, including the convergent Laurent series expansion, the distribution formula, the functional equation, the reflection formula, the derivative formula, the $p$-adic Raabe formula and so on. The definition for the $p$-adic Euler $L$…
In this paper we establish a new summation method by expanding $\prod_{k}(1-\frac{z}{a_{k}})^{-1}$ with two approaches: the Taylor expansion and the infinite partial fraction decomposition. Here we focus on the case when $a_{k}$ is arithmetic sequence. By this summation we obtain many equalities involve Hurwitz zeta function and Gammma function.
We study the moments $M_k(T;α) = \int_T^{2T} |ζ(s,α)|^{2k}\,dt$ of the Hurwitz zeta function $ζ(s,α)$ on the critical line, $s = 1/2 + it$ with a rational shift $α\in \mathbb Q$. We conjecture, in analogy with the Riemann zeta function, that $M_k(T;α) \sim c_k(α) T (\log T)^{k^2}$ . Using heuristics from analytic number theory and random matrix theory, we conjecturally compute $c_k(α)$. In the process, we investigate moments of products of Dirichlet $L$-functions on the critical line. We prove our conjectures for the cases $k = 1,2$.
Let $K$ be a compact set with connected complement on the half-plane Re$(s)>0$, and let $f$ be a continuous function on $K$ which is analytic in its interior. We prove that for any parameter $0<α<1, α\neq \frac 1 2$ then $f(s)$ may be uniformly approximated arbitrarily closely by $ζ(1+iT+iδs,α)$ on $K$ for some $T,δ>0$, where $ζ(s,α)$ denote the Hurwitz zeta-function. This is the first known universality result that is also known to hold for the Hurwitz zeta-function with an algebraic irrational parameter.
Rigorous high-precision computation of the Hurwitz zeta function and its derivatives
我们研究了使用欧拉-麦克劳林公式数值计算的赫维茨zeta函数$ζ(s,a)$的$s, a \in \mathbb{C}$，以及任意数量的衍生物相对于$s$，以任意精度与严格的误差界。技术，导致快速实施进行了讨论。我们提出了新的记录计算的Stieltjes常数，Keiper-Li系数和第一个非平凡零的黎曼zeta函数，获得使用本文所述的算法的开源实现。
We consider some closed-form evaluations of certain infinite sums involving the Hurwitz zeta function $ζ(s,α)$ of the form \[\sum_{k=1}^\infty (\pm 1)^k k^m ζ(s,k),\] where $m$ is a non-negative integer. For the sums with $m=0$ and the argument $k$ in $ζ(s,k)$ replaced by $ka+b$, where $a$ and $b$ are positive parameters, we also obtain a transformation formula suitable for computation in the limit $a\to0$.
We study the theta function and the Hurwitz-type zeta function associated to the Lucas sequence $U=\{U_n(P,Q)\}_{n\geq 0}$ of the first kind determined by the real numbers $P,Q$ under certain natural assumptions on $P$ and $Q$. We deduce an asymptotic expansion of the theta function $θ_U(t)$ as $t\downarrow 0$ and use it to obtain a meromorphic continuation of the Hurwitz-type zeta function $ζ_{U}\left( s,z\right) =\sum\limits_{n=0}^{\infty }\left(z+U_{n}\right) ^{-s}$ to the whole complex $s-$plane. Moreover, we identify the residues of $ζ_{U}\left( s,z\right)$ at all poles in the half-plane …
