Related papers: A domain containing all zeros of the partial theta…

In this paper we set up a bivariate representation of partial theta functions which not only unifies some famous identities for partial theta functions due to Andrews and Warnaar, et al. but also unveils a new characteristic of such…

For a given entire function $f(z)=\sum_{j=0}^{\infty}a_{j}z^{j}$, we study the zero distribution of $f_{r}(z)=\sum_{j\equiv r\pmod m}a_{j}z^{j}$ where $m\in\mathbb{N}$ and $0\le r<m$. We find conditions under which the zeros of $f_{r}(z)$…

We define a dynamical zeta function for nondegenerate Liouville domains, in terms of Reeb dynamics on the boundary. We use filtered equivariant symplectic homology to (i) extend the definition of the zeta function to a more general class of…

A Master equation has been previously obtained which allows the analytic integration of a fairly large family of functions provided that they possess simple properties. Here, the properties of this Master equation are explored, by extending…

Let $\mathcal{W}_{\beta}^\delta(\alpha,\gamma)$ be the class of normalized analytic functions $f$ defined in the domain $|z|<1$ satisfying \begin{align*} {\rm Re\,}…

We prove a functional limit theorem in a space of analytic functions for the random Dirichlet series $D(\alpha;z)=\sum_{n\geq 2}(\log n)^{\alpha}(\eta_n+{\rm i} \theta_n)/n^z$, properly scaled and normalized, where…

The finite Dirichlet series from the title are defined by the condition that they vanish at as many initial zeroes of the zeta function as possible. It turned out that such series can produce extremely good approximations to the values of…

We generalize the construction from arXiv:2102.09329 of theta series for quadratic forms of signature $(n-1,1)$ with homogeneous and spherical polynomials. Namely, we allow that the parameters $c_1,c_2$, which define the theta series and…

In this paper we study the function G(z) := int{0,infinity} y^{z-1}(1 + \exp(y))^{-1} dy, for z in C. We derive a functional equation that relates G(z) and G(1 - z) for all z in C, and we prove: -- That G and the Riemann Zeta function Zeta…

The following theorem is proven: Both real and imaginary parts of the function F(s) defined as F(s)=zeta(s)*Gamma(s/2)*pi**(-s/2)=xi(s)/(s*(s-1)), and whose zeroes exactly coincide with the non-trivial zeroes of the Riemann zeta-function,…

Let $0 < a \le 1$, $s,z \in {\mathbb{C}}$ and $0 < |z|\le 1$. Then the Hurwitz-Lerch zeta function is defined by $\Phi (s,a,z) := \sum_{n=0}^\infty z^n(n+a)^{-s}$ when $\sigma :=\Re (s) >1$. In this paper, we show that the Hurwitz zeta…

We intimate deeper connections between the Riemann zeta and gamma functions than often reported and further derive a new formula for expressing the value of $\zeta(2n+1)$ in terms of zeta at other fractional points. This paper also…

In this article, we study the zeta function $\zeta_q$ associated to the Laplace operator $\Delta_q$ acting on the space of the smooth $(0,q)$-forms with $q=0,\ldots,n$ on the complex projective space $\mathbb{P}^n(\mathbb{C})$ endowed with…

It is shown that, for every integer n>2, there exists \delta_{n}>0 such that the closure of the set of the real parts of the zeros of the nth partial sum of the Riemann zeta function \zeta_{n} contains to the interval [-\delta_{n},b^{n}].…

In the space of bounded real-valued functions on the interval $(0,1)$, we study the convergent sequences of $q$-analogues of multiple zeta values which do not converge to $0$. And we obtain the derived sets of the set of some $q$-analogue…

In this short note we prove the following result: If a completely multiplicative function $f:\mathbb{N}\to[-1,1]$ is small on average in the sense that $\sum_{n\leq x}f(n)\ll x^{1-\delta}$, for some $\delta>0$, and if the Dirichlet series…

We generalize a number of works on the zeros of certain level 1 modular forms to a class of weakly holomorphic modular functions whose $q$-expansions satisfy \[ f_k(A, \tau) \colon = q^{-k}(1+a(1)q+a(2)q^2+...) + O(q),\] where $a(n)$ are…

The bilateral series corresponding to many of the third-, fifth-, sixth- and eighth order mock theta functions may be derived as special cases of $_2\psi_2$ series \[ \sum_{n=-\infty}^{\infty}\frac{(a,c;q)_n}{(b,d;q)_n}z^n. \] Three…

We show the zero-density estimate \[ N(\zeta_{\mathcal{P}}; \alpha, T) \ll T^{\frac{4(1-\alpha)}{3-2\alpha-\theta}}(\log T)^{9} \] for Beurling zeta functions $\zeta_{\mathcal{P}}$ attached to Beurling generalized number systems with…

Using a different approach, we derive integral representations for the Riemann zeta function and its generalizations (the Hurwitz zeta, $\zeta(-k,b)$, the polylogarithm, $\mathrm{Li}_{-k}(e^m)$, and the Lerch transcendent,…