Related papers: Arctanh Sums: Analytic Continuation and Prime-Rest…

Let $A$ be a finite nonempty set of integers. An asymptotic estimate of several dilates sum size was obtained by Bukh. The unique known exact bound concerns the sum $|A+k\cdot A|,$ where $k$ is a prime and $|A|$ is large. In its full…

We present some novelties on the Riemann zeta function. Using an extended formula created for the polylogarithm in a previous paper, $\mathrm{Li}_{k}(e^{z})$, the zeta function's Dirichlet series is analytically continued from $\Re(k)>1$ to…

We analytically continue the Euler prime product for $\Re(s)>\tfrac{1}{2}$ (except for its pole $s=1$) assuming (RH) by introducing a new factor to the Euler product. We also discuss how to recover the Mertens's 3rd Theorem at $s=1$ case,…

We study the Diophantine equation $a^k + b^k = c^k + d^k$ with integer variables and exponent $k>1$, under the linear constraint $(c+d) - (a+b) = h$. We analyze the geometry and arithmetic of these linear slices. On the central slice $h=0$,…

This study presents explicit evaluations of the series \begin{equation*} \sum_{k=1}^\infty \frac{H_{k/n}^{(p)}}{k^q} \quad \text{and} \quad \sum_{k=1}^\infty \frac{(-1)^k H_{k/2n}^{(p)}}{k^q}, \quad p,q,n \in \mathbb{Z}_{\ge 1},\; q \ne 1,…

Combining probabilistic and analytic tools from potential theory, we investigate Dirichlet problems associated with the Dunkl Laplacian $\Delta_k$. We establish, under some conditions on the open set $D\subset\R^d$, the existence of a…

In this work we consider sums of primes that converging very slow. We set as a base, a reformulation of analytic prime number theorem and we use the values of Riemann Zeta function for the approximation. We also give the truncation error of…

Let a and b be non-zero rational numbers that are multiplicatively independent. We study the natural density of the set of primes p for which the subgroup of the multiplicative group of the finite field with p elements generated by (a\mod…

This paper presents a complete symplectic classification of $A_k$ Hamiltonians on $\mathbb R^2$, in the analytic and smooth categories. Precisely, consider the pair $(H, \omega)$ consisting of a Hamiltonian and a symplectic structure on…

Let $A$ be a finite set of $k$ integers. For $h \leq k$, the restricted $h$-fold sumset $h^{\wedge} A$ is the set of all sums of $h$ distinct elements of $A$. In additive combinatorics, much of the focus has traditionally been on finite…

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…

We show that the analytic continuations of Helson zeta functions $ \zeta_\chi (s)= \sum_1^{\infty}\chi(n)n^{-s} $ can have essentially arbitrary poles and zeroes in the strip $ 21/40 < \Re s < 1 $ (unconditionally), and in the whole…

For a positive irrational number $\alpha,$ we study the ordinary Dirichlet series $\zeta_\alpha(s) = \sum\limits_{n\geq1} \lfloor\alpha n\rfloor^{-s}$ and $S_\alpha(s) = \sum\limits_{n\geq1} (\left\lceil\alpha n\right\rceil - \left\lceil…

The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L^2-L^p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential…

Let $D$ be an open disk of radius $\le 1$ in $\mathbb C$, and let $(\epsilon_n)$ be a sequence of $\pm 1$. We prove that for every analytic function $f: D \to \mathbb C$ without zeros in $D$, there exists a unique sequence $(\alpha_n)$ of…

Given the asymptotic expansion for the logarithmic integral $\int_0^n \frac{dt}{\ln(t)}$, obtained from repeated integration by parts until the expansion terms reach a minimum; approaching zero. Which determines a cut-off for the number of…

The sum $S(h,k):=\sum_{j=1}^{k-1}(-1)^{j+1+[hj/k]}$ appears in the modular transformation formulae of the classical theta function $\vartheta_3(z)$. The double sum $S(k) := \sum_{h=1}^{k-1}S(h,k)$ has a remarkable distribution of values.…

It is shown that two conjectures put forward in the recent article Iksanov and Kostohryz (2025) are true. Namely, we prove a functional central limit theorem (FCLT) and a law of the iterated logarithm (LIL) for a random Dirichlet series…

In analytic number theory, several results make use of information regarding the prime values of a multiplicative function in order to extract information about its averages. Examples of such results include Wirsing's theorem and the…

In a recent work, Gun and co-workers have proposed that $\,\sum_{n=-\infty}^{\infty}{(n+\alpha)^{-k}}\,$ is a transcendental number for all integer $\,k$, $k > 1$, and $\,\alpha \in \mathbb{Q} \backslash \mathbb{Z}$. Here in this work, this…