Related papers: On linear relations for Dirichlet series formed by…

Let $r_1,\ldots,r_s:\mathbb{Z}_{n\geqslant 0}\to\mathbb{C}$ be linearly recurrent sequences whose associated eigenvalues have arguments in $\pi\mathbb{Q}$ and let $F(z):=\sum_{n\geqslant 0}f(n)z^n$, where $f(n)\in\{r_1(n),\ldots,$…

The \emph{Barnes $\zeta$-function} is \[ \zeta_n (z, x; \a) := \sum_{\m \in \Z_{\ge 0}^n} \frac{1}{\left(x + m_1 a_1 + \dots + m_n a_n \right)^z} \] defined for $\Re(x) > 0$ and $\Re(z) > n$ and continued meromorphically to $\C$.…

We give a new proof of Fatou's theorem: {\em if an algebraic function has a power series expansion with bounded integer coefficients, then it must be a rational function.} This result is applied to show that for any non--trivial completely…

Results are presented for some infinite series appearing in Feynman diagram calculations, many of which are similar to the Euler series. These include both one-, two- and three-dimensional series. The sums of these series can be evaluated…

Let $(a_n)_{n\geq 0}$ be an arbitrary sequence and $(a_{\lfloor n/k \rfloor})_{n\geq 0}$ its dual floor sequence. We study infinite series and finite generalized binomial sums involving $(a_{\lfloor n/k \rfloor})_{n\geq 0}$. As applications…

We present a numerical study of Riemann's formula for the oscillating part of the density of the primes and their powers. The formula is comprised of an infinite series of oscillatory terms, one for each zero of the zeta function on the…

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by the recurrence $F_{n+2}=F_{n+1}+F_n$, for $n\geq 0$, where $F_0=0$ and $F_1=1$. There are several generalizations of this sequence and also several interesting identities. In this…

For positive integers $i_1,...,i_k$ with $i_1 > 1$, we define the multiple $t$-value $t(i_1,...,i_k)$ as the sum of those terms in the usual infinite series for the multiple zeta value $\zeta(i_1,...,i_k)$ with odd denominators. Like the…

The bivariate series $\theta (q,x):=\sum _{j=0}^{\infty}q^{j(j+1)/2}x^j$ defines a {\em partial theta function}. For fixed $q$ ($|q|<1$), $\theta (q,.)$ is an entire function. We prove a property of stabilization of the coefficients of the…

This contribution presents all possible solutions to the Diophantine equations $F_k=L_mL_n$ and $L_k=F_mF_n$. To be clear, Fibonacci numbers that are the product of two arbitrary Lucas numbers and Lucas numbers that are the product of two…

We study the essential singularities of geometric zeta functions $\zeta_{\mathcal L}$, associated with bounded fractal strings $\mathcal L$. For any three prescribed real numbers $D_{\infty}$, $D_1$ and $D$ in $[0,1]$, such that…

We establish the mapping relations between analytic functions and periodic functions using the abstract operators $\cos(h\partial_x)$ and $\sin(h\partial_x)$, including the mapping relations between power series and trigonometric series,…

The Riemann zeta function $\zeta(s)$ is defined as the infinite sum $\sum_{n=1}^\infty n^{-s}$, which converges when ${\rm Re}\,s>1$. The Riemann hypothesis asserts that the nontrivial zeros of $\zeta(s)$ lie on the line ${\rm Re}\,s=…

We show that if $F(s)$ is a nondegenerate ordinary Dirichlet series with nonnegative coefficients and $F(k)$ is a rational number for all large enough positive integers $k$, then the denominators of those rational numbers are unbounded. In…

Nous \'etudions la nature arithm\'etique de $q$-analogues des valeurs $\zeta(s)$ de la fonction z\^eta de Riemann, notamment des valeurs des fonctions $\zeta_q(s)= \sum_{k=1} ^{\infty}q^k \sum_{d\mid k} ^{}d^{s-1}$, $s=1,2,...$, o{\`u} $q$…

We study sums of Dirichlet characters over polynomials in $\mathbb{F}_q[t]$ with a prescribed number of irreducible factors. Our main results are explicit formulae for these sums in terms of zeros of Dirichlet L-functions. We also exhibit…

We show that exponential sums (ES) of the form \begin{equation*} S(f, N)= \sum_{k=0}^{N-1} \sqrt{w_k} e^{2 \pi i f(k)}, \end{equation*} can be efficiently carried out with a quantum computer (QC). Here $N$ can be exponentially large, $w_k$…

Already in 1734 Euler found a short explicit formula for the value of Riemann zeta function Zeta(s) when the argument s equals a positive integer 2n where n=1,2,3,. No such formula exists for odd positive integer arguments of Zeta. The…

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_0 = 0, F_1 = 1$ and $F_{n+2} = F_{n+1}+F_n$ for $n \geq 0$. In this paper, we solve all powers of two which are sums of four Fibonacci numbers with a few exceptions that we…

Counting functions are constructed for sums of integers raised to a fixed positive rational power. That is, given values formed by $u_1^{j/k} + u_2^{j/k} + ... + u_l^{j/k}$, $u_i \in \mathbb{Z}^+$, the number of values less than or equal to…