Related papers: A note on odd zeta values

In this paper we prove the existence of infinitely many saddle-shaped positive solutions for non-cooperative nonlinear elliptic systems with bistable nonlinearities in the phase-separation regime. As an example, we prove that the system \[…

Formal multiple zeta values allow to study multiple zeta values by algebraic methods in a way that the open question about their transcendence is circumvented. In this note we show that Hoffman's basis conjecture for formal multiple zeta…

We examine the calculation of the second and fourth moments and shifted moments of the Riemann zeta-function on the critical line using long Dirichlet polynomials and divisor correlations. Previously this approach has proved unsuccessful in…

Using a novel rewriting problem, we show that several natural decision problems about finite automata are undecidable (i.e., recursively unsolvable). In contrast, we also prove three related problems are decidable. We apply one result to…

In this note, infinite series involving Fibonacci and Lucas numbers are derived by employing formulae similar to that which Roger Ap\'ery utilized in his seminal paper proving the irrationality of $\zeta(3)$.

We investigate the values of the Riemann zeta function at odd integers and the Dirichlet beta function at even integers, by collecting several distinct analytic frameworks converging to these values, thus providing a unifying perspective.…

The cotangent zeta function is a very interesting object, which is related to partial zeta functions and Hecke $L$-functions of real quadratic fields. Its special values at odd integers greater than 1 are explicitly evaluated by Berndt in…

Using elementary methods we find surprising connections between the values of the Riemann Zeta Function over integers and the fractional parts of rational powers, and a connection between the Riemann Zeta Function and the Prime Zeta…

In this paper we consider iterated integrals of multiple polylogarithm functions and prove some explicit relations of multiple polylogarithm functions. Then we apply the relations obtained to find numerous formulas of alternating multiple…

In this paper we introduce and study double tails of multiple zeta values. We show, in particular, that they satisfy certain recurrence relations and deduce from this a generalization of Euler's classical formula…

In this paper we will focus on the study of relationships that can exist between odd numbers and different traditional functions like the gamma function, Riemann zeta function or function of von Mangoldt. Number theory applies to this…

In this paper, Riemann's Zeta function with odd positive integer argument is represented as an infinite summation of integer powers of $\pi$ with rational coefficients. Specific values for Apery's Constant and Catalan's Constant are then…

In this paper, we construct an object of the abelian category of mixed Tate motive associated to multiple zeta values. as a consequence, we prove the inequality of the dimension of the vector space generated by multiple zeta values, which…

Conditionally on the Riemann Hypothesis we obtain bounds of the correct order of magnitude for the 2k-th moment of the Riemann zeta-function for all positive real k < 2.181. This provides for the first time an upper bound of the correct…

We study periodic points for endomorphisms $\sigma$ of abelian varieties $A$ over algebraically closed fields of positive characteristic $p$. We show that the dynamical zeta function $\zeta_\sigma$ of $\sigma$ is either rational or…

A cutset is a non-empty finite subset of $\mathbb{Z}^d$ which is both connected and co-connected. A cutset is odd if its vertex boundary lies in the odd bipartition class of $\mathbb{Z}^d$. Peled suggested that the number of odd cutsets…

We make plausible the existence of counterexamples to the Riemann hypothesis located in the neighbourhood of unusually large peaks of $\vert \zeta \vert$. The main ingredient in our argument is an identity which links the zeros of a…

In this article we obtain, using an expression of the digamma function $\psi(x)$ due to Mikolas, integral representations of the zeta function of odd arguments $\zeta(2p+1)$ for any positive value of $p$. The integrand consists of the…

We prove that the special value conjecture for the Zeta function of a proper, regular arithmetic scheme X that we formulated in our previous article [8] is compatible with the functional equation of the Zeta function provided that the…

Euler showed that if an odd perfect number exists, it must be of the form $N = p^\alpha q_{1}^{2\beta_{1}}$ $\ldots$ $q_{k}^{2\beta_{k}}$, where $p, q_{1}, \ldots, q_k$ are distinct odd primes, $\alpha$, $\beta_{i} \geq 1$, for $1 \leq i…