Related papers: The functional equation for $\zeta$

In this short note we give a characterization of ZM-groups that uses the functions defined and studied in [3,4]. This leads to a proof of Conjecture 6 in [4].

It is well known that the Riemann zeta function can be completed to the Riemann xi function $\xi(s)$ in the sense that its functional equation has a higher symmetric form $\xi(1-s)=\xi(s)$. In the previous paper (Tohoku Math. J. 72 (2020),…

The connection between Lefschetz formulae and zeta function is explained. As a particular example the theory of the generalized Selberg zeta function is presented. Applications are given to the theory of Anosov flows and prime geodesic…

Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation…

A key theorem formulated in the context of functional Mellin transforms generalizes the important relationship $\exp\mathrm{tr} M=\det\exp M$. Along with the involution symmetry of the zeta function, the theorem suggests a strategy for…

Some problems involving the classical Hardy function $$ Z(t) := \zeta(1/2+it)\bigl(\chi(1/2+it)\bigr)^{-1/2}, \quad \zeta(s) = \chi(s)\zeta(1-s) $$ are discussed. In particular we discuss the odd moments of $Z(t)$, the distribution of its…

This note studies the Laurent series of the inverse zeta function $1/\zeta(s)$ at any fixed nontrivial zero $\rho$ of the zeta function $\zeta(s)$, and its connection to the simplicity of the nontrivial zeros.

We solve an interpolation problem for computing $\zeta(2n)$ in a rather elementary way, by generalizing the main idea in \cite{SE}.

This paper discusses the simplest examples of spectral zeta functions, especially those associated with graphs, a subject which has not been much studied. The analogy and the similar structure of these functions, such as their parallel…

We define the functional LYZ ellipsoid of log-concave functions. Then we give notes appended to [6].

We present an operational semantics for the language MeTTa.

We prove under RH the existence of a very large positive and negative values of the argument of the Riemann zeta function on a very short intervals.

Using numerical, theoretical and general methods, we construct evaluation formulas for the Jacobi $\theta$ functions. Some of our results are conjectures, but are verified numerically.

We shall answer a question of Mez\H{o} on the $q$-analogue of the Raabe's integral formula for $0<q<1$ and we shall evaluate an integral involving the first theta function. Moreover, we will reproduce short proofs for some identities of…

We prove by an elementary method the Riemann hypothesis for the local Euler factor of the zeta function of quadratic orders.

We present an explicit formula for a weighted sum over the zeros of the Riemann zeta function. This weighted sum is evaluated in terms of a sum over the prime numbers, weighted with help of the Hermite polynomials. From the explicit formula…

This article provides a simple proof of the quadratic formula, which also produces an efficient and natural method for solving general quadratic equations. The derivation is computationally light and conceptually natural, and has the…

We approximate the Riemann Zeta-Function by polynomials and Dirichlet polynomials with restricted zeros.

We give a short Wiener measure proof of the Riemann hypothesis based on a surprising, unexpected and deep relation between the Riemann zeta $\zeta(s)$ and the trivial zeta $\zeta_{t}(s):=Im(s)(2Re(s)-1)$.

We give a formula for $f(\eta)$, where $f :\mathbb C \to \mathbb C$ is a continuously differentiable function satisfying $f(\bar z) = \overline{f(z)}$, and $\eta$ is a dual quaternion. Note this formula is straightforward or well known if…