Related papers: Special zeta Mahler functions

A connection between the theory of formal groups and arithmetic number theory is established. In particular, it is shown how to construct general Almkvist--Meurman--type congruences for the universal Bernoulli polynomials that are related…

We prove some results connecting the zeta functions of varieties over finite fields with the big Witt ring over $\mathbb Z$. We explore relations with motivic measures and a classical formula of Macdonald on invariants of symmetric products…

Siegel defined zeta functions associated with indefinite quadratic forms, and proved their analytic properties such as analytic continuations and functional equations. Coefficients of these zeta functions are called measures of…

In this paper we will establish functional equations for Mahler measures of families of genus-one two-variable polynomials. These families were previously studied by Beauville, and their Mahler measures were considered by Boyd,…

In this paper, motivated by studying special values of zeta functions attached to finite type F_p-schemes, we introduce a category of ``arithmetic C(S^1,R)-modules'' attached to any Dedekind ring R, and compute the 0th K-group of this…

We introduce the quaternionic Mahler measure for non-commutative polynomials, extending the classical complex Mahler measure. We establish the existence of quaternionic Mahler measure for slice regular polynomials in one and two variables.…

A recent result of Griffin, Ono, Rolen and Zagier on Jensen polynomials related with the Riemann zeta function is improved.

We prove an analogue of Kronecker's second limit formula for a continuous family of "indefinite zeta functions". Indefinite zeta functions were introduced in the author's previous paper as Mellin transforms of indefinite theta functions, as…

Let $\chi_{-f}$ be the odd quadratic Dirichlet character of conductor $f$, and let $\mathrm{m}(P)$ denote the Mahler measure of a polynomial $P$. In 1984, Chinburg conjectured that for any such $\chi_{-f}$ there exist an integral bivariate…

We conjecture results about the complex moments of the derivative of the Riemann zeta function, evaluated at the non-trivial zeros of the Riemann zeta function. We do this via two different random matrix computations. In the first, we find…

We explore the theory of multiple zeta values (MZVs) and some of their $q$-generalisations. Multiple zeta values are numerical quantities that satisfy several combinatorial relations over the rationals. These relations include two…

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…

We consider the Mahler measure of the polynomial 1+x_1+x_2+x_3+x_4, which is the first case not yet evaluated explicitly. A conjecture due to F. Rodriguez-Villegas represents this Mahler measure as a special value at the point 4 of the…

Laurent polynomials related to the Hahn-Exton $q$-Bessel function, which are $q$-analogues of the Lommel polynomials, have been introduced by Koelink and Swarttouw. The explicit strong moment functional with respect to which the Laurent…

We introduce new zeta functions related to an endomorphism $\phi$ of a discrete group $\Gamma$. They are of two types: counting numbers of fixed ($\rho\sim \rho\circ\phi^n$) irreducible representations for iterations of $\phi$ from an…

After Furstenberg had provided a first glimpse of remarkable rigidity phenomena associated with the joint action of several commuting automorphisms (or endomorphisms) of a compact abelian group, further key examples motivated the…

The idea of the metaplectic theta function was introduced by Tomio Kubota in the 1960s. These theta functions are constructed as residues of Eisenstein series and are only known completely in the case of double covers and, up to the…

If $\alpha$ is a non-zero algebraic number, we let $m(\alpha)$ denote the Mahler measure of the minimal polynomial of $\alpha$ over $\mathbb Z$. A series of articles by Dubickas and Smyth, and later by the author, develop a modified version…

In this paper we study the higher-order Euler numbers and polynomials and we introduce the mutiple zeta functions which interpolate higher-order Euler polynomials and numbers at negative integers

We derive several identities for the Hurwitz and Riemann zeta functions, the Gamma function, and Dirichlet $L$-functions. They involve a sequence of polynomials $\alpha_k(s)$ whose study was initiated in an earlier paper. The expansions…