Zeta-functions associated with modified Bessel functions are introduced as ordinary Dirichlet series whose coefficients are J-Bessel and K-Bessel functions. Integral representations, transformation formulas, a power series expansion involving the Riemann zeta-function and a recurrence formula are given. The inverse Laplace transform of Weber's first exponential integral is the basic tool to derive the integral representations. As an application, we give a new proof of the Fourier series expansion...

In a recent paper, Freitas and Siksek proved an asymptotic version of Fermat’s Last Theorem for many totally real fields. We prove an extension of their result to generalized Fermat equations of the form $A{x}^p+B{y}^p+C{z}^p=0$, where A, B, C are odd integers belonging to a totally real field.

Let $f$ be a non-CM newform of weight $k\ge 2$. Let $L$ be a subfield of the coefficient field of $f$. We completely settle the question of the density of the set of primes $p$ such that the $p$-th coefficient of $f$ generates the field $L$. This density is determined by the inner twists of $f$. As a particular case, we obtain that in the absence of nontrivial inner twists, the density is $1$ for $L$ equal to the whole coefficient field. We also present some new data on reducibility of Hecke polynomials, which suggest questions...

Let ${\lambda }_f\left(n\right)$ be the nth normalized Fourier coefficient of a holomorphic or Maass cusp form f for SL(2,ℤ). We establish the asymptotic formula for the summatory function ${\sum }_{\begin{array}{c}n\le x\\ n\equiv l\left(modq\right)\end{array}}{\left|{\lambda }_f\left(n\right)\right|}^{2j}$ as x → ∞, where q grows with x in a definite way and j = 2,3,4.

Let $f$ be a normalized primitive holomorphic cusp form of even integral weight for the full modular group $\Gamma =\mathrm{SL}(2,\mathbb{Z})$. Denote by ${\lambda }_f\left(n\right)$ the $n$th normalized Fourier coefficient of $f$. We are interested in the average behaviour of the sum $$\sum _{a^2+b^2\le x}{\lambda }_f^j(a^2+b^2)$$ for $x\ge 1$, where $a,b\in \mathbb{Z}$ and $j\ge 9$ is any fixed positive integer. In a similar manner, we also establish analogous results for the normalized coefficients of Dirichlet expansions of associated symmetric power $L$-functions and Rankin-Selberg $L$-functions.

We explore the question of how big the image of a Galois representation attached to a $\Lambda$-adic modular form with no complex multiplication is and show that for a “generic” set of $\Lambda$-adic modular forms (normalized, ordinary eigenforms with no complex multiplication), all have a large image.

Let $E$ be a CM number field, $p$ an odd prime totally split in $E$, and let $X$ be the $p$-adic analytic space parameterizing the isomorphism classes of $3$-dimensional semisimple $p$-adic representations of $\mathrm{Gal}(\overline{E}/E)$ satisfying a selfduality condition “of type $\mathrm{U}\left(3\right)$”. We study an analogue of the infinite fern of Gouvêa-Mazur in this context and show that each irreducible component of the Zariski-closure of the modular points in $X$ has dimension at least $3[E:\mathbb{Q}]$. As important steps, and in any rank, we prove that any first order...