This thesis consists of an introduction and four papers. All four papers are devoted to problems in Number Theory. In Paper I, a special class of local ζ-functions is studied. The main theorem states that the functions have all zeros on the line Re(s)=1/2.This is a natural generalization of the result of Bump and Ng stating that the zeros of the Mellin transform of Hermite functions have Re(s)=1/2.In Paper II and Paper III we study eigenfunctions of desymmetrized quantized cat maps.If N denotes the inverse of Planck's constant, we show that the behavior of the eigenfunctions is very dependent on the arithmetic properties of N. If N is a square, then there are normalized eigenfunctions with supremum norm equal to <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?N%5E%7B1/4%7D" />, but if N is a prime, the supremum norm of all eigenfunctions is uniformly bounded. We prove the sharp estimate <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5C%7C%5Cpsi%5C%7C_%5Cinfty=O(N%5E%7B1/4%7D)" /> for all normalized eigenfunctions and all $N$ outside of a small exceptional set. For normalized eigenfunctions of the cat map (not necessarily desymmetrized), we also prove an entropy estimate and show that our functions satisfy equality in this estimate.We call a special class of eigenfunctions newforms and for most of these we are able to calculate their supremum norm explicitly.For a given <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?N=p%5Ek" />, with k&gt;1, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?2/%5Csqrt%7B1%5Cpm%201/p%7D" /> and the supremum norm of the newforms in the second half have at most three different values, all of the order <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?N%5E%7B1/6%7D" />. The only dependence of A is that the normalization factor is different if A has eigenvectors modulo p or not. We also calculate the joint value distribution of the absolute value of n different newforms.In Paper IV we prove a generalization of Mertens' theorem to Beurling primes, namely that \lim_{n \to \infty}\frac{1}{\ln n}\prod_{p \leq n} \left(1-p^{-1}\right)^{-1}=Ae^{\gamma}<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Clim_%7Bn%20%5Cto%20%5Cinfty%7D%5Cfrac%7B1%7D%7B%5Cln%20n%7D%5Cprod_%7Bp%20%5Cleq%20n%7D%0A%5Cleft(1-p%5E%7B-1%7D%5Cright)%5E%7B-1%7D=Ae%5E%7B%5Cgamma%7D," />where γ is Euler's constant and Ax is the asymptotic number of generalized integers less than x. Thus the limit <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?M=%5Clim_%7Bn%5Cto%5Cinfty%7D%5Cleft(%5Csum_%7Bp%5Cle%20n%7Dp%5E%7B-1%7D-%5Cln(%5Cln%20n)%5Cright)" />exists. We also show that this limit coincides with <img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Clim_%7B%5Calpha%5Cto%200%5E+%7D%0A%5Cleft(%5Csum_p%20p%5E%7B-1%7D(%5Cln%20p)%5E%7B-%5Calpha%7D-1/%5Calpha%5Cright)" /> ; for ordinary primes this claim is called Meissel's theorem. Finally we will discuss a problem posed by Beurling, namely how small |N(x)-[x] | can be made for a Beurling prime number system Q≠P, where P is the rational primes. We prove that for each c&gt;0 there exists a Q such that |N(x)-[x] | / QC 20100902

The study of zeta functions is one of the primary aspects of modern number theory. Hecke was the first to prove that the Dedekind zeta function of any algebraic number field has an analytic continuation over the whole plane and satisfies a simple functional equation. He soon realized that his method would work, not only for Dedekind zeta functions and L–series, but also for a zeta function formed with a new type of ideal character which, for principal ideals depends not only on the residue class of the number(representing the principal ideal) modulo the conductor, but also on the position of the conjugates of the number in the complex field. He then showed that these “Hecke” zeta functions satisfied the same type of functional equation as the Dedekind zeta function, but with a much more complicated factor. In his doctoral thesis, John Tate replaced the classical notion of zeta function, as a sum over integral ideals of a certain type of ideal character, by the integral over the idele group of a rather general weight function times an idele character which is trivial on field elements. He derived a Poisson Formula for general functions over the adeles, summed over the discrete subgroup of field elements. This was then used to give an analytic continuation for all of the generalized zeta functions and an elegant functional equation was established for them. The mention of the Poisson Summation Formula immediately reminds one of the Theta function and the proof of the functional equation for the Riemann zeta function. The two proofs share close analogues with the functional equation for the Theta function now replaced by the number theoretic Riemann–Roch Theorem. Translating the results back into classical terms one obtains the Hecke functional equation, together with an interpretation of the complicated factor in it as a product of certain local factors coming form the archimedean primes and the primes of the conductor. This understanding of Tate’s results in the classical framework essentially boils down to constructing the generalized weight function and idele group characters which are trivial on field elements. This is facilitated by the understanding of the local zeta functions. We explicitly compute in both cases, the local and the global, illustrating the working of the ideas in a concrete setup. I have closely followed Tate’s original thesis in this exposition.