A. Speiser'is parodė, kad Riemann'o hipotezė yra ekvivalenti tam, kad Riemann'o dzeta funkcijos išvestinė neturi netrivialių nulių į kairę nuo kritinės tiesės. Kiekybinis šio fakto rezultatas buvo pasiektas N. Levinsono ir H. Montgomerio. Šie rezultatai buvo apibendrinti daugeliui dzeta funkcijų, kurioms tikimasi, kad Riemann'o hipotezė galioja. Šiame darbe mes apibendriname Speiser'io ekvivalentą dzeta-funkcijoms. Mes tiriame sąryšį tarp netrivialių nulių išplėstinės Selbergo klasės funkcijoms ir jų išvestinėms šiame regione. Šiai klasei priklauso ir funkcijos, kurioms Riemann'o hipotezė neteisinga. Kaip pavyzdį, mes skaitiniu būdu tiriame sąryšius tarp Dirichlet L-funkcijų ir jų išvestinių tiesinių kombinacijų. / A. Speiser showed that the Riemann hypothesis is equivalent to the absence of non-trivial zeros of the derivative of the Riemann zeta-function left of the critical line. The quantitative version of this result was obtained by N. Levinson and H. Montgomery. This result (or the quantitative version of this result proved by N. Levinson and H. Montgomery) were generalized for many zeta-functions for which the Riemann hypothesis is expected. Here we generalize the Speiser equivalent for zeta-functions. We also investigate the relationship between the on-trivial zeros of the extended Selberg class functions and of their derivatives in this region. This class contains zeta functions for which Riemann hypothesis is not true. As an example, we study the relationship between the trajectories of zeros of linear combinations of Dirichlet $L$-functions and of their derivatives computationally.

No description available.

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.

No description available.

No description available.

We present results evolving from established connections between zeta functions and different systems of polynomials, particularly the Riemann and Hurwitz zeta functions and the Bernoulli and Euler polynomials. In particular we develop certain results related to Apostol's deformation of the Bernoulli polynomials and obtain identities of Carlitz by a novel approach using generating functions instead of difference equations. In the last two chapters we work out new rapidly convergent series expansions of the Riemann zeta function, find coefficient symmetries of a polynomial sequence obtained from the cyclotomic polynomials by a linear fractional transformation of argument and obtain an expression for the constant term in an identity involving the gamma function.

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

L-functions form a connection between elliptic curves and modular forms. The goals of this thesis will be to discuss this connection, and to see similar connections for arithmetic functions.

No description available.

Le polynôme caractéristique d'une variété abélienne sur un corps fini est défini comme étant celui de son endomorphisme de Frobenius. La première partie de cette thèse est consacrée à l'étude des polynômes caractéristiques de variétés abéliennes de petite dimension. Nous décrivons l'ensemble des polynômes intervenant en dimension 3 et 4, le problème analogue pour les courbes elliptiques et surfaces abéliennes ayant été résolu par Deuring, Waterhouse et Rück.Dans la deuxième partie, nous établissons des bornes supérieures et inférieures sur le nombre de points rationnels des variétés abéliennes sur les corps finis. Nous donnons ensuite des bornes inférieures spécifiques aux variétés jacobiennes. Nous déterminons aussi des formules exactes pour les nombres maximum et minimum de points rationnels sur les surfaces jacobiennes. / The characteristic polynomial of an abelian variety over a finite field is defined to be the characteristic polynomial of its Frobenius endomorphism. The first part of this thesis is devoted to the study of the characteristic polynomials of abelian varieties of small dimension. We describe the set of polynomials which occur in dimension 3 and 4; the analogous problem for elliptic curves and abelian surfaces has been solved by Deuring, Waterhouse and Rück.In the second part, we give upper and lower bounds on the number of points on abelian varieties over finite fields. Next, we give lower bounds specific to Jacobian varieties. We also determine exact formulas for the maximum and minimum number of points on Jacobian surfaces.