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This work outcomes with the proof that Dirichlet L-functions are correct for two-dimensional limit theorem .

This thesis presents the following: (i) A detailed exposition of Rankin's classical work on the convulsion of two modular L-functions is given; (ii) Let S be the calss Dirichlet series with Euler product on Re(s) > 1 that can be continued analytically to Re(s) = 1 with a possible pole at s = 1. For F,G E S, let F X G be the Euler product convolution of F and G. Assuming the existence of analytic continuation for certain Dirichlet series and some other conditions, it is proved that F x G is non-vanishing on the line Re(s) = 1; (iii) Let Fn be the set of newforms of weight 2 and level N. For f E Fn, let L(sym2f,s) be the associated symmetric square L-function. Let s0=0o + ito with 1 - 1/46 < 0o <1. It is proved that Cs0,EN1-E<#{f E Fn; L (sym2 f, so)=0} for prime N large enough. Here E>0 and Cso,E is a constant depending only on So and E. / vii, 78 leaves ; 29 cm.

Thesis (doctoral)-- Rheinische Friedrich-Wilhelm-Universität zu Bonn, 1989. / Includes bibliographical references (p. 99-102).

Thesis (Ph. D.)--University of Texas at Austin, 2005. / Supervisor: Fernando Rodriguez-Villegas. Vita. Includes bibliographical references.

No description available.

In this thesis we extend a method of Hall $[30, 34]$ which he used to show the existence of large gaps between consecutive zeros, on the critical line, of the Riemann zeta-function $zeta(s)$. Our modification involves introducing an "amplifier" and enables us to show the existence of gaps between consecutive zeros, on the critical line at height $T,$ of $zeta(s)$ of length at least $2.766 x (2pi/log{T})$. To handle some integral-calculations, we use the article $[44]$ by Hughes and Young. Also, we show that Hall's strategy can be applied not only to $zeta(s),$ but also to Dirichlet $L$-functions $L(s,chi),$ where $chi$ is a primitive Dirichlet character. This also enables us to use stronger integral-results, the article $[14]$ by Conrey, Iwaniec and Soundararajan is used. An unconditional result here about large gaps between consecutive zeros, on the critical line, of some Dirichlet $L$-functions $L(s,chi),$ with $chi$ being an even primitive Dirichlet character, is found. However, we will need to use the Generalised Riemann Hypothesis to make sense of the average gap-length between such zeros. Then the gaps, whose existence we show, have a length of at least $3.54$ times the average.

Traditionally number theorists have studied, both theoretically and computationally, elliptic curves and their L-functions over number fields, in particular over the rational numbers. Much less work has been done over function fields, especially computationally, where the underlying geometry of the function field plays an intimate role in the arithmetic of elliptic curves. We make use of this underlying geometry to develop a method to compute the L-function of an elliptic curve and its twists over the function field of the projective line over a finite field. This method requires computing the number of points on an elliptic curve over a finite field, for which we present a novel algorithm. If the j-invariant of an elliptic curve over a function field is non-constant, its L-function is a polynomial, hence its analytic rank and value at a given point can be computed exactly. We present data in this direction for a family of quadratic twists of four fixed elliptic curves over a few function fields of differing characteristic. First we present analytic rank data that confirms a conjecture of Goldfeld, in stark contrast to the corresponding data in the number field setting. Second, we present data on the integral moments of the value of the L-function at the symmetry point, which on the surface appears to refute random matrix theory conjectures. / text

Thesis (Ph. D.)--University of Chicago, Dept. of Mathematics, June 1999. / Includes bibliographical references. Also available on the Internet.

Thesis--University of Turku, 2008. / Includes bibliographical references (p. 49-50).