<p>This thesis in analytic number theory consists of 3 parts and 13 individual papers.</p><p>In the first part we prove some results in Turán power sum theory. We solve a problem of Paul Erdös and disprove conjectures of Paul Turán and K. Ramachandra that would have implied important results on the Riemann zeta function.</p><p>In the second part we prove some new results on moments of the Hurwitz and Lerch zeta functions (generalized versions of the Riemann zeta function) on the critical line.</p><p>In the third and final part we consider the following question: What is the natural generalization of the classical Poisson summation formula from the Fourier analysis of the real line to the matrix group SL(2,R)? There are candidates in the literature such as the pre-trace formula and the Selberg trace formula.</p><p>We develop a new summation formula for sums over the matrix group SL(2,Z) which we propose as a candidate for the title "The Poisson summation formula for SL(2,Z)". The summation formula allows us to express a sum over SL(2,Z) of smooth functions f on SL(2,R) with compact support, in terms of spectral theory coming from the full modular group, such as Maass wave forms, holomorphic cusp forms and the Eisenstein series. In contrast, the pre-trace formula allows us to get such a result only if we assume that f is also SO(2) bi-invariant.</p><p>We indicate the summation formula's relationship with additive divisor problems and the fourth power moment of the Riemann zeta function as given by Motohashi. We prove some identities on Kloosterman sums, and generalize our main summation formula to a summation formula over integer matrices of fixed determinant D. We then deduce some consequences, such as the Kuznetsov summation formula, the Eichler-Selberg trace formula and the classical Selberg trace formula.</p>

This thesis in analytic number theory consists of 3 parts and 13 individual papers. In the first part we prove some results in Turán power sum theory. We solve a problem of Paul Erdös and disprove conjectures of Paul Turán and K. Ramachandra that would have implied important results on the Riemann zeta function. In the second part we prove some new results on moments of the Hurwitz and Lerch zeta functions (generalized versions of the Riemann zeta function) on the critical line. In the third and final part we consider the following question: What is the natural generalization of the classical Poisson summation formula from the Fourier analysis of the real line to the matrix group SL(2,R)? There are candidates in the literature such as the pre-trace formula and the Selberg trace formula. We develop a new summation formula for sums over the matrix group SL(2,Z) which we propose as a candidate for the title "The Poisson summation formula for SL(2,Z)". The summation formula allows us to express a sum over SL(2,Z) of smooth functions f on SL(2,R) with compact support, in terms of spectral theory coming from the full modular group, such as Maass wave forms, holomorphic cusp forms and the Eisenstein series. In contrast, the pre-trace formula allows us to get such a result only if we assume that f is also SO(2) bi-invariant. We indicate the summation formula's relationship with additive divisor problems and the fourth power moment of the Riemann zeta function as given by Motohashi. We prove some identities on Kloosterman sums, and generalize our main summation formula to a summation formula over integer matrices of fixed determinant D. We then deduce some consequences, such as the Kuznetsov summation formula, the Eichler-Selberg trace formula and the classical Selberg trace formula.

S-boxes are key components of many symmetric cryptographic primitives. Among them, some block ciphers and hash functions are vulnerable to attacks based on differential cryptanalysis, a technique introduced by Biham and Shamir in the early 90’s. Resistance against attacks from this family depends on the so-called differential properties of the S-boxes used. When we consider S-boxes as functions over finite fields of characteristic 2, monomials turn out to be good candidates. In this Master’s Thesis, we study the differential properties of a particular family of monomials, namely those with exponent 2ͭᵗ-1 In particular, conjectures from Blondeau’s PhD Thesis are proved. More specifically, we derive the differential spectrum of monomials with exponent 2ͭᵗ-1 for several values of t using a method similar to the proof Blondeau et al. made of the spectrum of x -<img src="http://www.diva-portal.org/cgi-bin/mimetex.cgi?%5Crightarrow" /> x⁷. The first two chapters of this Thesis provide the mathematical and cryptographic background necessary while the third and fourth chapters contain the proofs of the spectra we extracted and some observations which, among other things, connect this problem with the study of particular Dickson polynomials.