This is an extension and background to a talk I gave on 9 October 2013 to the Brown Graduate Student Seminar, called `A friendly intro to sieves with a look towards recent progress on the twin primes conjecture.’ During the talk, I mention several sieves, some with a lot of detail and some with very little detail. I also discuss several results and built upon many sources. I’ll provide missing details and/or sources for additional reading here.

In this survey I discuss A. Buium’s theory of “differential equations in the p-adic direction” ([Bu05]) and its interrelations with “geometry over fields with one element”, on the background of various approaches to p-adic models in theoretical physics.

We provide lower bounds for p-adic valuations of multisums of factorial ratios which satisfy an Ap\’ery-like recurrence relation: these include Ap\’ery, Domb, Franel numbers, the numbers of abelian squares over a finite alphabet, and constant terms of powers of certain Laurent polynomials. In particular, we prove Beukers’ conjectures on the p-adic valuation of Ap\’ery numbers. Furthermore, we give an effective criterion for a sequence of factorial ratios to satisfy the p-Lucas property for almost all primes p.

For sufficiently large n Ramanujan gave a sufficient condition for the truth Robin’s InEquality $X(n):=\frac{\sigma(n)}{n\ln\ln n}

By a classical result of Weyl, for any increasing sequence $(n_k)_{k \geq 1}$ of integers the sequence of fractional parts $(\{n_k x\})_{k \geq 1}$ is uniformly distributed modulo 1 for almost all $x \in [0,1]$. Except for a few special cases, e.g. when $n_k=k, k \geq 1$, the exceptional set cannot be described explicitly. The exact asymptotic order of the discrepancy of $(\{n_k x\})_{k \geq 1}$ is only known in a few special cases, for example when $(n_k)_{k \geq 1}$ is a (Hadamard) lacunary sequence, that is when $n_{k+1}/n_k \geq q > 1, k \geq 1$. In this case of quickly increasing $(n_k)_{k \geq 1}$ the system $(\{n_k x\})_{k \geq 1}$ (or, more general, $(f(n_k x))_{k \geq 1}$ for a 1-periodic function $f$) shows many asymptotic properties which are typical for the behavior of systems of \emph{independent} random variables. Precise results depend on a fascinating interplay between analytic, probabilistic and number-theoretic phenomena.

We study Kummer’s approach towards proving the Fermat’s last Theorem for regular primes. Some basic algebraic prerequisites are also discussed in this report, and also a brief history of the problem is mentioned. We review among other things the Class number formula, and use this formula to conclude our study.

In this paper we propose a definition of a recurrence relation homomorphism and illustrate our definition with a few examples. We then define the period of a k-th order of recurrence relation and deduce certain preliminary results associated with them.

We describe a computation that confirms the ternary Goldbach Conjecture up to 8,875,694,145,621,773,516,800,000,000,000 (>8.875e30).

This paper has two parts. The first part surveys Euler’s work on the constant gamma=0.57721… bearing his name, together with some of his related work on the gamma function, values of the zeta function and divergent series. The second part describes various mathematical developments involving Euler’s constant, as well as another constant, the Euler-Gompertz constant. These developments include connections with arithmetic functions and the Riemann hypothesis, and with sieve methods, random permutations and random matrix products. It includes recent results on Diophantine approximation and transcendence related to Euler’s constant.

This is a survey of methods developed in the last few years to prove results on growth in non-commutative groups. These techniques have their roots in both additive combinatorics and group theory, as well as other fields. We discuss linear algebraic groups, with SL_2(Z/pZ) as the basic example, as well as permutation groups. The emphasis lies on the ideas behind the methods.