We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, Zhang and Maynard-Tao, respectively.[-]

We will talk about recent work showing there are infinitely many primes with no $7$ in their decimal expansion. (And similarly with $7$ replaced by any other digit.) This shows the existence of primes in a 'thin' set of numbers (sets which contain at most $X^{1-c}$ elements less than $X$) which is typically vey difficult.
The proof relies on a fun mixture of tools including Fourier analysis, Markov chains, Diophantine approximation, combinatorial geometry as well as tools from analytic number theory.[-]

All previous methods of showing the existence of large gaps between primes have relied on the fact that smooth numbers are unusually sparse. This feature of the argument does not seem to generalise to showing large gaps between primes in subsets, such as values of a polynomial. We will talk about recent work which allows us to show large gaps between primes without relying on smooth number estimates. This then generalizes naturally to show long strings of consecutive composite values of a polynomial. This is joint work with Ford, Konyagin, Pomerance and Tao.[-]

Bourgain (2015) estimated the number of prime numbers with a proportion $c$ > 0 of preassigned digits in base 2 ($c$ is an absolute constant not specified). We present a generalization of this result in any base $g$ ≥ 2 and we provide explicit admissible values for the proportion $c$ depending on $g$. Our proof, which adapts, develops and refines Bourgain's strategy, is based on the circle method and combines techniques from harmonic analysis together with results on zeros of Dirichlet $L$-functions, notably a very sharp zero-free region due to Iwaniec.[-]

It is well known that the every letter $\alpha$ of an automatic sequence $a(n)$ has a logarithmic density -- and it can be decided when this logarithmic density is actually adensity. For example, the letters $0$ and $1$ of the Thue-Morse sequences $t(n)$ have both frequences $1/2$. The purpose of this talk is to present a corresponding result for subsequences of general automatic sequences along primes and squares. This is a far reaching of two breakthroughresults of Mauduit and Rivat from 2009 and 2010, where they solved two conjectures by Gelfond on the densities of $0$ and $1$ of $t(p_n)$ and $t(n^2)$ (where $p_n$ denotes thesequence of primes). More technically, one has to develop a method to transfer density results for primitive automatic sequences to logarithmic-density results for general automatic sequences. Then asan application one can deduce that the logarithmic densities of any automatic sequence along squares $(n^2){n\geq 0}$ and primes $(p_n)_{n\geq 1}$ exist and are computable. Furthermore, if densities exist then they are (usually) rational. [-]

This is joint work with Jörg Thuswaldner from University of Leoben.

A linear recurrent number system is a generalization of the $q$-adic number system, where we replace the sequence of powers of $q$ by a linear recurrent sequence $G_{k+d}=a_1G_{k+d-1}+\cdots+a_dG_k$ for $k\geq 0$. Under some mild conditions on the recurrent sequence every positive integer $n$ has a representation of the form \[n=\sum_{j=0}^k \varepsilon_j(n)G_j.\]

The $q$-adic number system corresponds to the linear recursion $G_{k+1}=qG_k$ and $G_0=1$. The first example of a real generalization is due to Zeckendorf who showed that the Fibonacci sequence $G_0=1$, $G_1=2$, $G_{k+2}=G_{k+1}+G_k$ for $k\geq0$ yields a representation for each positive integer. This is unique if we additionally suppose that no two consecutive ones exist in the representation. Similar restrictions hold for different recurrent sequences and they build the essence of these number systems.

In the present talk we investigate the representation of primes and almost primes in linear recurrent number systems. We start by showing the different results due to Fouvry, Mauduit and Rivat in the case of $q$-adic number systems. Then we shed some light on their main tools and techniques. The heart of our considerations is the following Bombieri-Vinogradov type result
\[\sum_{q < x^{\vartheta-\varepsilon}}\max_{y < x}\max_{1\leq a\leq q} \left\vert\sum_{\substack{n< y,s_G(n)\equiv b\bmod d\\ n\equiv b\bmod q}}1 -\frac1q\sum_{n < y,s_G(n)\equiv b\bmod d}1\right\vert \ll x(\log 2x)^{-A},\]
which we establish under the assumption that $a_1\geq30$. This lower bound is due to numerical estimations. With this tool in hand we are able to show that \[ \left\vert\{n\leq x\colon s_G(n)\equiv b\bmod d, n=p_1\text{ or }n=p_1p_2\}\right\vert\gg \frac{x}{\log x}.\][-]

In the lecture we prove a lower estimate for the average of the absolute value of the remainder term of the prime number theorem which depends in an explicit way on a given zero of the Riemann Zeta Function. The estimate is only interesting if this hypothetical zero lies off the critical line which naturally implies the falsity of the Riemann Hypothesis. (If the Riemann Hypothesis is true, stronger results areobtainable by other metods.) The first explicit results in this direction were proved by Turán and Knapowski in the 1950s, answering a problem of Littlewood from the year 1937. They used the power sum method of Turán. Our present approach does not use Turán's method and gives sharper results.[-]