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y
Following Zhang's breakthrough on bounded gaps between primes, much work has gone into improving upper bounds on the smallest integer which appears infinitely often as the gap between a given number of primes. Equidistribution estimates for primes in certain arithmetic progressions are a key ingredient of Zhang's proof and later work of Polymath. In this talk, I will highlight how bounded gaps and primes in arithmetic progressions are linked, and I will discuss obstacles and recent successes in using various types of old and new equistribution estimates to improve on the results of Polymath for bounded gaps between primes.
[-] Following Zhang's breakthrough on bounded gaps between primes, much work has gone into improving upper bounds on the smallest integer which appears infinitely often as the gap between a given number of primes. Equidistribution estimates for primes in certain arithmetic progressions are a key ingredient of Zhang's proof and later work of Polymath. In this talk, I will highlight how bounded gaps and primes in arithmetic progressions are linked, ...
[+] 11N05 ; 11N36 ; 11L07
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y
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.
[-] 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 ...
[+] 11M26 ; 11N05 ; 11N30
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y
In this talk, I will discuss new bounds on constrained sets of fractions. Specifically, I will discuss the answer to the following question, which arises in several areas of number theory: For an integer $k\geq2$, consider the set of $k$-tuples of reduced fractions $\frac{a1}{q1} , . . . , \frac{ak}{qk} \in I$, where $I$ is an interval around 0. How many $k$-tuples are there with $\sum_{i} \frac{ai}{qi} \in \mathbb{Z} $? When $k$ is even, the answer is well-known: the main contribution to the number of solutions comes from “diagonal” terms, where the fractions $\frac{ai}{qi}$ cancel in pairs. When $k$ is odd, the answer is much more mysterious! In joint work with Bloom, we prove a near-optimal upper bound on this problem when $k$ is odd. I will also discuss applications of this problem to estimating moments of the distributions of primes and reduced residues.
[-] In this talk, I will discuss new bounds on constrained sets of fractions. Specifically, I will discuss the answer to the following question, which arises in several areas of number theory: For an integer $k\geq2$, consider the set of $k$-tuples of reduced fractions $\frac{a1}{q1} , . . . , \frac{ak}{qk} \in I$, where $I$ is an interval around 0. How many $k$-tuples are there with $\sum_{i} \frac{ai}{qi} \in \mathbb{Z} $? When $k$ is even, the ...
[+] 11D68 ; 11D79 ; 11N05
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y
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 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 ...
[+] 11N05 ; 11B05
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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.
[-] 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, com...
[+] 11N05 ; 11A63
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2 y
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.
[-] 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 ...
[+] 11N05 ; 11N35 ; 11N36
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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.
[-] 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 ...
[+] 11N05 ; 11A41 ; 11A63
English
