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When the sieve works II
[article]
2015
arXiv
pre-print
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This task was recently initiated by Granville, Koukoulopoulos and Matomäki and their main conjecture is proved in this paper. ...
The first ones to study what happens if one also sieves out some primes from [x 1/2 , x] were Granville, Koukoulopoulos and Matomäki [6] . ...
Granville, Koukoulopoulos and Matomäki [6, have reduced a slightly weaker form of the conjecture to an additive combinatorial problem similar to the following hypothesis. ...
arXiv:1509.02371v1
fatcat:ffxw34w5zbbrrnfxdnzaqmdy4e
Multiplicative functions in short intervals II
[article]
2020
arXiv
pre-print
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We determine the behavior of multiplicative functions vanishing at a positive proportion of prime numbers in almost all short intervals. Furthermore we quantify "almost all" with uniform power-saving upper bounds, that is, we save a power of the suitably normalized length of the interval regardless of how long or short the interval is. Such power-saving bounds are new even in the special case of the Möbius function. These general results are motivated by several applications. First, we
arXiv:2007.04290v1
fatcat:qlxfevuuuncbvhkq7y6w7aifyq
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... n work of Hooley on sums of two squares by establishing an asymptotic for the number of integers that are sums of two squares in almost all short intervals. Previously only the order of magnitude was known. Secondly, we extend this result to general norm forms of an arbitrary number field K (sums of two squares are norm-forms of Q(i)). Thirdly, Hooley determined the order of magnitude of the sum of (s_n + 1 - s_n)^γ with γ∈ (1, 5/3) where s_1 < s_2 < ... denote integers representable as sums of two squares. We establish a similar results with γ∈ (1, 3/2) and s_n the sequence of integers representable as norm-forms of an arbitrary number field K. This is the first such result for a number field of degree greater than two. Assuming the Riemann Hypothesis for all Hecke L-functions we also show that γ∈ (1,2) is admissible. Fourthly, we improve on a recent result of Heath-Brown about gaps between x^ε-smooth numbers. More generally, we obtain results about gaps between multiplicative sequences. Finally our result is useful in other contexts aswell, for instance in our forthcoming work on Fourier uniformity (joint with Terence Tao, Joni Teraväinen and Tamar Ziegler).
A new geometric approach to Sturmian words
[article]
2012
arXiv
pre-print
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We introduce a new geometric approach to Sturmian words by means of a mapping that associates certain lines in the n x n -grid and sets of finite Sturmian words of length n. Using this mapping, we give new proofs of the formulas enumerating the finite Sturmian words and the palindromic finite Sturmian words of a given length. We also give a new proof for the well-known result that a factor of a Sturmian word has precisely two return words.
arXiv:1201.4468v1
fatcat:vsxd7652pjgyppkd5wjvxo3oza
DIOPHANTINE APPROXIMATION BY PRIMES
2009
Glasgow Mathematical Journal
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We show that whenever δ > 0 and constants λ i satisfy some necessary conditions, there are infinitely many prime triples p 1 , p 2 , p 3 satisfying the inequality |λ 0 + λ 1 p 1 + λ 2 p 2 + λ 3 p 3 | < (max p j ) −2/9+δ . The proof uses Davenport-Heilbronn adaption of the circle method together with a vector sieve method. 2000 Mathematics Subject Classification. 11D75, 11N36, 11P32.
doi:10.1017/s0017089509990176
fatcat:jsveuu2iffhqbajase7eawwese
A note on the Liouville function in short intervals
[article]
2015
arXiv
pre-print
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In this note we give a short and self-contained proof that, for any δ > 0, ∑_x ≤ n ≤ x+x^δλ(n) = o(x^δ) for almost all x ∈ [X, 2X]. We also sketch a proof of a generalization of such a result to general real-valued multiplicative functions. Both results are special cases of results in our more involved and lengthy recent pre-print.
arXiv:1502.02374v1
fatcat:tsz3r43iijcjvmghvtkvrelnhm
Sign changes of Hecke eigenvalues
[article]
2015
arXiv
pre-print
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2014 pre-print arXiv:1405.7671v1 fatcat:tlwstrj6rraaxkki5ln4lij4zu
Let f be a holomorphic or Maass Hecke cusp form for the full modular group and write λ_f(n) for the corresponding Hecke eigenvalues. We are interested in the signs of those eigenvalues. In the holomorphic case, we show that for some positive constant δ and every large enough x, the sequence (λ_f(n))_n ≤ x has at least δ x sign changes. Furthermore we show that half of non-zero λ_f(n) are positive and half are negative. In the Maass case, it is not yet known that the coefficients are
arXiv:1405.7671v2
fatcat:di7wavbm4ndrjkum3ac4cklrgu
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... , but our method is robust enough to show that on the relative set of non-zero coefficients there is a positive proportion of sign changes. In both cases previous lower bounds for the number of sign changes were of the form x^δ for some δ < 1.
Singmaster's conjecture in the interior of Pascal's triangle
[article]
2021
arXiv
pre-print
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Singmaster's conjecture asserts that every natural number greater than one occurs at most a bounded number of times in Pascal's triangle; that is, for any natural number t ≥ 2, the number of solutions to the equation nm = t for natural numbers 1 ≤ m < n is bounded. In this paper we establish this result in the interior region exp(log^2/3+ε n) ≤ m ≤ n-exp(log^2/3 + ε n) for any fixed ε > 0. Indeed, when t is sufficiently large depending on ε, we show that there are at most four solutions (or at
arXiv:2106.03335v1
fatcat:5unt3ziclvf3xo2xpgd7swesmu
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... ost two in either half of Pascal's triangle) in this region. We also establish analogous results for the equation (n)_m = t, where (n)_m := n(n-1)...(n-m+1) denotes the falling factorial.
Multiplicative functions in short intervals
2016
Annals of Mathematics
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This improves on earlier work of Matomäki [27] and unpublished work of Hafner [13] . ...
Our result also improves on earlier work of Croot [4] , Matomäki [28] , [27] and Balog [1] . ...
doi:10.4007/annals.2016.183.3.6
fatcat:v7rmhixtzzbp5d5ybwmupdvpd4
Sums of positive density subsets of the primes
2013
Acta Arithmetica
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We show that if A and B are subsets of the primes with positive relative lower densities α and β, then the lower density of A + B in the natural numbers is at least (1 − o(1))α/(e γ log log(1/β)) which is asymptotically best possible. This improves results of Ramaré and Ruzsa and of Chipeniuk and Hamel. As in the latter work the problem is reduced to a similar problem for subsets of Z * m using techniques of Green and Green-Tao. Concerning this new problem we show that, for any square-free m
doi:10.4064/aa159-3-1
fatcat:ggujpgsd3zbgpmajuhv3pzhp5a
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... any A, B ⊆ Z * m of densities α and β, the density of A + B in Zm is at least (1 − o(1))α/(e γ log log(1/β)), which is asymptotically best possible when m is a product of small primes. We also discuss an inverse question.
A new geometric approach to Sturmian words
2012
Theoretical Computer Science
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We introduce a new geometric approach to Sturmian words by means of a mapping that associates certain lines in the n ×n-grid and sets of finite Sturmian words of length n. Using this mapping, we give new proofs of the formulas enumerating the finite Sturmian words and the palindromic finite Sturmian words of a given length. We also give a new proof for the well-known result that a factor of a Sturmian word has precisely two return words.
doi:10.1016/j.tcs.2012.01.040
fatcat:wwhqywl3u5eapa53h4xxre5m2e
Multiplicative Functions in Short Intervals, with Applications
2020
EMS Newsletter
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Matomäki (University of Turku, Finland)
1 and 1 ) 11 4P j )<m≤4X/P j λ(m) m s , with P 1 = H, P j+1 = P log P j j for 1 ≤ j ≤ J − 1 and P J = exp((log X)3/4 ). ...
Bibliography Feature Before discussing the Liouville function further, let us define another important object: write ζ : C → C for the Riemann Multiplicative Functions in Short Intervals, with Applications Kaisa ...
doi:10.4171/news/118/9
fatcat:2yjikgzx2be4zpdxee5zlyp5u4
Almost primes in almost all very short intervals
[article]
2021
arXiv
pre-print
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We show that as soon as h→∞ with X →∞, almost all intervals (x-hlog X, x] with x ∈ (X/2, X] contain a product of at most two primes. In the proof we use Richert's weighted sieve, with the arithmetic information eventually coming from results of Deshouillers and Iwaniec on averages of Kloosterman sums.
arXiv:2012.11565v2
fatcat:mcksokn3zrap7jklcghc5is2nm
CARMICHAEL NUMBERS IN ARITHMETIC PROGRESSIONS
2013
Journal of the Australian Mathematical Society
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We prove that when (a, m) = 1 and a is a quadratic residue mod m, there are infinitely many Carmichael numbers in the arithmetic progression a mod m. Indeed the number of them up to x is at least x 1/5 when x is large enough (depending on m).
doi:10.1017/s1446788712000547
fatcat:qqsilopapvfjxc7iusylqczgmm
On the Möbius function in all short intervals
[article]
2020
arXiv
pre-print
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2019 pre-print arXiv:1911.09076v1 fatcat:humxmnwo25d6pobgejllsxxovq
We show that, for the Möbius function μ(n), we have ∑_x < n≤ x+x^θμ(n)=o(x^θ) for any θ>0.55. This improves on a result of Ramachandra from 1976, which is valid for θ>7/12. Ramachandra's result corresponded to Huxley's 7/12 exponent for the prime number theorem in short intervals. The main new idea leading to the improvement is using Ramaré's identity to extract a small prime factor from the n-sum. The proof method also allows us to improve on an estimate of Zhan for the exponential sum of the
arXiv:1911.09076v2
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... öbius function as well as some results on multiplicative functions and almost primes in short intervals.
Fourier uniformity of bounded multiplicative functions in short intervals on average
[article]
2018
arXiv
pre-print
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Let λ denote the Liouville function. We show that as X →∞, ∫_X^2X_α | ∑_x < n ≤ x + Hλ(n) e(-α n) | dx = o ( X H) for all H ≥ X^θ with θ > 0 fixed but arbitrarily small. Previously, this was only known for θ > 5/8. For smaller values of θ this is the first 'non-trivial' case of local Fourier uniformity on average at this scale. We also obtain the analogous statement for (non-pretentious) 1-bounded multiplicative functions. We illustrate the strength of the result by obtaining cancellations in
arXiv:1812.01224v1
fatcat:eh6zlsrul5a2fape2fth4u4f6q
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... e sum of λ(n) Λ(n + h) Λ(n + 2h) over the ranges h < X^θ and n < X, and where Λ is the von Mangoldt function.
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