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Jörg Brüdern

Publications and source records attributed to Jörg Brüdern.

14 recordsLinked to original sources
arXiv2024

The Manin-Peyre conjecture for smooth spherical Fano threefolds

The Manin-Peyre conjecture is established for smooth spherical Fano threefolds of semisimple rank one and type N. Together with the previously solved case T and the toric cases, this covers all types of smooth spherical Fano threefolds. The case N features a number of structural novelties; most notably, one may lose regularity of the ambient toric variety, the height conditions may contain fractional exponents, and it may be necessary to exclude a thin subset with exceptionally many rational points from the count, as otherwise Manin's conjecture in its original form would turn out to be incorrect.

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arXiv2023

On the distribution of powered numbers

Asymptotic formulae are established for the number of natural numbers $m$ with largest square-free divisor not exceeding $m^{\vartheta}$, for any fixed positive parameter $\vartheta$. Related counting functions are also considered.

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arXiv2023

On the sum of two powered numbers

Fix a positive real number $θ$. The natural numbers $m$ with largest square-free divisor not exceeding $m^θ$ form a set $\mathscr{A}$, say. It is shown that whenever $θ>1/2$ then all large natural numbers $n$ are the sum of two elements of $\mathscr{A}$. This is nearly best possible.

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arXiv2017

Local oscillations in moderately dense sequences of primes

The distribution of differences of consecutive members of sequences of primes is investigated. A quantitative measure for oscillations among these differences is the curvature of the sequence. If the sequence is not too sparse, then sharp estimates for its curvature are provided.

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arXiv2012

Random diophantine equations, I

We consider additive diophantine equations of degree $k$ in $s$ variables and establish that whenever $s\ge 3k+2$ then almost all such equations satisfy the Hasse principle. The equations that are soluble form a set of positive density, and among the soluble ones almost all equations admit a small solution. Our bound for the smallest solution is nearly best possible.

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arXiv2012

On a senary cubic form

A strong form of the Manin-Peyre conjecture with a power saving error term is proved for a certain cubic fourfold.

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arXiv2012

Piatetski-Shapiro sequences

We consider various arithmetic questions for the Piatetski-Shapiro sequences $\fl{n^c}$ ($n=1,2,3,...$) with $c>1$, $c\not\in\N$. We exhibit a positive function $θ(c)$ with the property that the largest prime factor of $\fl{n^c}$ exceeds $n^{θ(c)-\eps}$ infinitely often. For $c\in(1,\tfrac{149}{87})$ we show that the counting function of natural numbers $n\le x$ for which $\fl{n^c}$ is squarefree satisfies the expected asymptotic formula. For $c\in(1,\tfrac{147}{145})$ we show that there are infinitely many Carmichael numbers composed entirely of primes of the form $p=\fl{n^c}$.

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arXiv2011

Random Diophantine inequalities of additive type

Using the Davenport-Heilbronn circle method, we show that for almost all additive Diophantine inequalities of degree $k$ in more than $2k$ variables the expected asymptotic formula for the density of solutions holds true. This appears to be the first metric result on Diophantine inequalities.

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arXiv2010

Random diophantine equations of additive type

Using the circle method in combination with lattice point counting arguments, we show that for almost all homogeneous diophantine equations of additive type and degree $k$ in more than $4k$ variables, the Local-Global principle holds true. Moreover, our approach shows that almost all such equations having a non-trivial integer solution have a very small such solution, the bound being close to the best possible one.

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