J. Schlage-Puchta

activity

20032026

most citedOn triangular billiards

38 citations · 123 across the 49 of their papers we have counts for

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31 papers · 1 filter

math.NT2025

On a Problem by Erdős and Mirsky on the ratio of the number of divisors of consecutive integers

Jan-Christoph Schlage-Puchta

Let $L$ be the closure of the set of all real numbers $α$ , such that there exist infinitely many integers $n$ , such that $α = lo g \frac{d ( n + 1 )}{d ( n )}$ , where $d$ is the num…

math.NT2023★ 1 cited

Alternating state complexity of the set of primes and squarefree integers

Jan-Christoph Schlage-Puchta

We show that the set of prime numbers has exponential alternating complexity, proving a conjecture by Fijalkow. We further show that the set of squarefree integers has essentially…

math.NT2021

Continuous prime systems satisfying $N (x) = c (x - 1) + 1$

Jan-Christoph Schlage-Puchta

Hilberdink showed that there exists a constant $c_{0} > 2$ , such that there exists a continuous prim system satisfying $N (x) = c (x - 1) + 1$ if and only if $c \leq c_{0}$ . Here we determine $c_…

math.NT2021

Addendum to "On triangular Billard"

Jan-Christoph Schlage-Puchta

Let $n, s, t$ be integers satisfying $(n, s, t) = 1$ . We classify all cases such that there is no integer $a$ with $n /2 < a s mod n + a t mod n < 3 n /2$ . This closes a gap in previous work of…

math.NT2020★ 2 cited

On a criterion for Catalan's Conjecture

Jan-Christoph Schlage-Puchta

We give a new proof of a theorem by P. Mihailescu which states that the equation $x^{p} - y^{q} = 1$ is unsolvable with $x, y$ integral and $p, q$ odd primes, unless the congruences $p^q \…

math.NT2019

The non-existence of universal Carmichael numbers

Jan-Christoph Schlage-Puchta

We show that universal elliptic Carmichael numbers do not exist, answering a question of Silverman. Moreover, we show that the probability that an integer $n$ , which is not a prime…

math.NT2019

Oscillations of the error term in the prime number theorem

Jan-Christoph Schlage-Puchta

Let $σ + iγ$ be a zero of the Riemann zeta function to the right of the line $\frac{1}{2} + i t$ . We show that this zero causes large oscillations of the error term of the prime number…

math.NT2019

Weakly multiplicative arithmetic functions and the normal growth of groups

Jan-Christoph Schlage-Puchta

We show that an arithmetic function which satisfies some weak multiplicativity properties and in addition has a non-decreasing or $lo g$ -uniformly continuous normal order is close…

math.NT2019

The density of integers representable as the sum of four prime cubes

Christian Elsholtz, Jan-Christoph Schlage-Puchta

The set of integers which can be written as the sum of four prime cubes has lower density at least $0.009664$ . This improves earlier bounds of $0.003125$ by Ren and $0.005776$ by L…

math.NT2017

Davenport's constant for groups with large exponent

Gautami Bhowmik, Jan-Christoph Schlage-Puchta

Let $G$ be a finite abelian group. We show that its Davenport constant $D (G)$ satisfies $D (G) \leq exp (G) + \frac{∣ G ∣}{e x p ( G )} - 1$ , provided that $exp (G) \geq ∣ G ∣$ , and $D(G)\l…

math.NT2015★ 16 cited

Some examples in the theory of Beurling's generalized prime numbers

Gregory Debruyne, Jan-Christoph Schlage-Puchta, Jasson Vindas

Several examples of generalized number systems are constructed to compare various conditions occurring in the literature for the prime number theorem in the context of Beurling gen…

math.NT2011

A problem of Ramanujan, Erdos and Katai on the iterated divisor function

Yvonne Buttkewitz, Christian Elsholtz, Kevin Ford +1

We determine asymptotically the maximal order of log d(d(n)), where d(n) is the number of positive divisors of n. This solves a problem first put forth by Ramanujan in 1915.

math.NT2011★ 3 cited

An additive property of almost periodic sets

Jan-Christoph Schlage-Puchta

We show that a set is almost periodic if and only if the associated exponential sum is concentrated in the minor arcs. Hence binary additive problems involving almost periodic sets…

math.NT2011★ 4 cited

Primes in short arithmetic progressions

Jan-Christoph Schlage-Puchta

We give a large sieve type inequality for functions supported on primes. As application we prove a conjecture by Elliott, and give bounds for short character sums over primes. The…

math.NT2011★ 5 cited

The equation $ω (n) = ω (n + 1)$

Jan-Christoph Schlage-Puchta

We prove that there are infinitely many integers $n$ such that $n$ and $n + 1$ have the same number of distinct prime divisors.

math.NT2011★ 4 cited

Sign changes of $π (x, q, 1) - π (x, q, a)$

Jan-Christoph Schlage-Puchta

It is known, that under the assumption of the generalized Riemannian hypothesis, the function $π (x, q, 1) - π (x, q, a)$ has infinitely many sign changes. In this article we give an…

math.NT2011★ 5 cited

The exponential sum over squarefree integers

Jan-Christoph Schlage-Puchta

We give an upper bound for the exponential sum over squarefree integers. This establishes a conjecture by Brüdern and Perelli.

math.NT2011★ 8 cited

An estimate for Frobenius' diophantine problem in three dimensions

Jan-Christoph Schlage-Puchta

We give upper and lower bounds for the largest integer not representable as positive linear combination of three given integers, disproving an upper bound conjectured by Beck, Eins…

math.NT2011★ 1 cited

On Shanks' Algorithm for Modular Square Roots

Jan-Christoph Schlage-Puchta

Let $p$ be a prime number, $p = 2^{n} q + 1$ , where $q$ is odd. D. Shanks described an algorithm to compute square roots $(mod p)$ which needs $O (lo g q + n^{2})$ modular multiplications.…

math.NT2011★ 5 cited

The irrationality of a number theoretical series

Jan-Christoph Schlage-Puchta

Denote by $σ_{k} (n)$ the sum of the $k$ -th powers of the divisors of $n$ , and let $S_{k} = \sum_{n \geq 1} \frac{σ _{k} ( n )}{n !}$ . We prove that Schinzel's conjecture H implies that $S_{k}$ is i…

math.NT2011★ 3 cited

The irrationality of some number theoretical series

Jan-Christoph Schlage-Puchta

We prove the irrationality of some factorial series. To do so we combine methods from elementary and analytic number theory with methods from the theory of uniform distribution.

math.NT2011★ 1 cited

Sets with more differences than sums

Jan-Christoph Schlage-Puchta

We show that a random set of integers with density 0 has almost always more differences than sums. This proves a conjecture by Martin and O'Bryant.

math.NT2011

Lower Bounds for Expressions of Large Sieve Type

Jan-Christoph Schlage-Puchta

We show that the large sieve is optimal for almost all exponential sums, thus proving a conjecture by Erdös and Renyi.

math.NT2011★ 1 cited

Sumsets avoiding squarefree integers

Jan-Christoph Schlage-Puchta

We describe the structure of a set of integers $A$ of positive density $δ$ , such that $A + A$ contains no squarefree integer. It turns out that the behaviour changes abruptly at the…

On a Problem by Erdős and Mirsky on the ratio of the number of divisors of consecutive integers

Alternating state complexity of the set of primes and squarefree integers

Continuous prime systems satisfying N(x)=c(x−1)+1

Addendum to "On triangular Billard"

On a criterion for Catalan's Conjecture

The non-existence of universal Carmichael numbers

Oscillations of the error term in the prime number theorem

Weakly multiplicative arithmetic functions and the normal growth of groups

The density of integers representable as the sum of four prime cubes

Davenport's constant for groups with large exponent

Some examples in the theory of Beurling's generalized prime numbers

A problem of Ramanujan, Erdos and Katai on the iterated divisor function

An additive property of almost periodic sets

Primes in short arithmetic progressions

The equation ω(n)=ω(n+1)

Sign changes of π(x,q,1)−π(x,q,a)

The exponential sum over squarefree integers

An estimate for Frobenius' diophantine problem in three dimensions

On Shanks' Algorithm for Modular Square Roots

The irrationality of a number theoretical series

The irrationality of some number theoretical series

Sets with more differences than sums

Lower Bounds for Expressions of Large Sieve Type

Sumsets avoiding squarefree integers

Continuous prime systems satisfying $N (x) = c (x - 1) + 1$

The equation $ω (n) = ω (n + 1)$

Sign changes of $π (x, q, 1) - π (x, q, a)$