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Stephan Baier

Publications and source records attributed to Stephan Baier.

At least 37 records · Page 2Linked to original sources
arXiv2025

Small Solutions of generic ternary quadratic congruences

We consider small solutions of quadratic congruences of the form $x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q}$, where $q=p^m$ is an odd prime power. Here, $α_2$ is arbitrary but fixed and $α_3$ is variable, and we assume that $(α_2α_3,q)=1$. We show that for all $α_3$ modulo $q$ which are coprime to $q$ except for a small number of $α_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$ to the congruence $x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q}$ with $\max\{|x_1|,|x_2|,|x_3|\}\le N$ holds if $N\ge q^{11/24+\varepsilon}$ as $q$ tends to infinity over the set of all odd prime powers. It is of significance that we break the barrier 1/2 in the above exponent. If $q$ is restricted to powers $p^m$ of a {\it fixed} prime $p$ and $m$ tends to infinity, we obtain a slight improvement of this result using the theory of $p$-adic exponent pairs, as developed by Milićević, replacing the exponent $11/24$ above by $11/25$. Under the Lindelöf hypothesis for Dirichlet $L$-functions, we are able to replace the exponent $11/24$ above by $1/3$.

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arXiv2024

Small solutions of generic ternary quadratic congruences to general moduli

We study small non-trivial solutions of quadratic congruences of the form $x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q}$, with $q$ being an odd natural number, in an average sense. This extends previous work of the authors in which they considered the case of prime power moduli $q$. Above, $α_2$ is arbitrary but fixed and $α_3$ is variable, and we assume that $(α_2α_3,q)=1$. We show that for all $α_3$ modulo $q$ which are coprime to $q$ except for a small number of $α_3$'s, an asymptotic formula for the number of solutions $(x_1,x_2,x_3)$ to the congruence $x_1^2+α_2x_2^2+α_3x_3^2\equiv 0 \bmod{q}$ with $\max\{|x_1|,|x_2|,|x_3|\}\le N$ and $(x_3,q)=1$ holds if $N\ge q^{11/24+\varepsilon}$ and $q$ is large enough. It is of significance that we break the barrier 1/2 in the above exponent. Key tools in our work are Burgess's estimate for character sums over short intervals and Heath-Brown's estimate for character sums with binary quadratic forms over small regions whose proofs depend on the Riemann hypothesis for curves over finite fields. We also formulate a refined conjecture about the size of the smallest solution of a ternary quadratic congruence, using information about the Diophantine properties of its coefficients.

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arXiv2024

A Bombieri-Vinogradov-type theorem for moduli with small radical

In this article, we extend our recent work on a Bombieri-Vinogradov-type theorem for sparse sets of prime powers $p^N\le x^{1/4-\varepsilon}$ with $p\le (\log x)^C$ to sparse sets of moduli $s\le x^{1/3-\varepsilon}$ with radical rad$(s)\le x^{9/40}$. To derive our result, we combine our previous method with a Bombieri-Vinogradov-type theorem for general moduli $s\le x^{9/40}$ obtained by Roger Baker.

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arXiv2024

Variance of primes in short residue classes for function fields

Keating and Rudnick derived asymptotic formulas for the variances of primes in arithmetic progressions and short intervals in the function field setting. Here we consider the hybrid problem of calculating the variance of primes in intersections of arithmetic progressions and short intervals. Keating and Rudnick used an involution to translate short intervals into arithmetic progressions. We follow their approach but apply this involution, in addition, to the arithmetic progressions. This creates dual arithmetic progressions in the case when the modulus $Q$ is a polynomial in $\mathbb{F}_q[T]$ such that $Q(0)\not=0$. The latter is a restriction which we keep throughout our paper. At the end, we discuss what is needed to relax this condition.

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arXiv2023

Large sieve inequalities with power moduli and Waring's problem

We improve the large sieve inequality with $k$th-power moduli, for all $k\ge 4$. Our method relates these inequalities to a restricted variant of Waring's problem. Firstly, we input a classical divisor bound on the number of representations of a positive integer as a sum of two $k$th-powers. Secondly, we input a recent and general result of Wooley on mean values of exponential sums. Lastly, we state a conditional result, based on the conjectural Hardy-Littlewood formula for the number of representations of a large positive integer as a sum of $k+1$ $k$th-powers.

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arXiv2023

Diophantine approximation with prime denominator in real quadratic function fields

In the thirties of the last century, I. M. Vinogradov proved that the inequality $||pα||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to a nearest integer. This result has subsequently been improved by many authors. In particular, Vaughan (1978) replaced the exponent $1/5$ by $1/4$ using his celebrated identity for the von Mangoldt function and a refinement of Fourier analytic arguments. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality $||pα||\le p^{-1/3+\varepsilon}$. This exponent $1/3$ is considered the limit of the current technology. Recently, in \cite{BaMo}, the authors established an analogue of Matomäki's result for imaginary quadratic extensions of the function field $k=\mathbb{F}_q(T)$. In this paper, we consider the case of real quadratic extensions of $k$ of class number 1, for which we prove a function field analogue of Vaughan's above-mentioned result (exponent $θ=1/4$). Our method uses versions of Vaughan's identity and the Dirichlet approximation theorem for function fields. The latter was established by Arijit Ganguly in the appendix to our previous paper \cite{BaMo} on the imaginary quadratic case. We also simplify arguments in the paper \cite{BM} on the same problem for real quadratic number fields by D. Mazumder and the first-named author.

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arXiv2022

Solutions of $x_1^2+x_2^2-x_3^2=n^2$ with small $x_3$

Friedlander and Iwaniec investigated integral solutions $(x_1,x_2,x_3)$ of the equation $x_1^2+x_2^2-x_3^2=D$, where $D$ is square-free and satisfies the congruence condition $D\equiv 5\bmod{8}$. They obtained an asymptotic formula for solutions with $x_3\asymp M$, where $M$ is much smaller than $\sqrt{D}$. To be precise, their condition is $M\ge D^{1/2-1/1332}$. Their analysis led them to averages of certain Weyl sums. The condition of $D$ being square-free is essential in their work. We investigate the "opposite" case when $D=n^2$ is a square of an odd integer $n$. This case is different in nature and leads to sums of Kloosterman sums. We obtain an asymptotic formula for solutions with $x_3\asymp M$, where $M\ge D^{1/2-1/16+\varepsilon}$ for any fixed $\varepsilon>0$.

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arXiv2022

Asymptotic behavior of small solutions of quadratic congruences in three variables modulo prime powers

Let $p>5$ be a fixed prime and assume that $α_1,α_2,α_3$ are coprime to $p$. We study the asymptotic behavior of small solutions of congruences of the form $α_1x_1^2+α_2x_2^2+α_3x_3^2\equiv 0\bmod{q}$ with $q=p^n$, where $\max\{|x_1|,|x_2|,|x_3|\}\le N$ and $(x_1x_2x_3,p)=1$. (In fact, we consider a smoothed version of this problem.) If $α_1,α_2,α_3$ are fixed and $n\rightarrow \infty$, we establish an asymptotic formula (and thereby the existence of such solutions) under the condition $N\gg q^{1/2+\varepsilon}$. If these coefficients are allowed to vary with $n$, we show that this formula holds if $N\gg q^{11/18+\varepsilon}$. The latter should be compared with a result by Heath-Brown who established the existence of non-zero solutions under the condition $N \gg q^{5/8+\varepsilon}$ for odd square-free moduli $q$.

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arXiv2022

A Bombieri-Vinogradov-type theorem with prime power moduli

In 2020, Roger Baker \cite{Bak} proved a result on the exceptional set of moduli in the prime number theorem for arithmetic progressions of the following kind. Let $\mathcal{S}$ be a set of pairwise coprime moduli $q\le x^{9/40}$. Then the primes $l\le x$ distribute as expected in arithmetic progressions mod $q$, except for a subset of $\mathcal{S}$ whose cardinality is bounded by a power of $\log x$. We use a $p$-adic variant Harman's sieve to extend Baker's range to $q\le x^{1/4-\varepsilon}$ if $\mathcal{S}$ is restricted to prime powers $p^N$, where $p\le (\log x)^C$ for some fixed but arbitrary $C>0$. For large enough $C$, we thus get an almost all result. Previously, an asymptotic estimate for $π(x;p^N,a)$ of the expected kind, with $p$ being an odd prime, was established in the wider range $p^N\le x^{3/8-\varepsilon}$ by Barban, Linnik and Chudakov \cite{BLC}. Gallagher \cite{Gal} extended this range to $p^N\le x^{2/5-\varepsilon}$ and Huxley \cite{Hux2} improved Gallagher's exponent to $5/12$. A lower bound of the correct order of magnitude was recently established by Banks and Shparlinski \cite{BaS} for the even wider range $p^N\le x^{0.4736}$. However, all these results hold for {\it fixed} primes $p$, and the $O$-constants in the relevant estimates depend on $p$. Therefore, they do not contain our result. In a part of our article, we describe how our method relates to these results.

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arXiv2022

Diophantine approximation with prime restriction in function fields

In the thirties of the last century, I. M. Vinogradov established uniform distribution modulo 1 of the sequence $pα$ when $α$ is a fixed irrational real number and $p$ runs over the primes. In particular, he showed that the inequality $||pα||\le p^{-1/5+\varepsilon}$ has infinitely prime solutions $p$, where $||.||$ denotes the distance to the nearest integer. This result has subsequently been improved by many authors. The current record is due to Matomäki (2009) who showed the infinitude of prime solutions of the inequality $||pα||\le p^{-1/3+\varepsilon}$. This exponent $1/3$ is considered the limit of the current technology. We prove function field analogues of this result for the fields $k=\mathbb{F}_q(T)$ and imaginary quadratic extensions $K$ of $k$. Essential in our method is the Dirichlet approximation theorem for function fields which is established in general form in the appendix authored by Arijit Ganguly.

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arXiv2022

Small Pythagorean triples modulo prime powers

Let $p>5$ be a fixed prime. We obtain an asymptotic formula related to small solutions of quadratic congruences of the form $x_1^2+x_2^2\equiv x_3^2\bmod{p^n}$ where $\max\{|x_1|,|x_2|,|x_3|\}\le p^{νn}$ with $ν>1/2$.

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arXiv2021

Solving $p$-adic polynomial equations using Jarratt's Method

We implement an iterative numerical method to solve polynomial equations $f(x)=0$ in the $p$-adic numbers, where $f(x) \in\mathbb{Z}_p[x]$. This method is a simplified $p$-adic analogue of Jarratt's method for finding roots of functions over the real numbers. We establish that our method has a higher order of convergence than J.F.T. Rabago's $p$-adic version of Olver's method from 2016. Moreover, we weaken the initial conditions in Rabago's method, which allows us to start the iteration with a multiple root of the congruence $f(x) \equiv 0 \bmod{p}$.

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arXiv2021

The large sieve for square moduli in function fields

We prove a lower and an upper bound for the large sieve with square moduli for function fields. These bounds correspond to bounds for the classical large sieve with square moduli established in arXiv:1812.05844 by Baier, Lynch and Zhao and arXiv:math/0512271 by Baier and Zhao. Our lower bound in the function field setting contradicts an upper bound obtained in arXiv:1802.03131 by Baier and Singh. Indeed, we point out an error in arXiv:1802.03131.

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arXiv2021

On the distribution of $αp$ modulo one in quadratic number fields

We investigate the distribution of $αp$ modulo one in quadratic number fields $\mathbb{K}$ with class number one, where $p$ is restricted to prime elements in the ring of integers of $\mathbb{K}$. Here we improve the relevant exponent $1/4$ obtained by the first and third named authors for imaginary quadratic number fields \cite{BT} and by the first and second named authors for real quadratic number fields \cite{BM} to $7/22$. This generalizes a result of Harman \cite{HarZi} who obtained the same exponent $7/22$ for $\mathbb{Q}(i)$ by extending his method which gave this exponent for $\mathbb{Q}$ \cite{harman1996on-the-distribu}. Our proof is based on an extension of his sieve method to arbitrary number fields. Moreover, we need an asymptotic evaluation of certain smooth sums over prime ideals appearing in \cite{BM}, for which we use analytic properties of Hecke L-functions with Größencharacters.

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arXiv2021

On the distribution of $αp$ modulo one in imaginary quadratic number fields with class number one

We investigate the distribution of $αp$ modulo one in imaginary quadratic number fields $\mathbb{K}\subset\mathbb{C}$ with class number one, where $p$ is restricted to prime elements in the ring of integers $\mathcal{O} = \mathbb{Z}[ω]$ of $\mathbb{K}$. In analogy to classical work due to R. C. Vaughan, we obtain that the inequality $\lVertαp\rVert_ω< \mathrm{N}(p)^{-1/8+ε}$ is satisfied for infinitely many $p$, where $\lVert\varrho\rVert_ω$ measures the distance of $\varrho\in\mathbb{C}$ to $\mathscr{O}$ and $\mathrm{N}(p)$ denotes the norm of $p$. The proof is based on Harman's sieve method and employs number field analogues of classical ideas due to Vinogradov. Moreover, we introduce a smoothing which allows us to make conveniently use of the Poisson summation formula.

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