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Stephan Baier

Publications and source records attributed to Stephan Baier.

At least 91 records · Page 5Linked to original sources
arXiv2010

Mean values with cubic characters

We investigate various mean value problems involving order three primitive Dirichlet characters. In particular, we obtain an asymptotic formula for the first moment of central values of the Dirichlet L-functions associated to this family, with a power saving in the error term. We also obtain a large-sieve type result for order three (and six) Dirichlet characters.

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arXiv2009

On Hecke Eigenvalues at Piatetski-Shapiro Primes

Let $λ(n)$ be the normalized n-th Fourier coefficient of a holomorphic cusp form for the full modular group. We show that for some constant $C > 0$ depending on the cusp form and every fixed $c$ in the range $1 < c < 8/7$, the mean value of $λ(p)$ is $\ll \exp (-C \sqrt{\log N})$ as p runs over all (Piatetski-Shapiro) primes of the form $[n^c]$ with a natural number $n \leq N$.

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arXiv2008

A refined version of the Lang-Trotter Conjecture

Let $E$ be an elliptic curve defined over the rational numbers and $r$ a fixed integer. Using a probabilistic model consistent with the Chebotarev theorem for the division fields of $E$ and the Sato-Tate distribution, Lang and Trotter conjectured an asymptotic formula for the number of primes up to $x$ which have Frobenius trace equal to $r$, where $r$ is a {\it fixed} integer. However, as shown in this note, this asymptotic estimate cannot hold for {\it all} $r$ in the interval $|r|\le 2\sqrt{x}$ with a uniform bound for the error term, because an estimate of this kind would contradict the Chebotarev density theorem as well as the Sato-Tate conjecture. The purpose of this note is to refine the Lang-Trotter conjecture, by taking into account the "semicircular law", to an asymptotic formula that conjecturally holds for arbitrary integers $r$ in the interval $|r|\le 2\sqrt{x}$, with a uniform error term. We demonstrate consistency of our refinement with the Chebotarev theorem for a fixed division field, and with the Sato-Tate conjecture. We also present numerical evidence for the refined conjecture.

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arXiv2008

On the Low-lying zeros of Hasse-Weil L-functions for Elliptic Curves

In this paper, we obtain an unconditional density theorem concerning the low-lying zeros of Hasse-Weil L-functions for a family of elliptic curves. From this together with the Riemann hypothesis for these L-functions, we infer the majorant of 27/14 (which is strictly less than 2) for the average rank of the elliptic curves in the family under consideration. This upper bound for the average rank enables us to deduce that, under the same assumption, a positive proportion of elliptic curves have algebraic ranks equaling their analytic ranks and finite Tate-Shafarevic group. Statements of this flavor were known previously under the additional assumptions of GRH for Dirichlet L-functions and symmetric square L-functions which are removed in the present paper.

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arXiv2008

On Primes in Quadratic Progressions

We verify the Hardy-Littlewood conjecture on primes in quadratic progressions on average. The results in the present paper significantly improve those of a previous paper of the authors(arXiv:math.NT/0605563).

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arXiv2008

On Primes Represented by Quadratic Polynomials

This is a survey article on the Hardy-Littlewood conjecture about primes in quadratic progressions. We recount the history and quote some results approximating this hitherto unresolved conjecture.

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arXiv2006

Bombieri-Vinogradov Type Theorem for Sparse Sets of Moduli

In this paper, we establish theorems of Bombieri-Vinogradov type and Barban-Davenport-Halberstam type for sparse sets of moduli. As an application, we prove that there exist infinitely many primes of the form $p=am^2+1$ such that $a\leq p^{5/9+ε}$.

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arXiv2006

The Lang-Trotter Conjecture on Average

For an elliptic curve $E$ over $\ratq$ and an integer $r$ let $π_E^r(x)$ be the number of primes $p\le x$ of good reduction such that the trace of the Frobenius morphism of $E/\fie_p$ equals $r$. We consider the quantity $π_E^r(x)$ on average over certain sets of elliptic curves. More in particular, we establish the following: If $A,B>x^{1/2+ε}$ and $AB>x^{3/2+ε}$, then the arithmetic mean of $π_E^r(x)$ over all elliptic curves $E$ : $y^2=x^3+ax+b$ with $a,b\in \intz$, $|a|\le A$ and $|b|\le B$ is $\sim C_r\sqrt{x}/\log x$, where $C_r$ is some constant depending on $r$. This improves a result of C. David and F. Pappalardi. Moreover, we establish an ``almost-all'' result on $π_E^r(x)$.

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arXiv2005

On the $p^λ$ problem

We deal with the distribution of the fractional parts of $p^λ$, $p$ running over the prime numbers and $λ$ being a fixed real number lying in the interval $(0,1)$. Roughly speaking, we study the following question: Given a real $θ$, how small may $δ>0$ be choosen if we suppose that the number of primes $p\le N$ satisfying ${p^λ-θ<δ}$ is close to the expected one? We improve some results of Balog and Harman on this question for $λ<5/66$ if $θ$ is rational and for $λ<1/5$ if $θ$ is irrational. Our improvement is based on incorporating the zero detection argument into Harman's method and on using new mean value estimates for products of shifted and ordinary (unshifted) Dirichlet polynomials.

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arXiv2005

Addendum to "On the $p^λ$ problem"

We prove that the conditions $λ<5/19$ and $L\le T^{1/2}$ in Theorems 3 and 4 of our recent paper "On the $p^λ$ problem" can be omitted.

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arXiv2005

On the large sieve with sparse sets of moduli

Extending a method of D. Wolke, we establish a general result on the large sieve with sparse sets S of moduli which are in a sense well-distributed in arithmetic progressions. We then use this result together with Fourier techniques to obtain large sieve bounds for the case when S consists of squares. These bounds improve a recent result by L. Zhao.

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arXiv2005

The large sieve with sparse sets of moduli

Extending a method of D. Wolke, we establish a general result on the large sieve with sparse sets S of moduli which are in a sense well-distributed in arithmetic progressions. We then apply our result to the case when S consists of sqares. In this case we obtain an estimate which improves a recent result by L. Zhao.

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