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4 results

  • 4 - L-functions from number theory

  • Bruno Kahn
  • Book:
    Zeta and L-Functions of Varieties and Motives
    Published online:
    28 April 2020
    Print publication:
    07 May 2020, pp 73-103

  • Summary

    This chapter returns to more elementary mathematics, introducing Dirichlet, Hecke, and Artin L-functions. A proof of Dirichlet’s theorem on arithmetic progressions is given, by the method expounded by Serre; it would however be a shame to omit Dirichlet’s original method, which gave additional information and anticipated the analytic class number formulae. The two main generalisations of Dirichlet’s L-functions are then introduced: those of Hecke and Artin. Hecke’s main theorem is stated without proof: existence of an analytic continuation and a functional equation, and it is then explained how Artin and Brauer derived the same results for non-abelian L-functions.

  • The sets of Dirichlet non-improvable numbers versus well-approximable numbers

  • Part of
  • AYREENA BAKHTAWAR, PHILIP BOS, MUMTAZ HUSSAIN
  • Journal:
    Ergodic Theory and Dynamical Systems / Volume 40 / Issue 12 / December 2020
    Published online by Cambridge University Press:
    27 June 2019, pp. 3217-3235
    Print publication:
    December 2020

  • Let $\unicode[STIX]{x1D6F9}:[1,\infty )\rightarrow \mathbb{R}_{+}$ be a non-decreasing function, $a_{n}(x)$ the $n$th partial quotient of $x$ and $q_{n}(x)$ the denominator of the $n$th convergent. The set of $\unicode[STIX]{x1D6F9}$-Dirichlet non-improvable numbers,

    $$\begin{eqnarray}G(\unicode[STIX]{x1D6F9}):=\{x\in [0,1):a_{n}(x)a_{n+1}(x)>\unicode[STIX]{x1D6F9}(q_{n}(x))\text{ for infinitely many }n\in \mathbb{N}\},\end{eqnarray}$$
    is related with the classical set of $1/q^{2}\unicode[STIX]{x1D6F9}(q)$-approximable numbers ${\mathcal{K}}(\unicode[STIX]{x1D6F9})$ in the sense that ${\mathcal{K}}(3\unicode[STIX]{x1D6F9})\subset G(\unicode[STIX]{x1D6F9})$. Both of these sets enjoy the same $s$-dimensional Hausdorff measure criterion for $s\in (0,1)$. We prove that the set $G(\unicode[STIX]{x1D6F9})\setminus {\mathcal{K}}(3\unicode[STIX]{x1D6F9})$ is uncountable by proving that its Hausdorff dimension is the same as that for the sets ${\mathcal{K}}(\unicode[STIX]{x1D6F9})$ and $G(\unicode[STIX]{x1D6F9})$. This gives an affirmative answer to a question raised by Hussain et al [Hausdorff measure of sets of Dirichlet non-improvable numbers. Mathematika 64(2) (2018), 502–518].

  • An inhomogeneous Dirichlet theorem via shrinking targets

  • Part of
  • Dmitry Kleinbock, Nick Wadleigh
  • Journal:
    Compositio Mathematica / Volume 155 / Issue 7 / July 2019
    Published online by Cambridge University Press:
    25 June 2019, pp. 1402-1423
    Print publication:
    July 2019
    • Article
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  • We give an integrability criterion on a real-valued non-increasing function $\unicode[STIX]{x1D713}$ guaranteeing that for almost all (or almost no) pairs $(A,\mathbf{b})$, where $A$ is a real $m\times n$ matrix and $\mathbf{b}\in \mathbb{R}^{m}$, the system

    $$\begin{eqnarray}\Vert A\mathbf{q}+\mathbf{b}-\mathbf{p}\Vert ^{m}<\unicode[STIX]{x1D713}(T),\quad \Vert \mathbf{q}\Vert ^{n}<T,\end{eqnarray}$$
    is solvable in $\mathbf{p}\in \mathbb{Z}^{m}$, $\mathbf{q}\in \mathbb{Z}^{n}$ for all sufficiently large $T$. The proof consists of a reduction to a shrinking target problem on the space of grids in $\mathbb{R}^{m+n}$. We also comment on the homogeneous counterpart to this problem, whose $m=n=1$ case was recently solved, but whose general case remains open.