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2 results

  • TRANSCENDENTAL SUMS RELATED TO THE ZEROS OF ZETA FUNCTIONS

  • Part of
  • Sanoli Gun, M. Ram Murty, Purusottam Rath
  • Journal:
    Mathematika / Volume 64 / Issue 3 / 2018
    Published online by Cambridge University Press:
    01 August 2018, pp. 875-897
    Print publication:
    2018

  • While the distribution of the non-trivial zeros of the Riemann zeta function constitutes a central theme in Mathematics, nothing is known about the algebraic nature of these non-trivial zeros. In this article, we study the transcendental nature of sums of the form

    $$\begin{eqnarray}\mathop{\sum }_{\unicode[STIX]{x1D70C}}R(\unicode[STIX]{x1D70C})x^{\unicode[STIX]{x1D70C}},\end{eqnarray}$$
    where the sum is over the non-trivial zeros $\unicode[STIX]{x1D70C}$ of $\unicode[STIX]{x1D701}(s)$, $R(x)\in \overline{\mathbb{Q}}(x)$ is a rational function over algebraic numbers and $x>0$ is a real algebraic number. In particular, we show that the function
    $$\begin{eqnarray}f(x)=\mathop{\sum }_{\unicode[STIX]{x1D70C}}\frac{x^{\unicode[STIX]{x1D70C}}}{\unicode[STIX]{x1D70C}}\end{eqnarray}$$
     has infinitely many zeros in $(1,\infty )$, at most one of which is algebraic. The transcendence tools required for studying $f(x)$ in the range $x<1$ seem to be different from those in the range $x>1$. For $x<1$, we have the following non-vanishing theorem: If for an integer $d\geqslant 1$, $f(\unicode[STIX]{x1D70B}\sqrt{d}x)$ has a rational zero in$(0,1/\unicode[STIX]{x1D70B}\sqrt{d})$, then
    $$\begin{eqnarray}L^{\prime }(1,\unicode[STIX]{x1D712}_{-d})\neq 0,\end{eqnarray}$$
     where $\unicode[STIX]{x1D712}_{-d}$ is the quadratic character associated with the imaginary quadratic field $K:=\mathbb{Q}(\sqrt{-d})$. Finally, we consider analogous questions for elements in the Selberg class. Our proofs rest on results from analytic as well as transcendental number theory.

  • Classical and modular approaches to exponential Diophantine equations II. The Lebesgue–Nagell equation

  • Yann Bugeaud, Maurice Mignotte, Samir Siksek
  • Journal:
    Compositio Mathematica / Volume 142 / Issue 1 / January 2006
    Published online by Cambridge University Press:
    13 January 2006, pp. 31-62
    Print publication:
    January 2006
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  • This is the second in a series of papers where we combine the classical approach to exponential Diophantine equations (linear forms in logarithms, Thue equations, etc.) with a modular approach based on some of the ideas of the proof of Fermat's Last Theorem. In this paper we use a general and powerful new lower bound for linear forms in three logarithms, together with a combination of classical, elementary and substantially improved modular methods to solve completely the Lebesgue–Nagell equation x2 + D = yn, x, y integers, $n\geq 3$, for D in the range $1 \leq D \leq 100$.