Geodesic
Extension of the Riemann ξ-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip
Canadian mathematical bulletin, Tome 52 (2009) no. 2, pp. 186-194

Voir la notice de l'article provenant de la source Cambridge University Press

If $K$ is a number field with ${{n}_{k}}\,=\,\left[ k\,:\,\mathbb{Q} \right]$ , and ${{\xi }_{k}}$ the symmetrized Dedekind zeta function of the field, the inequality $$\Re \frac{\xi _{k}^{'}\left( \sigma \,+\,\text{i}t \right)}{{{\xi }_{k}}\left( \sigma \,+\,\text{i}t \right)}\,>\,\frac{\xi _{k}^{'}\left( \sigma\right)}{{{\xi }_{k}}\left( \sigma\right)}$$ for $t\,\ne \,0$ is shown to be true for $\sigma \,\ge \,1\,+\,8/n_{k}^{\frac{1}{3}}$ improving the result of Lagarias where the constant in the inequality was 9. In the case $k\,=\,\mathbb{Q}$ the inequality is extended to $\sigma \,\ge \,1$ for all $t$ sufficiently large or small and to the region $\sigma \,\ge \,1\,+\,1/\left( \log \,t\,-\,5 \right)$ for all $t\,\ne \,0$ . This answers positively a question posed by Lagarias.
DOI : 10.4153/CMB-2009-021-3
Mots-clés : 11M26, 11R42, Riemann zeta function, xi function, zeta zeros 
Broughan, Kevin A. Extension of the Riemann ξ-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip. Canadian mathematical bulletin, Tome 52 (2009) no. 2, pp. 186-194. doi: 10.4153/CMB-2009-021-3
@article{10_4153_CMB_2009_021_3,
     author = {Broughan, Kevin A.},
     title = {Extension of the {Riemann} {\ensuremath{\xi}-Function's} {Logarithmic} {Derivative} {Positivity} {Region} to {Near} the {Critical} {Strip}},
     journal = {Canadian mathematical bulletin},
     pages = {186--194},
     year = {2009},
     volume = {52},
     number = {2},
     doi = {10.4153/CMB-2009-021-3},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2009-021-3/}
}
TY  - JOUR
AU  - Broughan, Kevin A.
TI  - Extension of the Riemann ξ-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip
JO  - Canadian mathematical bulletin
PY  - 2009
SP  - 186
EP  - 194
VL  - 52
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-2009-021-3/
DO  - 10.4153/CMB-2009-021-3
ID  - 10_4153_CMB_2009_021_3
ER  - 
%0 Journal Article
%A Broughan, Kevin A.
%T Extension of the Riemann ξ-Function's Logarithmic Derivative Positivity Region to Near the Critical Strip
%J Canadian mathematical bulletin
%D 2009
%P 186-194
%V 52
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-2009-021-3/
%R 10.4153/CMB-2009-021-3
%F 10_4153_CMB_2009_021_3

[1] [1] Alzer, H., On some inequalities for the gamma and psi functions. Math. Comp. 66(1997), no. 217, 373–389. Google Scholar

[2] [2] Alzer, H., Sharp inequalities for the digamma and polygamma functions. Forum. Math. 16(2004), no. 2, 181–221. Google Scholar

[3] [3] Cheng, Y., An explicit upper bound for the Riemann zeta-function near the line σ = 1 . Rocky Mountain J. Math. 29(1999), no. 1, 115–140. Google Scholar

[4] [4] Delange, H., Une remarque sur la dérivée logarithmique de la fonction zêta de Riemann. Colloq. Math. 53(1987), no. 2, 333–335. Google Scholar

[5] [5] Garunkstis, R., On a positivity property of the Riemann ζ-function. Liet. Mat. Rink 42(2002), no. 2, 179–184. Google Scholar

[6] [6] Hinkkanen, A., On functions of bounded type. Complex Variables Theory Appl. 34(1997), no. 1–2, 119–139. Google Scholar

[7] [7] Ivić, A., The Riemann zeta-function. Theory and Applications. Wiley, New York, 1985. Google Scholar

[8] [8] Lagarias, J. C., On a positivity property of the Riemann ζ-function. Acta Arith. 89(1999), no. 3, 217–234. Google Scholar

[9] [9] Levinson, N. and Montgomery, H. L., Zeros of the derivatives of the Riemann zetafunction. Acta Math. 133(1974), 49–65. Google Scholar

[10] [10] Richert, H.-E., Zur Awchätzung der Riemannschen Zetafunktion in der Nähe der Vertikalen σ = 1 . Math. Ann. 169(1967), 97–101. Google Scholar

[11] [11] Titchmarsh, E. C., The theory of the Riemann Zeta-function. Second edition, Oxford University Press, New York, 1986. Google Scholar

[12] [12] Wedeniwski, S., Results connected with the first 100 billion zeros of the Riemann function. http://www.zetagrid.net/zeta/math/zeta.result.100billion.zeros.html. Google Scholar

Cité par Sources :