TY  - JOUR
ID  - Maślanka2004
LB  - Maślanka2004
AU  - Maślanka, Krzysztof
TI  - Li's criterion for the Riemann hypothesis - numerical approach
ST  - Li's criterion for the Riemann hypothesis - numerical approach
JO  - Opuscula Math.
JA  - Opuscula Math.
JF  - Opuscula Mathematica
PY  - 2004
VL  - 24
IS  - 1
SP  - 103
EP  - 114
DO  - http://dx.doi.org/10.7494/OpMath.2004.24.1.103
UR  - http://dx.doi.org/10.7494/OpMath.2004.24.1.103
AB  - There has been some interest in a criterion for the Riemann hypothesis proved recently by Xian-Jin Li [Li X.-J.: The Positivity of a Sequence of Numbers and the Riemann Hypothesis. J. Number Theory 65 (1997), 325-333]. The present paper reports on a numerical computation of the first 3300 of Li's coefficients which appear in this criterion. The main empirical observation is that these coefficients can be separated in two parts. One of these grows smoothly while the other is very small and oscillatory. This apparent smallness is quite unexpected. If it persisted till infinity then the Riemann hypothesis would be true.
KW  - Riemann zeta function
KW  - Riemann hypothesis
KW  - Li's criterion
KW  - numerical methods in analytic number theory
SN  - 1232-9274
ER  -