Numerical computation of the Riemann zeta function and prime counting function by using Gauss–Hermite and Gauss–Laguerre quadratures

Abstract

In this paper, we transform ζ(s) to appropriate integral forms and for numerical computing of these integrals, we introduce a method based on Gauss–Hermite and Gauss–Laguerre quadratures. By using the zeta function, we compute the prime counting function π(x) numerically. Some relations are new and three examples are given to show the good accuracy of the method.

Keywords:

• Gauss–Laguerre quadrature
• Gauss–Hermite quadrature
• Riemann zeta function
• prime counting function
• prime numbers

2000 AMS Subject Classifications :

• 65D30
• 65D32
• 65D20
• 11M06