A physical method is developed to analyze the Riemann zeta function and the L-function.
First, a physical model of an elastic bar fixed at both ends and subjected to symmetric axial
loads with respect to the mid-span is constructed. The axial force of the bar at the left end is
calculated with two methods: the equilibrium of forces and the principle of minimum
potential energy. With the help of the orthogonality of {sinix} (i 1, 2, 3, ) , the Parseval
identity and the Bessel inequality of the physical model are obtained. Further, it is proven that,
in all the possible displacements which satisfy the boundary conditions, the real one
minimizes the total potential energy of the bar. With the equivalence of the two methods, an
identity of a type of infinite series is derived. Based on the identity, L(1) , L(3) , a recurrence
formula of L(2k 1) (k 3) , (2) , (4) , and a relationship between (2k) (k 2) and
L(2i 1) (i 1, 2, ..., k) are then deduced with power axial load functions. The upper and
lower bounds for the Riemann zeta function (2k -1) and the L-function L(2k) are given.
The numerical results show that the upper and lower bounds are in excellent agreement with
the accurate value. Finally, two improvement methods are proposed for estimating the
circumference ratio. The numerical results show that the two methods can estimate the
circumference ratio with high accuracy and that the L -function is more efficient than the
Riemann zeta function in estimating the circumference ratio.