# Pi from String Theory Amplitudes

From the paper by Arnab Priya Saha and Aninda Sinha. A new series for $\pi$ and its convergence.

$$\pi = 4 + \sum_{n=1}^{\infty} \frac{1}{n!} \left( \frac{1}{n + \lambda} - \frac{4}{2n + 1} \right) \left( \frac{(2n + 1)^2}{4(n + \lambda)} - n \right)_{n-1}$$

Where $(a)_{n-1}$ is the Pochhammer symbol.

$$(a)_{n-1} = \frac{\Gamma(a + n - 1)}{\Gamma(a)}$$

$\lambda$ is a free parameter for optimizing convergence. As $\lambda$ approaches infinity, the series reduces to the classic Madhava series. With $\lambda$ between 10 and 100, the series takes 30 terms to converge to ten decimal places. Which is far more efficient than the classic Madhava series, which takes 5 billion terms to converge to ten decimal places.

This can be implemented in Python as follows:

import mpmath as mp

def mp_calculate_pi(lambda_val, iters):
mp.dps = 100
pi = mp.mpf(4)
n = 1
prev_pi = 0
factorial = mp.mpf(n)

for i in range(iters):
term1 = 1.0 / factorial
term2 = 1.0 / (n + lambda_val) - 4 / (2 * n + 1)
term3 = ((2 * n + 1) ** 2) / (4 * (n + lambda_val)) - n

pochhammer = 1
for i in range(n - 1):
pochhammer *= term3 + i

prev_pi = pi
pi += term1 * term2 * pochhammer

# print to 30 decimal places using mpmath
approx = mp.nstr(pi, 100)

# Compare to mp.pi to calcualte the delta
delta = abs(pi - mp.pi)

# numbers of digits that pi approximate is correct to
accuracy = round(abs(mp.log(abs(pi - mp.pi), 10)))

# Print a line with the current iteration, pi, delta
print(f'{n} {approx} {delta} {accuracy}')

n += 1
factorial = factorial * n

return pi, n - 1

lambda_val = 42
pi, n = mp_calculate_pi(lambda_val, 30)