Step by Step solution about How to resolve the algorithm Meissel–Mertens constant step by step in the Mathematica / Wolfram Language programming language

Purpose: The code computes the Mertens function (M(x)) for (x=100,000,000), which is defined as the sum of the logarithmic integral (\text{li}(x)) over all primes (p \le x).

Implementation:

1. The code first generates a list of prime numbers up to (100,000,000) using the Select and PrimeQ functions.
2. It then calculates the Mertens function using the Total and Log functions.
3. The EulerGamma constant is added to the result. The precision of the calculation is set to 10 decimal places using the N function.
4. The code also calculates an analytic approximation of the Mertens function using the EulerGamma constant and a sum over the Mobius mu function and the logarithmic derivative of the Riemann zeta function.
5. The precision of the analytic approximation is set to 305 decimal places for the first calculation and to 1001 decimal places for the second calculation using the N function.

Output: The output of the code is the value of the Mertens function (M(100,000,000)) and the analytic approximations of the Mertens function.

Notes:

1. The calculation of the Mertens function for large values of (x) can be computationally expensive.
2. The analytic approximation of the Mertens function is more efficient to compute and can be used to obtain accurate approximations of the Mertens function for large values of (x).

PrimeNumbers = Select[Range[100000000], PrimeQ[#] &]; (*all primes in the first 100 000 000 numbers, this takes a toll on my computer's CPU and RAM*)
MM = N[Total[Log[1 - 1/PrimeNumbers] + 1/PrimeNumbers] + EulerGamma, 10] (*Calculating it up to a precision of 10, this is correct up to 8 digits*)
AnalyticMMto305 = N[EulerGamma + Sum[MoebiusMu[n]/n Log[Zeta[n]], {n, 2, 1000}], 1000] (*Precise up to 305 digits*)
AnalyticMM = N[EulerGamma + Sum[MoebiusMu[n]/n Log[Zeta[n]], {n, 2, 10000}], 1001] (*Precise up to at least 1000 digits*)