Proving the Solution of the Prime Zeta Function (P_x 1)

The prime zeta function, denoted as (P_x), is a fascinating mathematical concept deeply intertwined with number theory. It is defined as the sum of the reciprocals of the powers of prime numbers, specifically:

Define the Prime Zeta Function

The prime zeta function (P_x) is given by:

[ P_x sum_{p in text{primes}} frac{1}{p^x} ]

Where (p) represents prime numbers, and (x) is the exponent of these primes. This function converges for (text{Re}(s) > 1).

Approximating with Finite Sums

To verify the solution numerically, we can approximate the sum using a finite range of prime numbers. Let's use Mathematica to compute the partial sum of the prime zeta function for a large but finite upper bound of summation.

Using Mathematica to Compute Numerical Values

Let us denote:

[ S_q^x sum_{n1}^q frac{1}{p_n^x} ]

Where (p_n) represents the (n^{text{th}}) prime number. For (q 5,000,000) and (x 1.4), we can use Mathematica to compute the partial sum:

[ S_{5000000}^{1.4} approx 0.998938 ]

We can extend this process to yield:

[ S_{16,000,000}^{1.4} approx 0.999489 ]

and

[ S_{17,000,000}^{1.399432} approx 0.999956 ]

Verifying with Wolfram Alpha

Using Wolfram Alpha, it has been verified that the value of the infinite sum of the prime zeta function (P_x) is closest to 1 for (x) in the range of 1.399433 to 1.399431. This range is crucial for the numerical approximation to converge towards 1.

Computation and Solution Verification

The problem can be expressed as solving the equation:

[ P_x 1 ; text{for} ; x 1.399433 ]

To find the precise value of (x) that satisfies this equation, we use Mathematica’s built-in symbol PrimeZetaP. By using this function and solving the equation numerically, we obtain:

[ x approx 1.399433328726330318202807214745644327905 ]

This solution indicates that the prime zeta function (P_x) converges to 1 at this specific value of (x).