Summation involving exponents of Mersenne primes

Question

We all know the Leibniz formula for $\pi$ : $\dfrac{\pi}{4}=\displaystyle\sum_{k=0}^{\infty}\dfrac{(-1)^k}{2k+1}$ .

Now observe the following summation:

$$s=1+\displaystyle\sum_{p \in \mathbb{M}}\dfrac{(-1)^{(p-1)/2}}{p}$$

where $\mathbb{M}=\{p | p \in \mathbb{P} ,p \text{ is odd and } 2^p-1 \in \mathbb{P}\}$.

For the first $51$ values of odd exponents of Mersenne primes, I am getting the following value $s=0.78530826\ldots$ , while the real value of $\dfrac{\pi}{4}$ is $0.78539816\ldots$

PARI/GP code.

Is this just a coincidence? Can someone shed more light on this? Note that if $s$ were exactly equal to $\dfrac{\pi}{4}$ that would imply the infinitude of Mersenne primes.

Are you aware that the expression $(-1)^{\operatorname{mod}((p-1)/2,2)}$ is the same as $(-1)^{(p-1)/2}$? Is there any reason to write it in a convoluted way? — jjagmath, Jan 20 '25 at 23:13

score 5 · Accepted Answer · answered Jan 16 '25 at 16:33

I don't know any heuristic reason to expect $s$ to be $\pi/4$. While the expected growth rate of Mersenne primes implies $s$ converges, I can't think of an unconditional proof that $s$ converges.

Currently the 52-nd known Mersenne prime (= 51st one with an odd exponent) is 136,279,841, but the full range up to there has not been double checked to be sure no smaller Mersenne primes have been missed: see https://t5k.org/mersenne/#known and the explanation below the table of the question marks in the last few rows.

Since the terms in the series are $\pm 1/p$ and 1/136,279,841 is around 0.00000000733, which is far smaller than the difference between your "estimate" for $s$ and $\pi/4$, my hunch is that what you found is a coincidence.

Summation involving exponents of Mersenne primes

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