A282468 Decimal expansion of the zeta function at 2 of every second prime number.

1, 4, 4, 7, 1, 5, 5, 8, 6, 6, 8, 8, 7

Offset: 0

Author

Terry D. Grant, Apr 14 2017

Keywords

• nonn
• cons
• hard
• more

Comments

From Husnain Raza, Aug 30 2023: (Start)

Note that since p_n > n*log(n), we can place a bound on the tail of the sum:

Sum_{n >= N} (prime(2n))^(-2) <= Sum_{n >= N} (2*n*log(2n))^(-2) <= Integral_{x=N..oo} (2*x*log(2x))^(-2) dx.

Taking the sum over all primes < 10^12, we see that the constant lies between 0.14471558668870 and 0.14471558668873. (End)

Examples

			1/3^2 + 1/7^2 + 1/13^2 + 1/19^2 + 1/29^2 + ... = 0.14471558...
		

Links

• Husnain Raza, PARI program for calculating more digits.

Crossrefs

Cf. A031215, A085548.

Zeta functions at 2: A085548 (for primes), A275647 (for nonprimes), A013661 (for natural numbers), A117543 (for semiprimes), A131653 (for triprimes), A222171 (for even numbers), A111003 (for odd numbers).

Programs

PARI

sum(n=1, 2500000, 1./prime(2*n)^2)

PARI

\\ see Raza link

Formula

Equals Sum_{n>=1} 1/A031215(n)^2 = Sum_{n>=1} 1/prime(2n)^2.

Extensions

a(8)-a(12) from Husnain Raza, Aug 31 2023