A190124 Decimal expansion of Ramanujan prime constant: Sum_{n>=1} (1/R_n)^2, where R_n is the n-th Ramanujan prime, A104272(n).

2, 6, 5, 5, 6, 3, 2, 7, 5, 8, 0

Offset: 0

Author

John W. Nicholson, May 04 2011

Keywords

• nonn
• cons
• more

Comments

By computing all Ramanujan primes less than 10^9, we find that about 9 decimal places of the sum should be correct: 0.265563275 (truncated, not rounded). The following table shows the number of Ramanujan primes between powers of 10 and the sum of the squared reciprocals of those primes.

1 1 0.25000000000000000

2 9 0.01477600368240514

3 62 0.00072814919125266

4 487 0.00005457480850461

5 3900 0.00000417097012694

6 32501 0.00000034491619098

7 279106 0.00000002943077197

8 2444255 0.00000000255829675

9 21731345 0.00000000022619762

Total: 0.26556327578374667 - T. D. Noe, May 05 2011

From Jonathan Sondow, May 06 2011: (Start)

Since R_n > n, the bound Sum_{n > N} 1/(R_n)^2 < 1/N holds, by the integral test. Taking N = #{R_n < 10^9} = 24491666, the error is < 4.09 x 10^-8.

Using the stronger inequality R_n > 2n log 2n (from "Ramanujan primes and Bertrand's postulate"), the error is actually < 2.94 * 10^-11. So the sum 0.265563275... is correct. The next digit is either 7 or 8. (End)

A190124 and A085548 (Prime Zeta(2)) converge by comparison with A013661 (Zeta(2)), which converges by the integral test. As real numbers, A190124 < A085548 < A013661. - Robert G. Wilson v, May 08 2011

Prime Zeta(2) - (this constant) = 0.4522474200 - 0.2655632757 = 0.186684144 (truncated, not rounded). - John W. Nicholson, May 24 2011

From Dana Jacobsen, Jul 27 2015: (Start)

Calculating more Ramanujan primes, we can expand on the earlier table, which should give us more terms.

1 1 0.25000000000000000000 0.25000000000000000000

2 9 0.26477600368240513652 0.01477600368240513652

3 62 0.26550415287365779725 0.00072814919125266073

4 487 0.26555872768216240627 0.00005457480850460902

5 3900 0.26556289865228934691 0.00000417097012694064

6 32501 0.26556324356848032844 0.00000034491619098153

7 279106 0.26556327299925229431 0.00000002943077196587

8 2444255 0.26556327555754904279 0.00000000255829674847

9 21731345 0.26556327578374665897 0.00000000022619761618

10 195606622 0.26556327580402332096 0.00000000002027666198

11 1778301947 0.26556327580586060071 0.00000000000183727975

12 16301375641 0.26556327580602856045 0.00000000000016795974. (End)

Examples

			0.265563275...
		

Links

• J. Sondow, Ramanujan primes and Bertrand's postulate, Amer. Math. Monthly 116 (2009), 630-635.
• J. Sondow, J. W. Nicholson, and T. D. Noe, Ramanujan Primes: Bounds, Runs, Twins, and Gaps, arXiv:1105.2249 [math.NT], 2011.

Crossrefs

Cf. A078437, A085548, A104272.

Programs

Perl

use ntheory ":all";
use Math::MPFR qw/Rmpfr_get_str Rmpfr_set_default_prec Rmpfr_printf/;
Rmpfr_set_default_prec(500);
my $limit = shift || 9;
my($maxexp, $sum) = (9, Math::MPFR->new(0));
for my $e (1..$limit) {
  my($numrp, $psum) = (0, Math::MPFR->new(0));
  if ($e <= $maxexp) {
    my $rp = ramanujan_primes(10**($e-1),10**$e);
    $numrp += scalar @$rp;
    $psum += (1/Math::MPFR->new("$_"))**2 for @$rp;
  } else {
    for my $k (10**($e-$maxexp-1) .. 10**($e-$maxexp)-1) {
      my $rp = ramanujan_primes($k*10**$maxexp,($k+1)*10**$maxexp);
      $numrp += scalar @$rp;
      $psum += (1/Math::MPFR->new("$_"))**2 for @$rp;
    }
  }
  Rmpfr_printf("%2d ", $e);
  Rmpfr_printf("%14lu   ", $numrp);
  Rmpfr_printf("%.20Rf  ", $sum += $psum);
  Rmpfr_printf("%.20Rf\n", $psum);
} # Dana Jacobsen, Jul 27 2015

Extensions

a(10) and a(11) (from data above by Dana Jacobsen_, Jul 27 2015) added by John W. Nicholson, Dec 17 2015