Lal's Constant

Let  denote the number of primes of the form  for , then

(1)

where  is the logarithmic integral (Shanks 1960, pp. 321-332). Let  denote the number of primes of the form  for , then

(2)

(Shanks 1961, 1962). Let  denote the number of pairs of primes  and  for , then

(3)

where

(4)

(Shanks 1960, pp. 201-203). Finally, let  denote the number of pairs of primes  and  for , then

(5)

(Lal 1967), where  is called Lal's constant. Shanks (1967) showed that .

REFERENCES:

Lal, M. "Primes of the Form ." Math. Comput. 21, 245-247, 1967.

Shanks, D. "On the Conjecture of Hardy and Littlewood Concerning the Number of Primes of the Form ." Math. Comput. 14, 321-332, 1960.

Shanks, D. "On Numbers of the Form ." Math. Comput. 15, 186-189, 1961.

Shanks, D. Corrigendum to "On the Conjecture of Hardy and Littlewood Concerning the Number of Primes of the Form ." Math. Comput. 16, 513, 1962.

Shanks, D. "Lal's Constant and Generalization." Math. Comput. 21, 705-707, 1967.