Prime Sums

Let

(1)

be the sum of the first  primes (i.e., the sum analog of the primorial function). The first few terms are 2, 5, 10, 17, 28, 41, 58, 77, ... (OEIS A007504). Bach and Shallit (1996) show that

(2)

and provide a general technique for estimating such sums.

The first few values of  such that  is prime are 1, 2, 4, 6, 12, 14, 60, 64, 96, 100, ... (OEIS A013916). The corresponding values of  are 2, 5, 17, 41, 197, 281, 7699, 8893, 22039, 24133, ... (OEIS A013918).

The first few values of  such that  are 1, 23, 53, 853, 11869, 117267, 339615, 3600489, 96643287, ... (OEIS A045345). The corresponding values of  are 2, 874, 5830, 2615298, 712377380, 86810649294, 794712005370, 105784534314378, 92542301212047102, ... (OEIS A050247; Rivera), and the values of  are 2, 38, 110, 3066, 60020, 740282, 2340038, 29380602, 957565746, ... (OEIS A050248; Rivera).

In 1737, Euler showed that the harmonic series of primes, (i.e., sum of the reciprocals of the primes) diverges

(3)

(Nagell 1951, p. 59; Hardy and Wright 1979, pp. 17 and 22), although it does so very slowly.

A rapidly converging series for the Mertens constant

(4)

is given by

(5)

where  is the Euler-Mascheroni constant,  is the Riemann zeta function, and  is the Möbius function (Flajolet and Vardi 1996, Schroeder 1997, Knuth 1998).

Dirichlet showed the even stronger result that

(6)

(Davenport 1980, p. 34). Despite the divergence of the sum of reciprocal primes, the alternating series

(7)

(OEIS A078437) converges (Robinson and Potter 1971), but it is not known if the sum

(8)

does (Guy 1994, p. 203; Erdős 1998; Finch 2003).

There are also classes of sums of reciprocal primes with sign determined by congruences on , for example

(9)

(OEIS A086239), where

(10)

(Glaisher 1891b; Finch 2003; Jameson 2003, p. 177),

(11)

(OEIS A086240; Glaisher 1893, Finch 2003), and

(12)

(OEIS A086241), where

(13)

(Glaisher 1891c; Finch 2003; Jameson 2003, p. 177).

Although  diverges, Brun (1919) showed that

(14)

where

(15)

(OEIS A065421) is Brun's constant.

The function defined by

(16)

taken over the primes converges for  and is a generalization of the Riemann zeta function known as the prime zeta function.

Consider the positive integers  with prime factorizations

(17)

such that there are an odd number of (not necessarily distinct) prime factors, i.e.,  is odd. The first few such numbers are 2, 3, 5, 7, 8, 11, 12, 13, 17, 18, 19, 20, 23, 27, 28, 29, ... (OEIS A026424). Then

			(18)
			(19)
			(20)

(Gourdon and Sebah), where  is the Riemann zeta function. The first few terms are then

			(21)
			(22)
			(23)
			(24)

(OEIS A093597 and A093598).

Consider the analogous sum where, in addition, the terms included must have an odd number of distinct prime factors, i.e.,  is odd and . The first few such numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, ... (OEIS A030059), which include the composite numbers 30, 42, 66, 70, 78, 102, ... (OEIS A093599). Then

			(25)
			(26)
			(27)

(Gourdon and Sebah). The first few terms are then

			(28)
			(29)
			(30)
			(31)

(OEIS A093595 and A093596).

The sum

			(32)
			(33)
			(34)

(OEIS A086242) is also finite (Glaisher 1891a; Cohen; Finch 2003), where

(35)

 is the totient function, and  is the Riemann zeta function.

Some curious sums satisfied by primes  include

(36)

giving the sequence 0, 2, 18, 60, 270, 462, 1080, ... (OEIS A078837; Doster 1993) for , 3, 5, ..., and

(37)

giving the sequence 0, 2, 30, 120, 630, 1122, 2760, ... (OEIS A078838; Doster 1993),

			(38)
			(39)

where  is the Mangoldt function, and

(40)

(Berndt 1994, p. 114).

Let  be the number of ways an integer  can be written as a sum of two or more consecutive primes. For example, , so  and , so . The sequence of values of  for , 2, ... is given by 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, ... (OEIS A084143). The following table gives the first few  such that  for small .

	OEIS	values of  such that
1	A050936	5, 8, 10, 12, 15, 17, 18, 23, 24, 26, 28, 30, 31, 36, ...
2	A067372	36, 41, 60, 72, 83, 90, 100, 112, 119, ...
3	A067373	240, 287, 311, 340, 371, 510, 660, 803, ...

Similarly, the following table gives the first few  such that  for small .

	OEIS	values of  such that
1	A084146	5, 8, 10, 12, 15, 17, 18, 23, 24, 26, 28, 30, 31, 39, ...
2	A084147	36, 41, 60, 72, 83, 90, 100, 112, 119, 120, 138, ...

Now consider instead the number  of ways in which a number  can be represented as a sum of one or more consecutive primes (i.e., the same sequence as before but one larger for each prime number). Amazingly, it then turns out that

(41)

(Moser 1963; Le Lionnais 1983, p. 30).

REFERENCES:

Bach, E. and Shallit, J. §2.7 in Algorithmic Number Theory, Vol. 1: Efficient Algorithms. Cambridge, MA: MIT Press, 1996.

Berndt, B. C. "Ramanujan's Theory of Prime Numbers." Ch. 24 in Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994.

Brun, V. "La serie , les dénominateurs sont nombres premiers jumeaux est convergente où finie." Bull. Sci. Math. 43, 124-128, 1919.

Cohen, H. "High Precision Computation of Hardy-Littlewood Constants." Preprint. https://www.math.u-bordeaux.fr/~cohen/hardylw.dvi.

Davenport, H. Multiplicative Number Theory, 2nd ed. New York: Springer-Verlag, 1980.

Doster, D. "Problem 10346." Amer. Math. Monthly 100, 951, 1993.

Erdős, P. "Some of My New and Almost New Problems and Results in Combinatorial Number Theory." In Number Theory: Diophantine, Computational and Algebraic Aspects. Proceedings of the International Conference Held in Eger, July 29-August 2, 1996 (Ed. K. Győry, A. Pethő and V. T. Sós). Berlin: de Gruyter, pp. 169-180, 1998.

Finch, S. R. "Meissel-Mertens Constants." §2.2 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 94-98, 2003.

Finch, S. "Two Asymptotic Series." December 10, 2003. https://algo.inria.fr/bsolve/.

Flajolet, P. and Vardi, I. "Zeta Function Expansions of Classical Constants." Unpublished manuscript. 1996. https://algo.inria.fr/flajolet/Publications/landau.ps.

Glaisher, J. W. L. "On the Sums of the Inverse Powers of the Prime Numbers." Quart. J. Pure Appl. Math. 25, 347-362, 1891a.

Glaisher, J. W. L. "On the Series ." Quart. J. Pure Appl. Math. 25, 375-383, 1891b.

Glaisher, J. W. L. "On the Series ." Quart. J. Pure Appl. Math. 25, 48-65, 1891c.

Glaisher, J. W. L. "On the Series ." Quart. J. Pure Appl. Math. 26, 33-47, 1893.

Gourdon, X. and Sebah, P. "Collection of Series for ." https://numbers.computation.free.fr/Constants/Pi/piSeries.html.

Guy, R. K. "A Series and a Sequence Involving Primes." §E7 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 203, 1994.

Hardy, G. H. and Wright, E. M. "Prime Numbers" and "The Sequence of Primes." §1.2 and 1.4 in An Introduction to the Theory of Numbers, 5th ed. Oxford, England: Clarendon Press, pp. 1-4, 17, 22, and 251, 1979.

Jameson, G. J. O. The Prime Number Theorem. Cambridge, England: Cambridge University Press, p. 177, 2003.

Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1998.

Le Lionnais, F. Les nombres remarquables. Paris: Hermann, pp. 26, 30, and 46, 1983.

Moree, P. "Approximation of Singular Series and Automata." Manuscripta Math. 101, 385-399, 2000.

Moser, L. "Notes on Number Theory III. On the Sum of Consecutive Primes." Can. Math. Bull. 6, 159-161, 1963.

Nagell, T. Introduction to Number Theory. New York: Wiley, 1951.

Ramanujan, S. "Irregular Numbers." J. Indian Math. Soc. 5, 105-106, 1913. Ramanujan, S. Collected Papers of Srinivasa Ramanujan (Ed. G. H. Hardy, P. V. S. Aiyar, and B. M. Wilson). Providence, RI: Amer. Math. Soc., pp. 20-21, 2000.

Rivera, C. "Problems & Puzzles: Puzzle 031-The Average Prime Number, ." https://www.primepuzzles.net/puzzles/puzz_031.htm.

Robinson, H. P. and Potter, E. Mathematical Constants. Report UCRL-20418. Berkeley, CA: University of California, 1971.

Schroeder, M. R. Number Theory in Science and Communication, with Applications in Cryptography, Physics, Digital Information, Computing, and Self-Similarity, 3rd ed. New York: Springer-Verlag, 1997.

Sloane, N. J. A. Sequences A007504/M1370, A013916, A013918, A030059, A045345, A046024, A050247, A050248, A050936, A065421, A067372, A067373, A078437, A078837, A078838, A084143, A084146, A084147, A086239, A086240, A086241, A086242, A093595, A093596, A093597, A093598, and A093599 in "The On-Line Encyclopedia of Integer Sequences."