From Sums over Natural Numbers to Sums over Primes

Overview

Objectives/Goals Any sum over the naturals such as 1 + 2 + 3 . . . n = F(n) can be converted to a sum over the primes such as 2 + 3 + 5 + . . p_n = P(n) where p_n is the n-th prime. I started by proposing an Expand-Sum-Prune (ESP) heuristic in which P(n) is approximated as F (n ln n)/ ln n. ESP provides correct asymptotic results for sums of prime powers, duplicating a result of Salat-Znam. The goals of this project are: 1. To examine the hypothesis that ESP fails when any one term is too large a fraction of the whole sum. 2. To find new sums over primes never published earlier 3. When ESP fails, to find better summation methods. Methods/Materials 1. Series: I studied several sums over primes including the alternating series (2 - 3 +5 -7 + ...), reciprocal sums (1/(2*3*5)+ 1 (3*5*7) +. . .), and sums of prime powers (2^2 + 3^2 + 5 ^2 + . . .) 2. All estimates were checked for accuracy using a Visual C program that uses the sieve of Eratosthenes to produce (and sum) all primes up to 1000000. Results 1. Alternating sum: Consider A = 2 # 3 + 5. I provide a new estimate of |A| = 0.5 p_n with errors less than 2% for 500<n<78,401 by summing half the prime gaps using a modified ESP method. When my estimate was posed on Math Overflow (viewed 386 times, 11 votes +1 badge for "good question"), mathematicians felt my new result was "almost certainly true". However, using current sieve techniques they can only prove unconditionally that |A|< p_n /64. My method generalizes to alternating series of prime powers. I published a new series for alternating primes squared in the Online Encyclopedia on Integer Sequences (OEIS) as A240860. 2. Reciprocal sums: I prove that S=1/(2*3*5) + 1/ (3 *5*7) + ... converges and 0.0474 < S < 0.0475, published in the OEIS as A242187 using a Bound-Reduce that applies to the infinite series for e. 3. Better estimates for sums of prime powers: I found a better approximation than the Salat-Znam estimate using a balancing constant c. I found experimentally that the best values of c are roughly 0.6 for prime sums, 0.7 for squared sums, and 0.9 for cubed sums 4. New estimates from old: I found a new asymptotic estimate for prime products two at a time with 2% error, added to the OEIS as A024447. Conclusions/Discussion The hypothesis that ESP method fails if any term dominates (limit of ratio of largest term to sum does not tend to zero) is supported by results.

Competition history

• CSEF 2015 Mathematics & Software · Entry S1430

Resources

• CSEF project summary ↗

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