An improved bound on the sum of prime numbers

We derive two asymptotic formulae, for the upper bound on the sum of the first n primes. Both the Supremum and the Estimate of the sum are superior to known bounds. The Estimate bound had been derived to promote the efficiency of estimation of the sum.

The conjectured inequality by R. Mandl, has been proven by J.B. Rosser and L. Schoenfeld in their 1975 paper [8]. The Mandl's inequality asserts that: The efforts directed to refine and sharpen the Mandl's estimate include for example Dusart [2] who proved that: M. Hassani [7] in 2006 presented a considerably improved refinement: These estimates however, diverge relatively quickly from the actual sum of the prime numbers. In contrast to that, theory based on the Gauss' offset integral provides for a significantly improved bounds. Let's therefore define: and γ ≈ 0.57721566490153286 is the Euler-Mascheroni constant.
In this section we present the Supremum SUP (pn) upper bound, as well as the Sharp Estimate S.E. (pn) . Both these functions depend on the theory of the primorial function and its bounds, presented here: The natural logarithm of the primorial function is strictly less than the respective prime number p ∈ N: In particular the natural logarithm of the primorial function is asymptotic (from below) to the respective prime number: The error of estimation of the primorial function by the use of the value of p (n) imposes the following lower bound: where γ ≈ 0.57721566490153286060651209 is the Euler-Mascheroni constant.

Lemma 1.7 (Upper Estimation Error Bound On The Difference p n − log p n ♯).
The error of estimation of the primorial function by the use of the value of p (n) imposes the following upper bound: For the proofs of Lemmas 1.5, 1.6 and 1.7, please consult Feliksiak [5].

Supremum Bound on the Sum of Prime Numbers.
The Supremum bound and the Sharp Estimate, both implement a variant of the Gauss' offset logarithmic integral. The Supremum is a very accurate bound, its computation however will become challenging very quickly. The degree of computational difficulty of the Supremum is about the same as computing the sum itself. Consequently, the Sharp Estimate S.E. (pn) is so to say, the "golden mean" to achieve the goal.
The Sharp Estimate S.E. (pn) is quick and straightforward to compute. Thus, it attenuates the computational difficulty, while at the same time it is sufficiently accurate to rely on. The S.E. (pn) cuts through the meanders of the True Sum of the prime numbers, producing alternatively positive/negative estimation error values. By taking the Absolute value of the Estimation Error, we can demonstrate that the estimation error, at the prime p n = 19 373 already falls below 1 percent of the value that the True Sum attains. The estimation error improves as p n → ∞. The

Lemma 1.8 (The Supremum Bound on the Sum of Prime Numbers).
The Supremum bound on the sum of primes up to p n , for all p n ∈ N | p n ≥ 5 is given by: Where T .S. and θ s are given by the Definitions: 1.1 and 1.2 respectively.

Proof.
Suppose that for p n ∈ N | p n ≥ 17 the inequality is false: This implies that: (1.10) However, at p n = 17 the difference of terms of the inequality 1.10 attains approximately −1.3563587 and diverges at every step n, at the rate ∝ k p n with k → 1 as p n → ∞. Consequently, we have a contradiction. Necessarily therefore, < p n for all p n ∈ N | p n ≥ 17 the inequality 1.11 holds. This in turn implies that for all p n ∈ N | p n ≥ 17, Consequently, Lemma 1.8 holds for all p n ∈ N | p n ≥ 17. For all 5 ≤ p n ≤ 17 Table  1 lists all values of p n within the range, demonstrating that Lemma 1.8 holds in this range as well. Thus, necessarily Lemma 1.8 holds as stated for all p n ∈ N | p n ≥ 5, concluding the proof.

Proof.
Suppose that for p n ∈ N | p n ≥ 2, the following inequality is false: This implies that it must be true: However, at p n = 2 the inequality 1.16 attains ∼ −0.910865 and diverges rapidly at every step, at a rate ∝ p 2 n . Consequently we have a contradiction to the initial hypothesis. Necessarily therefore, for all p n ∈ N | p n ≥ 2: the inequality 1.18 must be true. Now suppose, that for p n ∈ N | p n ≥ 127 the following inequality is false: This implies that: However, at p n = 127 inequality 1.19 attains ∼ 45.218, and diverges at every step, at a rate exceeding 3 √ p n . The divergence rate further increases as p n increases unboundedly. Consequently we have a contradiction to the initial hypothesis. Thus necessarily, for all p n ∈ N | p n ≥ 127: (1.20) Direct computer evaluation demonstrates that for p n in the range 2 ≤ p n ≤ 127, inequality 1.20 holds. The output in graphical form is given in Figure 5. Thus Lemma 1.10 holds as stated. Concluding the proof.