On the sum of the first $n$ prime numbers

Abstract

In this paper we establish a general asymptotic formula for the sum of the first n prime numbers, which leads to a generalization of the most accurate asymptotic formula given by Massias and Robin in 1996.

An Arithmetic Equivalence of the Riemann hypothesis

“…where δ a satisfies −3.15 < δ a < −2.88. Further, by using inequalities (2.7) and (2.10), for x ≥ x 0 , we obtain li(x (a+2)/2 ) = 2 x (a+2)/2 (a + 2) log x + η 4 x (a+2)/2 (a + 2) 2 log 2 x , with 1 < η < 1.101, whence we get (5.11).…”

“…• Similarly, inequalities (5.18) and (5.44) imply b n < 2 3 + c + 0.77 (log log n/ log n) for n ≥ 157 933 210. which proves the upper bound of (vi) and concludes the proof of Theorem 1.1.…”

“…Therefore, from the definition (1.9) of b n , we have The correctness of ok rec(n1, n2) is proved by recursion about the size of n 2 − n 1 . The largest n which does not satisfy (5.44), n = 157 933 209, is given by the call ok rec (2,305926023). It computed four values of ok(n) and 11 395 values of good interval(n1,n2), and took 35.27 seconds.…”

Some D-finite and some Possibly D-finite Sequences in the OEIS

“…It just chose the coefficients of the recurrence in such a way that it matches the finitely many irregular terms in the beginning. Incidentally, the sequence is indeed constant for n ≥ 8, so after all, the guessed recurrence happens to be correct; see the recent work of Axler [2] and the references given in his paper.…”

Alternative Proof of the Infinitude Primes and Prime Properties

“…Improving the security of data access or currency transactions electronic improvement can use a combination of two methods, namely digital face and the product of two prime numbers [10]. Modification Additive Generator Fibonacci (AGF) by using prime numbers as an iterative equation module that describes the operation of the generator to be implemented in hardware that allows providing high performance [13].…”