L. Euler, Variae observationes circa series infinites. Comm. Acad. Petropolitanae 9 (1737), 222-236. Reprinted: Opera omnia 14, pp. 216-244. Home/Search Document Not in Database Summary Related Articles |
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On the remainder in a series of Mertens - Lindqvist, Peetre (Correct)
....Mertens,rest On the remainder in a series of Mertens Peter Lindqvist and Jaak Peetre 1. Introduction. The infinite series P 1 p , where the summation is extended over all prime numbers p = 2; 3; 5; 7; 11; was first considered by Euler [3] in 1737; Euler wrote 1 2 1 3 1 5 1 7 1 11 etc: l:l1 ; which assertion has to be understood as a vague expression of the basic insight that the partial sums P px 1 p grow as log log x. Also young Gauss did encounter the same sum as early as in 1796. In his Nachlass one has ....
....root. 2. Mertens s series via a Lambert series. Let us start with the formula log(1 Gamma p Gammas ) Z 1 p s dt t(t s Gamma 1) Summation over p = 2; 3; 5; 7; 11; now gives (1) 1 s log i(s) Z 1 2 (t) dt t(t s Gamma 1) which is just Euler s famous product formula ([3], Theorem 7) in slight disguise. We recall that this formula, in contemporary reinterpretation, states that i(s) 1 X n=1 1 n s = Y p 1 Gamma 1 p s Gamma1 for Re s 1: In the same fashion, using 1 mp m = Z 1 p dt t m 1 we can obtain a proof of formula (3) in ....
L. Euler, Variae observationes circa series infinites. Comm. Acad. Petropolitanae 9 (1737), 222-236. Reprinted: Opera omnia 14, pp. 216-244.
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