The gamma constant is defined by an equation where the harmonic series is subtracted by the natural logarithm:
γ=limn→∞(n∑k=11k−ln(n))
It is well known that both the harmonic series by itself and the sum of reciprocals of primes are divergent.
Is there a well known function that when subtracted from the sum of reciprocals of primes makes the resultant series convergent?
Is there a function f(x) that makes the following series convergent:
limn→∞(n∑p is a prime 1p−f(n))
Answer
Yes there is a constant associated with the sum of the reciprocals of the primes. In particular, Mertens showed that ∑p prime ≤x1p−loglog(x)
I found the result in a paper:
EULER’S CONSTANT: EULER’S WORK AND MODERN DEVELOPMENTS - By JEFFREY C. LAGARIAS
Mertens' paper is titled: Ein Beitrag zur analytischen Zahlentheorie
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