On the constant in the Mertens product for arithmetic progressions. II: Numerical values

Abstract

Abstract. We give explicit numerical values with 100 decimal digits for the constant in the Mertens product over primes in the arithmetic progressions a mod q, for q ∈ {3, . . . , 100} and (a, q) = 1.

Values of the Euler 𝜙-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields

“…Put a(1) = 0 and inductively define a(n) to be the smallest integer exceeding a(n − 1) such that, for every prime r, the set {a(i)(mod r) : 1 i n} has at most r − 1 elements (using the Chinese remainder theorem it is easily seen that the sequence is infinite). Given the prime k-tuples conjecture an equivalent statement is that a(n) is minimal such that there are infinitely many primes q with q + a(i) prime for 1 i n. We have 2,6,8,12,18,20,26,30,32, . .…”

“…For recent work on this theme, the reader is referred to the papers by Languasco and Zaccagnini [28,29,30,31].…”

Accurate computations of Euler products over primes in arithmetic progressions

“…We havé If finding this identity has not been immediate, checking it is only a matter of calculations that we reproduce in Section 2. A partial identity of this sort has already been used by K. Williams in [14] and more recently by A. Languasco and A. Zaccagnini in [5,7], and [6, (2-5)] is a related formula. It is worth noticing that, with our conventions, we have the obvious log L P ps, χq " log Lps, χq´ÿ păP logp1´χppq{p s q.…”

Biases in prime factorizations and Liouville functions for arithmetic progressions

“…Note that this constant has been studied in greater detail, for example [14,15,18], which may be useful in further analysis of the partial sums of ω(n; q, a). 3 One may also consider higher moments, such as…”