Computing the Mertens and Meissel–Mertens Constants for Sums over Arithmetic Progressions

Abstract

ABSTRACT. We give explicit numerical values with 100 decimal digits for the Mertens constant involved in the asymptotic formula for ∑ p≤x p≡a mod q

Large Bias for Integers With Prime Factors in Arithmetic Progressions

“…In particular, with the values of M (q, a) calculated by Languasco and Zaccagnini [12], by (1.4), we have C(3, 2) ≈ 0.641945, C(3, 1) ≈ −0.641945;…”

“…Lemma 1 ([9], Chapter IX, §2, Theorem 2, [3], page 96, (12)). The Dirichlet L-function L(s, χ) has no zeros in the domain…”

“…where γ is Euler's constant, B := p log 1 − 1 p + 1 p is Mertens' constant, and M (q, a) is a number depending on q and a. Languasco and Zaccagnini [11] investigated the value of M (q, a) and other related constants. By (1.2), (1.3), and the orthogonality of Dirichlet characters, letting x → ∞, we get…”

An analog of perfect numbers involving the unitary totient function

“…If P 4 = 11, then P 5 = 13 or 17 by Lemma 2.9. If (P 4 , P 5 ) = (11,13), then 2 11 | N by Lemma 2.1 and therefore 2047/1028 ≤ h(3 2 · 5 · 7 · 11 · 13 · P 6 ) < 2. Hence, we must have 43 < P 6 < 47, which is impossible.…”

On Erdős and Sárközy’s sequences with Property P

“…which was studied by Languasco and Zaccagnini in [12]. 1 The computational results of Languasco and Zaccagnini imply that 0.0482 < M(3, 4) < 0.0483 and hence allow for the following lower bound for C (3,4):…”