# Moscow Mathematical Journal

Volume 15, Issue 2, April–June 2015 pp. 283–291.

A Note on Artin’s Constant

**Authors**:
Ivan Cherednik (1)

**Author institution:**(1) Department of Mathematics, UNC Chapel Hill, North Carolina 27599, USA

**Summary: **

We suggest new heuristic summation formulas for the Artin
constant and its analogs in higher ranks, for the density of prime numbers *p* such that a given set of integers generates the corresponding
multiplicative groups modulo powers of *p*. Depending on the choice of
these integers, certain interesting rational corrections emerge, which are
calculated in rank one case. The rate of convergence of these sums to
the corresponding constants, connected with the generalized Riemann
Hypothesis, is analyzed numerically. We also discuss the Stephens constant.

2010 Math. Subj. Class. 11M06, 11N69, 11Z05.

**Keywords:**Artin constant, Riemann hypothesis.

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