Improved results on the Mertens conjecture

ar X iv : 1 21 0 . 09 45 v 2 [ m at h . N T ] 2 O ct 2 01 3 ON THE MERTENS CONJECTURE FOR FUNCTION FIELDS

We study the natural analogue of the Mertens conjecture in the setting of global function fields. Building on the work of Cha, we show that most hyperelliptic curves do not satisfy the Mertens conjecture, but that if we modify the Mertens conjecture to have a larger constant, then this modified conjecture is satisfied by a positive proportion of hyperelliptic curves. 1. The Mertens Conjecture L...

Disproof of the Mertens Conjecture

The Mertens conjecture states that  M(x)  < x ⁄2 for all x > 1, where M(x) = n ≤ x Σ μ(n) , and μ(n) is the Mo bius function. This conjecture has attracted a substantial amount of interest in its almost 100 years of existence because its truth was known to imply the truth of the Riemann hypothesis. This paper disproves the Mertens conjecture by showing that x → ∞ lim sup M(x) x − ⁄2 > 1. 06 ....

1 1 Ja n 20 13 ON THE MERTENS CONJECTURE FOR ELLIPTIC CURVES OVER FINITE FIELDS

(1) |M(x)| ≤ √x for all x ≥ 1. This conjecture stems from the work of Mertens [8], who in 1897 calculated M(x) from x = 1 up to x = 10 000 and arrived at the conjecture (1). Notably, this conjecture implies that all of the nontrivial zeroes of the Riemann zeta function ζ(s) lie on the line R(s) = 1/2 (that is, that the Riemann hypothesis is true), and also that all such zeroes are simple. Howev...

Some difference results on Hayman conjecture and uniqueness

In this paper, we show that for any finite order entire function $f(z)$, the function of the form $f(z)^{n}[f(z+c)-f(z)]^{s}$ has no nonzero finite Picard exceptional value for all nonnegative integers $n, s$ satisfying $ngeq 3$, which can be viewed as a different result on Hayman conjecture. We also obtain some uniqueness theorems for difference polynomials of entire functions sharing one comm...

A proof of the Thompson moonshine conjecture