TY - JOUR
T1 - A heuristic for discrete mean values of the derivative of the Riemann zeta function
AU - Hughes, Christopher
AU - Martin, Greg
AU - Pearce-Crump, Andrew
PY - 2024/9/16
Y1 - 2024/9/16
N2 - Shanks conjectured that ζ ′ (ρ), where ρ ranges over non-trivial zeros of the Riemann zeta function, is real and positive in the mean. We present a history of this problem and its proof, including a generalization to all higher-order derivatives ζ (n) (s), for which the sign of the mean alternatives between positive for odd n and negative for even n. Furthermore, we give a simple heuristic that provides the leading term (including its sign) of the asymptotic formula for the average value of ζ (n) (ρ).
AB - Shanks conjectured that ζ ′ (ρ), where ρ ranges over non-trivial zeros of the Riemann zeta function, is real and positive in the mean. We present a history of this problem and its proof, including a generalization to all higher-order derivatives ζ (n) (s), for which the sign of the mean alternatives between positive for odd n and negative for even n. Furthermore, we give a simple heuristic that provides the leading term (including its sign) of the asymptotic formula for the average value of ζ (n) (ρ).
U2 - 10.48550/arXiv.2305.14253
DO - 10.48550/arXiv.2305.14253
M3 - Article (Academic Journal)
SN - 1553-1732
VL - 24
JO - INTEGERS: Electronic Journal of Combinatorial Number Theory
JF - INTEGERS: Electronic Journal of Combinatorial Number Theory
M1 - #A83
ER -