Timeline for Rate of convergence of the Riemann zeta function and the Euler product formula

8 events

when toggle format	what		by	license	comment
Jul 27 at 11:45	vote	accept	Ali Taghavi
Jul 20 at 1:37	history	became hot network question
Jul 19 at 21:19	history	edited	GH from MO	CC BY-SA 4.0	edited tags
Jul 19 at 17:43	answer	added	Will Sawin		timeline score: 5
Jul 19 at 17:42	comment	added	Ali Taghavi		@WillSawin the rate of convergence has a standard definition in numerical analysis: Let $a_n$ positive goes to zero then the rate is unique q for which the limit $\frac{a_{n+1}}{a_n^q}$ is finite and non zero. In case of non convergence of the later limit you may replace by limsup or lim inf appropriately. now if $a_n$ goes to $\ell$ you consider $\|a_n-\ell$ instead of $a_n$. An interesting example is that rate of convergence of secant algorithm is the golden number
Jul 19 at 17:41	comment	added	Will Sawin		If you interpret both sides as sequences with partial products and partial sums, right-hand side is still indexed by the primes rather than the natural numbers, so you still have to decide whether the $n$th member of the sequence is the $n$th prime or the product over primes up to $n$. But I guess this doesn't matter since in all cases the rate of convergence will be $1$.
Jul 19 at 17:35	comment	added	Will Sawin		How do you define the rate of convergence? It's not clear if "rate of convergence" can be defined in such a way that the rate of convergence of a sum can be compared to a product, nor two sums/products whose index sets are different.
Jul 19 at 17:32	history	asked	Ali Taghavi	CC BY-SA 4.0