关于求和式$N(T) := \sum_{0<\gamma\leq T} 1$的读法及求和含义的技术问询

Hey there! Let's unpack this summation clearly—this notation is super common in analytic number theory, especially when working with the Riemann zeta function, which is almost certainly the context here.

读法 #

You can read this expression in a couple of natural ways:

• Formal, literal reading: "N of T is defined as the sum over all gamma where gamma is greater than 0 and less than or equal to T of 1"
• More intuitive, functional reading: "N of T counts the number of gamma values satisfying 0 < γ ≤ T"

求和含义（关键疑问解答） #

Great question about summing "just 1"—here's the core idea: when you sum the constant 1 over every element in a set of values (in this case, the set of $\gamma$s meeting the inequality), you're effectively counting how many elements are in that set.

Each $\gamma$ here refers to the imaginary part of a non-trivial zero of the Riemann zeta function (the standard definition for this $N(T)$ function). For every such $\gamma$ that falls in the range $(0, T]$, you add 1 to the total sum. The final result is exactly the total number of those non-trivial zeros with imaginary part between 0 and T.

To make it concrete: suppose there are 4 valid $\gamma$s in the range—say 1.414, 2.606, 3.385, 4.018—then the sum becomes $1+1+1+1=4$, which is exactly the count of those zeros up to T.

备注：内容来源于stack exchange，提问作者zeynel