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Is there any sort of standard notation for the value represented by $f(n) = \frac{\Lambda(n)}{\log n}$, where $\Lambda(n)$ is the Mangoldt function? So essentially a function that is $\frac {1}{a}$ when $n$ is equal to some prime $p$ to the power of $a$, and 0 otherwise.

And while on the subject, is there any consensus about notation for its summatory function, which is also equal to $\pi(n) + \frac{1}{2}\pi(n^\frac{1}{2})+ \frac{1}{3}\pi(n^\frac{1}{3})+\cdots$, where $\pi(n)$ is the prime counting function? Riemann called it $f(n)$, I believe, Edwards calls it $J(n)$, Crandall and Pomerance seem to refer to it as $\pi^*(n)$, and I've found several other references that refer to it as $\Pi(n)$. I feel like I've seen $\Pi(n)$ more often, but not enough to feel confident...

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    On the second question I've seen $\pi^*(n)$ most often.2011-12-14

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Bateman and Diamond, Analytic Number Theory, use $\kappa(n)$ for the function in your title (page 28), but I don't know that I'd call it standard. They use $\Pi$ for the summatory function (page 218).

Jameson, The Prime Number Theorem, uses $c(n)$ for the function in the title, but I think that's only in an exercise on page 81. In the same exercise, he uses $\Pi$ for the summatory function.

Stopple, A Primer of Analytic Number Theory, also uses $\Pi$ for the summatory function. Ditto Montgomery and Vaughan, Multiplicative Number Theory, page 416.

Davenport, Multiplicative Number Theory, page 112, calls the summatory function $\pi_1$.

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    Yes, but I wouldn't expect readers to know what $\Pi(n)$ is without giving them a definition.2011-12-15