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where is the prime power counting function introduced . It's high time we applied this!

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Prime-counting function - Wikipedia

When it comes to counting the number of divisors(for the time being let's count both 1 and N as divisors),only the sequence of exponents a,b,c,... matters(not the sequence of prime factors A,B,C,...).To get a divisor of N you should pick one exponent for the first prime among the(a+1) integers from 0 to a, one exponent for the second prime among the (b+1)integers between 0 and b, etc.

we can again interpret it as an error-correcting formula for counting the primes.

as the basis for Heisenberg's Uncertainty Principle);several important lemmas of analysis are due to Cauchy;the famous Burnside's Counting Theorem was first discovered by Cauchy; etc.

One is Riemann's prime-counting function, ..

We define our new prime counting function, usually denoted by , as follows.

Prime numbers, the indivisible atoms of arithmetic, seem to be strewn haphazardly along the number line, starting with 2, 3, 5, 7, 11, 13, 17 and ad infinitum. But in 1859, the great German mathematician Bernhard Riemann hypothesized that the spacing of the primes logically follows from other numbers, now known as the “nontrivial zeros” of the Riemann zeta function.

After all this playing with the -function it is time to return to the overall objective of this whole exercise: counting prime numbers. The idea behind analytic number theory is that primes are unpredictable on the small scale, but actually surprising regular on the large scale. This is why we'll look at certain functions that behave pretty erratically when we look at every single value, but become smooth and "easy" to calculate once we "zoom out" and consider the global properties, the so-called asymptotic.

is the prime-counting function, ..

Here are a few other applets related to the zeta function and the prime number theorem:

Assuming the Riemann Hypothesis, you can use a smooth approximation to the characteristic function of an interval in the Guinand-Weil explicit formula to approximately count the number of zeros of the zeta-function in an interval on the critical line. This expresses the approximate number of such zeros in terms of an integral of your test function and a sum over primes, as you seek. In fact, this can be set-up in such a way that the sum over primes is finite. (This method can be used to give upper and lower bounds for the number of zeros, but not an exact formula.)

It looks like the exact formula being sought here can be found in A.P. Guinand, "A summation formula in the theory of prime numbers", Proceedings of the London Mathematical Society (2) 50 (1948) 107--119. The first page is visible here without subscription:

He won the Order of Lenin three times and several prizes fromWestern countries.
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  • Prime counting function | Understanding the Riemann Hypothesis

    th prime count

  • Zeta Function - Part 6 - The Prime Counting Function - …

    And in this particular case, when $m$ is even you do in fact count all odd primes in the sum, yes. –

  • Euler’s Pi Prime Product and Riemann’s Zeta Function - Duration: ..

    Other prime-counting functions are also used because they are more convenient to work with

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Two Representations of the Prime Counting ..

However, Riemann knew that his formula would be valid only if the zeros of the zeta function satisfied a certain property: Their real parts all had to equal ½. Otherwise the formula made no sense. Riemann calculated the first few nontrivial zeros of the zeta function and confirmed that their real parts were equal to ½. The calculation supported his hypothesis that all zeros had this property, and thus that the spacing of all prime numbers followed from his function. But he noted that “without doubt it would be desirable to have a rigorous proof of this proposition.”

Prime Numbers and the Riemann Zeta Function, Edwin …

I've mentioned the function before which is defined as the number of primes not exceeding some real . (Never mind the overloading of the symbol for different and completely unrelated functions, constants, etc. It's been like that for ever, and always will be. Mathematicians are pretty stubborn.) If we can calculate exactly, we would have perfect information about the primes since the function counts the primes exactly. Theoretically. But as mentioned before, we are only interested in capturing the overall behaviour, forgetting "local" irregularities.

[Think of this as a generalization of the prime counting function ..

The modern of primality is that"a prime number is a positive integer with exactly two positive divisors".However, this may seem unconvincing (and/or arbitrary) by itself,until you stop to consider why we define things the way we do in mathematics,physics or other sciences.A relevant quote from Henri Poincaré has been given a superb conciseEnglish translation by, namely:

Explicit formula for Riemann zeros counting function

So, what does look like? Most of the time, it is just a flat line for most of the values when the count does not change as they are not primes. At every prime value it jumps up by one step. That's it.

Because there are various explicit formulae whereby prime counting ..

for a complex number. This function is analytic for real part of greater than and is related to the prime numbers by the Euler Product Formula

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