## Approximating pi(x) using the zeros of the Riemann zeta function

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• I m trying to understand how the zeros of the zeta function can be used to approximate pi(x). From Tomás Oliveira e Silva s webpage I understand that if we
Message 1 of 2 , Apr 28, 2008
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I'm trying to understand how the zeros of the zeta function can be
used to approximate pi(x).

From Tomás Oliveira e Silva's webpage I understand that if we define
H(x)=2*(li(x)-pi(x))/pi(sqrt(x))
which can be thought of as way of displaying the normalised
irregularities in the distribution. It can be approximated by
H(x)=1+2*sum(sin(t*ln(x))/t) where t are the zeros of the zeta
function
I've plotted H(x) and its approximation using the zeroes for primes
up to 1000000 and the wiggly shape of both functions is the same but
there's an offset between the two graphs which is neither a constant
increment or a constant ratio.

Is this correct or am I missing something?

Perhaps if I could get hold of a copy of "Zeroes of Dirichlet L-
functions and irregularities in the distribution of primes" by Carter
Bays and Richard H. Hudson I could work out where I went wrong. Does
anyone have a copy?

Cheers!

Richard
• ... http://en.wikipedia.org/wiki/Prime-counting_function#Formulas_for_prime-counting_functions
Message 2 of 2 , Apr 28, 2008
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> Is this correct or am I missing something?

http://en.wikipedia.org/wiki/Prime-counting_function#Formulas_for_prime-counting_functions
http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics.htm

:-)

Hope that helps,

Andrey

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