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:

http://plms.oxfordjournals.org/content/s2-50/1/107.extract

You can see the explicit formula in the abstract, although hard to see the fine detail without access to the full PDF. This is a general form involving a function $f$ and an integral transform thereof. The abstract mentions "appropriate conditions" on $f$: I can't see what these are, but with an appropriate choice of this function, the formula displayed would presumably reduce to a fairly straightforward relation between a sum over the primes and a sum over the nontrivial zeros.

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:

http://plms.oxfordjournals.org/content/s2-50/1/107.extract

You can see the explicit formula in the abstract, although hard to see the fine detail without access to the full PDF. This is a general form involving a function $f$ and an integral transform thereof. The abstract mentions "appropriate conditions" on $f$: I can't see what these are, but with an appropriate choice of this function, the formula displayed would presumably reduce to a fairly straightforward relation between a sum over the primes and a sum over the nontrivial zeros.

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:

http://plms.oxfordjournals.org/content/s2-50/1/107.extract

You can see the explicit formula in the abstract, although hard to see the fine detail without access to the full PDF. This is a general form involving a function f and an integral transform thereof. The abstract mentions "appropriate conditions" on f: I can't see what these are, but with an appropriate choice of this function, the formula displayed would presumably reduce to a fairly straightforward relation between a sum over the primes and a sum over the nontrivial zeros.

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:

http://plms.oxfordjournals.org/content/s2-50/1/107.extract

You can see the explicit formula in the abstract, although hard to see the fine detail without access to the full PDF. This is a general form involving a function $f$ and an integral transform thereof. The abstract mentions "appropriate conditions" on $f$: I can't see what these are, but with an appropriate choice of this function, the formula displayed would presumably reduce to a fairly straightforward relation between a sum over the primes and a sum over the nontrivial zeros.