I do not know what is exactly the KS philosophy, or much number theory for that matter, but maybe I can tell you a few things. Take the Riemann zeta function, for instance. It was discovered by Montgomery, with some help from Dyson, that the Riemann zeros have the same correlation function as the eigenvalues of unitary random matrices. Some averages over the zeta functions agree with averages over the unitary group. There was a major result in this area by Jon Keating and Nina Snaith, for example, when they were able to arrive at a formula for the quantity $\int |\zeta(\frac{1}{2}+it)|^{2k}dt$ for general integer $k$, which was previously unavailable.

(if you look at papers that cite JP Keating, NC Snaith, Random matrix theory and ζ (1/2+ it). Communications in Mathematical Physics 214, p. 57 (2000), you will find a lot of information about this).

The general idea is that the Riemann zeros have some universal statistical properties combined with some specific properties which are related to the prime numbers. On the RMT side there are no prime numbers, so only the universality aspect can be recovered. There is however another connection, which is to the quantum mechanics of chaotic systems. Energy levels of such systems have some universal statistical properties, described by RMT, and also some specific properties, related to the corresponding classical dynamics, so that periodic orbits are the analogue of prime numbers in this case. There is some nice work by Michael Berry and Jon Keating about this (such as M. Berry, J. Keating, The Riemann Zeros and Eigenvalue Asymptotics. SIAM Rev., 41(2), 236 (1999)) .

The relation between number theory averages and RMT averages extends in some ways (to other $L$-functions and their derivatives, to non-integer moments, etc.) I would venture that the KS philosophy is the idea that there exist universal statistical properties underlying some number theory questions, and that these are well described by RMT.

I do not know what is exactly the KS philosophy, or much number theory for that matter, but maybe I can tell you a few things. Take the Riemann zeta function, for instance. It was discovered by Montgomery, with some help from Dyson, that the Riemann zeros have the same correlation function as the eigenvalues of unitary random matrices. Some averages over the zeta function agree with averages over the unitary group. There was a major result in this area by Jon Keating and Nina Snaith, for example, when they were able to arrive at a formula related to the quantity $\int |\zeta(\frac{1}{2}+it)|^{2k}dt$ for positive integers $k$, which was previously unavailable.

(if you look at papers that cite JP Keating, NC Snaith, Random matrix theory and ζ (1/2+ it). Communications in Mathematical Physics 214, p. 57 (2000), you will find a lot of information about this).

The general idea is that the Riemann zeros have some universal statistical properties combined with some specific properties which are related to the prime numbers. On the RMT side there are no prime numbers, so only the universality aspect can be recovered. There is however another connection, which is to the quantum mechanics of chaotic systems. Energy levels of such systems have some universal statistical properties, described by RMT, and also some specific properties, related to the corresponding classical dynamics, so that periodic orbits are the analogue of prime numbers in this case. There is some nice work by Michael Berry and Jon Keating about this (such as M. Berry, J. Keating, The Riemann Zeros and Eigenvalue Asymptotics. SIAM Rev., 41(2), 236 (1999)) .

The relation between number theory averages and RMT averages extends in some ways (to other $L$-functions and their derivatives, to non-integer moments, etc.) I would venture that the KS philosophy is the idea that there exist universal statistical properties underlying some number theory questions, and that these are well described by RMT.

I do not know what is exactly the KS philosophy, or much number theory for that matter, but maybe I can tell you a few things. Take the Riemann zeta function, for instance. It was discovered by Montgomery, with some help from Dyson, that the Riemann zeros have the same correlation function as the eigenvalues of unitary random matrices. Some averages over the zeta functions agree with averages over the unitary group. There was a major result in this area by Jon Keating and Nina Snaith, for example, when they were able to arrive at a formula for the quantity $\int |\zeta(\frac{1}{2}+it)|^{2k}dt$ for general integer $k$, which was previously unavailable.

(if you look at papers that cite JP Keating, NC Snaith, Random matrix theory and ζ (1/2+ it). Communications in Mathematical Physics 214, p. 57 (2000), you will find a lot of information about this).

The general idea is that the Riemann zeros have some universal statistical properties combined with some specific properties which are related to the prime numbers. On the RMT side there are no prime numbers, so only the universality aspect can be recovered. There is however another connection, which is to the quantum mechanics of chaotic systems. Energy levels of such systems have some universal statistical properties, described by RMT, as also some specific properties, related to the corresponding classical dynamics, so that periodic orbits are the analogue of prime numbers in this case. There is some nice work by Michael Berry and Jon Keating about this (such as M. Berry, J. Keating, The Riemann Zeros and Eigenvalue Asymptotics. SIAM Rev., 41(2), 236 (1999)) .

The relation between number theory averages and RMT averages extends in some ways (to other $L$-functions and their derivatives, to non-integer moments, etc.) I would venture that the KS philosophy is the idea that there exist universal statistical properties underlying some number theory questions, and that these are well described by RMT.

I do not know what is exactly the KS philosophy, or much number theory for that matter, but maybe I can tell you a few things. Take the Riemann zeta function, for instance. It was discovered by Montgomery, with some help from Dyson, that the Riemann zeros have the same correlation function as the eigenvalues of unitary random matrices. Some averages over the zeta functions agree with averages over the unitary group. There was a major result in this area by Jon Keating and Nina Snaith, for example, when they were able to arrive at a formula for the quantity $\int |\zeta(\frac{1}{2}+it)|^{2k}dt$ for general integer $k$, which was previously unavailable.

(if you look at papers that cite JP Keating, NC Snaith, Random matrix theory and ζ (1/2+ it). Communications in Mathematical Physics 214, p. 57 (2000), you will find a lot of information about this).

The general idea is that the Riemann zeros have some universal statistical properties combined with some specific properties which are related to the prime numbers. On the RMT side there are no prime numbers, so only the universality aspect can be recovered. There is however another connection, which is to the quantum mechanics of chaotic systems. Energy levels of such systems have some universal statistical properties, described by RMT, and also some specific properties, related to the corresponding classical dynamics, so that periodic orbits are the analogue of prime numbers in this case. There is some nice work by Michael Berry and Jon Keating about this (such as M. Berry, J. Keating, The Riemann Zeros and Eigenvalue Asymptotics. SIAM Rev., 41(2), 236 (1999)) .

The relation between number theory averages and RMT averages extends in some ways (to other $L$-functions and their derivatives, to non-integer moments, etc.) I would venture that the KS philosophy is the idea that there exist universal statistical properties underlying some number theory questions, and that these are well described by RMT.