# Mean values of L-functions and symmetry

@article{Conrey1999MeanVO, title={Mean values of L-functions and symmetry}, author={J. Brian Conrey and David W. Farmer}, journal={International Mathematics Research Notices}, year={1999}, volume={2000}, pages={883-908} }

Recently Katz and Sarnak introduced the idea of a symmetry group attached to a family of L-functions, and they gave strong evidence that the symmetry group governs many properties of the distribution of zeros of the L-functions. We consider the mean-values of the L-functions and the mollified mean-square of the L-functions and find evidence that these are also governed by the symmetry group. We use recent work of Keating and Snaith to give a complete description of these mean values. We find a…

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## References

SHOWING 1-10 OF 58 REFERENCES

Zeroes of zeta functions and symmetry

- Mathematics
- 1999

Hilbert and Polya suggested that there might be a natural spectral interpretation of the zeroes of the Riemann Zeta function. While at the time there was little evidence for this, today the evidence…

Pair correlation of zeros of the zeta function.

- Mathematics
- 1985

s T— »oo and U-+Q in such a way that UL = A; here L = ̂ —logT is the average 2n density of zeros up to T and A is an arbitrary positive constant. If the same number of points were distributed at…

Characteristic Polynomials of Random Matrices

- Mathematics
- 2000

Abstract: Number theorists have studied extensively the connections between the distribution of zeros of the Riemann ζ-function, and of some generalizations, with the statistics of the eigenvalues of…

More than two fifths of the zeros of the Riemann zeta function are on the critical line.

- Mathematics
- 1989

In this paper we show that at least 2/5 of the zeros of the Riemann zeta-function are simple and on the critical line. Our method is a refinement of the method Levinson [11] used when he showed that…

Evidence for a Spectral Interpretation of the Zeros of

- Education
- 1998

By looking at the average behavior (n-level density) of the low lying zeros of certain families of L-functions, we nd evidence, as predicted by function eld analogs, in favor of a spectral…

On the Mean Value of L(1/2, χ ) FW Real Characters

- Mathematics, Philosophy
- 1981

Asymptotic formulae are derived for the sums ZL''(i,x), where k=l or 2 and x runs over all Dirichlet characters defined by Kronecker's symbol ^ with d being restricted to fundamental discriminants in…

High moments of the Riemann zeta-function

- Mathematics
- 1999

The authors describe a general approach which, in principal, should produce the correct (conjectural) formula for every even integer moment of the Riemann zeta function. They carry it out for the…

Random matrices, Frobenius eigenvalues, and monodromy

- Mathematics
- 1998

Statements of the main results Reformulation of the main results Reduction steps in proving the main theorems Test functions Haar measure Tail estimates Large $N$ limits and Fredholm determinants…

Simple zeros of the Riemann zeta-function

- Mathematics, Philosophy
- 1993

Assuming the Riemann Hypothesis, Montgomery and Taylor showed that at least 67.25% of the zeros of the Riemann zeta-function are simple. Using Montgomery and Taylor's argument together with an…

On mean values of the zeta-function

- Mathematics
- 1984

It is conjectured that for any k, Ik(T)~ckT(\ogTf (2) for some constant ck. It is well known that (2) holds for k = 0, 1 and 2 with c0 = 1, Cj = 1, and c2 = (2n )~, but there is not even a…