# Pair correlation of zeros of the zeta function.

@article{Gallagher1985PairCO, title={Pair correlation of zeros of the zeta function.}, author={Patrick X. Gallagher}, journal={Journal f{\"u}r die reine und angewandte Mathematik (Crelles Journal)}, year={1985}, volume={1985}, pages={72 - 86} }

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 random (uniformly and independently on an interval of length Γ), then the ratio on the left would tend to A. According to the conjecture, the zeros have on the average fewer near neighbors than they would have if they were distributed at random. This feature is most striking for small A.

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## References

SHOWING 1-10 OF 18 REFERENCES

A Note on Gaps between Zeros of the Zeta Function

- Mathematics
- 1984

Recently Montgomery and Odlyzko [1] showed, assuming the Riemann Hypothesis, that infinitely often consecutive zeros of the zeta function differ by at least 1.9799 times the average spacing and…

Primes and zeros in short intervals.

- Mathematics
- 1978

A few years ago, H. L. Montgomery [8], [9], [10] made a conjecture, which he supported with some theoretical evidence, on the distribution of pairs of nearby zeros of the Riemann zeta function. The…

ON THE DISTRIBUTION OF GAPS BETWEEN ZEROS OF THE ZETA-FUNCTION

- Mathematics
- 1985

It is expected that D(a) = D~(a) (=D(a)) for all a and that D(0) = 0, D ( a ) < l for all a, and D(a) is continuous. In fact, from a "multiple correlation" conjecture as in Montgomery [5] it is…

On the difference between consecutive zeros of the Riemann zeta function

- Mathematics, Philosophy
- 1982

On the difference between consecutive primes

- Mathematics
- 1966

Abstract The way in which the primes are distributed among the integers have attracted the curiosity of mathematicians and even actuaries as long as the problems have been recognised. It has been…

Arithmetic equivalent of essential simplicity of zeta zeros

- Mathematics
- 1983

Let R(x) and S(t) be the remainder terms in the prime number theorem and the Riemann-von Mangoldt formula respectively, that is 4{(x) = x + R(x) and N(t) = (1/27T)f0log(T/27T) dT + S(t) + 7/8 +…

A class of extremal functions for the Fourier transform

- Mathematics
- 1981

We determine a class of real valued, integrable functions f(x) and corresponding functions M$x) such that f(x) 1, and the value of MfO) is miimi. Several applications of these functions to number…

Pair Correlation of Zeros and Primes in Short Intervals

- Mathematics
- 1987

In 1943, A. Selberg [15] Deduced From The Riemann Hypothesis (Rh) that
$$\int\limits_{\rm{1}}^{\rm{X}} {{{\left( {\psi \left( {\left( {{\rm{1 + }}\delta } \right){\rm{x}}} \right){\rm{ - }}\psi…