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}
}
  • P. Gallagher
  • Published 1985
  • Mathematics
  • Journal für die reine und angewandte Mathematik (Crelles Journal)
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. 
Maximum of the Riemann Zeta Function on a Short Interval of the Critical Line
We prove the leading order of a conjecture by Fyodorov, Hiary, and Keating about the maximum of the Riemann zeta function on random intervals along the critical line. More precisely, as T → ∞ for a
Small gaps between zeros of twisted L-functions
We use the asymptotic large sieve, developed by the authors, to prove that if the Generalized Riemann Hypothesis is true, then there exist many Dirichlet L-functions that have a pair of consecutive
On Repeated Values of the Riemann Zeta Function on the Critical Line
TLDR
This paper studies repeated values of ζ(s) on the critical line, and gives evidence to support the conjecture that for every nonzero complex number z, the equation ζ (1/2+it)=z has at most two solutions.
On the pair correlation conjecture and the alternative hypothesis
A Central Limit Theorem for Linear Combinations of Logarithms of Dirichlet $L$-functions
an approximate Gaussian distribution with mean 0 and variance a1 2 + · · ·+ an 2 2 log log T . Here a1, . . . , an ∈ R, each of the χi is a primitive Dirichlet character modulo Mi with Mi ≤ T , and 0
A note on small gaps between zeros of the Riemann zeta-function
Assuming the Riemann Hypothesis, we improve on previous results by proving there are infinitely many zeros of the Riemann zeta-function whose differences are smaller than 0.50412 times the average
Correlations of sieve weights and distributions of zeros
  • A. Walker
  • Mathematics
    International Journal of Number Theory
  • 2022
In this note we give two small results concerning the correlations of the Selberg sieve weights. We then use these estimates to derive a new (conditional) lower bound on the variance of the primes in
Low-lying zeros of L-functions for Quaternion Algebras
The density conjecture of Katz and Sarnak predicts that, for natural families of L-functions, the distribution of zeros lying near the real axis is governed by a group of symmetry. In the case of the
Some Observations concerning the Distribution of the Zeros of the Zeta Functions (I)
Let Z ( s) be a zeta function which has nice properties like the Riemann zeta function ((s ). Let a0 be the critical point of Z(s) and suppose that the Riemann Hypothesis (R.H.) holds for Z ( s),
On the distribution of imaginary parts of zeros of the Riemann zeta function, II
We continue our investigation of the distribution of the fractional parts of αγ, where α is a fixed non-zero real number and γ runs over the imaginary parts of the non-trivial zeros of the Riemann
...
...

References

SHOWING 1-10 OF 18 REFERENCES
A Note on Gaps between Zeros of the Zeta Function
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.
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
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 primes
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
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
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
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
Notes on small class numbers
...
...