Zeta Zeros Research Articles

Large spaces between the zeros of the Riemann zeta-function and random matrix theory

On the hypothesis that the 2 k-th mixed moments of Hardy's Z-function and its derivative are correctly predicted by random matrix theory, it is established that large gaps (depending on, and apparently increasing with k) exist between the zeta zeros. The case k=3 has been worked out in an earlier paper (in this journal) and the cases k=4,5,6 are considered here. When k=6 the gaps obtained have >4 times the average gap length. This depends on calculations involving Jacobi-Schur functions and formulae for these functions due to Jacobi, Trudi and Aitken in the classical theory of equations.

Random matrix theory and the zeros of (s)

We study the density of the roots of the derivative of the characteristic polynomial Z(U,z) of an N x N random unitary matrix with distribution given by Haar measure on the unitary group. Based on previous random matrix theory models of the Riemann zeta function zeta(s), this is expected to be an accurate description for the horizontal distribution of the zeros of zeta'(s) to the right of the critical line. We show that as N --> infinity the fraction of roots of Z'(U,z) that lie in the region 1-x/(N-1) <= |z| < 1 tends to a limit function. We derive asymptotic expressions for this function in the limits x --> infinity and x --> 0 and compare them with numerical experiments.

The Riemann Hypothesis, simple zeros and the asymptotic convergence degree of improper Riemann sums

The Riemann Hypothesis, simple zeros and the asymptotic convergence degree of improper Riemann sums

The k-functions in multiplicative number theory. III. Uniform distribution of zeta zeros; discrepancy

The k-functions in multiplicative number theory. III. Uniform distribution of zeta zeros; discrepancy

The k-functions in multiplicative number theory. II. Uniform distribution of zeta zeros

The k-functions in multiplicative number theory. II. Uniform distribution of zeta zeros

Gram's law for the zeta zeros and the eigenvalues of Gaussian unitary ensembles

Gram's law for the zeta zeros and the eigenvalues of Gaussian unitary ensembles

Arithmetic equivalent of essential simplicity of zeta zeros

Let R ( x ) R(x) and S ( t ) S(t) be the remainder terms in the prime number theorem and the Riemann-von Mangoldt formula respectively, that is ψ ( x ) = x + R ( x ) \psi (x) = x + R(x) and N ( t ) = ( 1 / 2 π ) ∫ 0 t log ⁡ ( τ / 2 π ) d τ + S ( t ) + 7 / 8 + O ( 1 / t ) N(t) = (1/2\pi )\int _0^t {\log (\tau /2\pi )\,d\tau + S(t) + 7/8 + O(1/t)} . We are interested in the following integrals: J ( T , β ) = ∫ 1 T β ( R ( x + x / T ) − R ( x ) ) 2 d x / x 2 J(T,\beta ) = \int _1^{{T^\beta }} {{{(R(x + x/T) - R(x))}^2}dx/{x^2}} and I ( T , α ) = ∫ 1 T ( S ( t + α / L ) − S ( t ) ) 2 d t I(T,\alpha ) = \int _1^T {{{(S(t + \alpha /L) - S(t))}^2}dt} , where L = ( 2 π ) − 1 log T L = {(2\pi )^{ - 1}}\log \,T . Furthermore, denote by N ( T , α ) ( N ∗ ( T ) ) N(T,\alpha )(N^{\ast }(T)) the number of pairs of zeros 1 2 + i Υ , 1 2 + i Υ ′ \frac {1} {2} + i\Upsilon ,\frac {1} {2} + i\Upsilon ’ with 0 &gt; Υ ⩽ T 0 &gt; \Upsilon \leqslant T and 0 &gt; ( Υ ′ − Υ ) L ⩽ α ( ( Υ ′ − Υ ) L = 0 ) 0 &gt; (\Upsilon ’ - \Upsilon )L \leqslant \alpha \,((\Upsilon ’ - \Upsilon )L = 0) —i.e., off-diagonal and diagonal pairs. Theorem. Assume the Riemann hypothesis. The following three hypotheses (A), (B) and ( C 1 , C 2 ) ({{\text {C}}_1},{{\text {C}}_2}) are equivalent: for β → ∞ \beta \to \infty and α → 0 \alpha \to 0 as T → ∞ T \to \infty we have (A) J ( T , β ) ∼ β T − 1 log 2 T J(T,\beta ) \sim \beta {T^{ - 1}}{\log ^2}T , (B) I ( T , α ) ∼ α T I(T,\alpha ) \sim \alpha T and ( C 1 ) N ∗ ( T ) ∼ T L , ( C 2 ) N ( T , α ) = o ( T L ) ({{\text {C}}_1})\;N^{\ast }(T) \sim TL,({{\text {C}}_2})N(T,\alpha ) = o(TL) . Hypothesis ( C 1 , C 2 ) ({{\text {C}}_1},{{\text {C}}_2}) is called the essential simplicity hypothesis.

Arithmetic Equivalent of Essential Simplicity of Zeta Zeros

Arithmetic Equivalent of Essential Simplicity of Zeta Zeros

Zeta Zeros on the Critical Line

Zeta Zeros on the Critical Line

Zeta Zeros on the Critical Line

Zeta Zeros on the Critical Line

A note on recent papers of Ramachandra and Huxley

The methods of the two authors on the zeros of zeta and L-functions are compared.

The size of the Riemann zeta-function at places symmetric with respect to the point 1/2

Abstract : Improvement is made on and a simpler proof provided of a result of R. Spira which is to appear in the Duke Mathematical Journal. This result is that if s = sigma + it and zeta is the Riemann zeta-function, then absolute value (zeta (1 - s))> absolute value (zeta (s)) for all s other than zeros of zeta provided t > 1/2. The proof uses Stirling's formula, as did Spira's.