It has been conjectured that the statistical properties of zeros of the Riemann zeta function near $z = 1/2 + \ui E$ tend, as $E \to \infty$, to the distribution of eigenvalues of large random matrices from the Unitary Ensemble. At finite $E$ numerical results show that the nearest-neighbour spacing distribution presents deviations with respect to the conjectured asymptotic form. We give here arguments indicating that to leading order these deviations are the same as those of unitary random matrices of finite dimension $N_{\rm eff}=\log(E/2π)/\sqrt{12 Λ}$, where $Λ=1.57314 ...$ is a well defined constant.
We show that the density functions of nearest neighbor spacing distributions for zeros of the real or imaginary part of the Riemann xi-function on vertical lines are described by the M-function which is appeared in value distributions of the logarithmic derivative of the Riemann zeta-function on vertical lines.
Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 2.7327 times the average spacing and infinitely often they differ by at most 0.5154 times the average spacing.
We show that the generalized Riemann hypothesis implies that there are infinitely many consecutive zeros of the Riemann zeta function whose spacing is 2.9125 times larger than the average spacing. This is deduced from the calculation of the second moment of the Riemann zeta function multiplied by a Dirichlet polynomial averaged over the zeros of the zeta function.
Assuming the Riemann Hypothesis, we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at most 0.5155 times the average spacing and infinitely often they differ by at least 2.69 times the average spacing.
Assuming the Riemann hypothesis, we investigate the distribution of gaps between the zeros of ξ'(s). We prove that a positive proportion of gaps are less than 0.796 times the average spacing and, in the other direction, a positive proportion of gaps are greater than 1.18 times the average spacing. We also exhibit the existence of infinitely many normalized gaps smaller (larger) than 0.7203 (1.5, respectively).
Assuming the Riemann Hypothesis we show that there exist infinitely many consecutive zeros of the Riemann zeta-function whose gaps are greater than 2.9 times the average spacing.
Assuming the Generalized Riemann Hypothesis(GRH), we show that infinitely often consecutive non-trivial zeros of the Riemann zeta-function differ by at least 3.072 times the average spacing.
This paper studies the local spacings of deformations of the Riemann zeta function under certain averaging and differencing operations. For real h it considers A_h(s)= 1/2(xi(s+h)+ xi(s-h)) and B_h(s)=1/(2i)(xi(s+h)-xi(s-h)), where xi(s) is the Riemann xi-function. For |h|\ge 1/2 all zeros of these functions lie on the critical line and are simple zeros. Assuming the Riemann hypothesis, the same holds for all nonzero h. The number of zeros to height T has the same asymptotics as for the zeta function in its main terms, so one can define normalized zero spacings as for the xi-function. It is sh…
In this paper, we will employ the Opial and Wirtinger type inequalities to derive some conditional and unconditional lower bounds for the gaps between the zeros of the Riemann zeta-function. First, we prove (unconditionally) that the consecutive nontrivial zeros often differ by at least 1.9902 times the average spacing. This value improves the value 1.9 due to Mueller and the value 1.9799 due to Montogomery and Odlyzko. Second, on the hypothesis that the 2k-th mixed moments of the Hardy Z-function and its derivative are correctly predicted by random matrix theory, we derive some explicit formu…
We derive strong and effective lower bounds for the class number h(q) of the imaginary quadratic field Q(\sqrt{-q}), conditionally subject to the existence of many small (subnormal) gaps between zeros of the L-function associated with a character of the class group associated with this field. In particular, we prove that if the gap between consecutive zeros of the L-function is somewhat smaller than the average for sufficiently many pairs of zeros on the critical line, then h >> \sqrt q (log q)^{-A} for some constant A > 0. For the trivial character, the L-function is the Dedekind zeta-functio…
Combining the mollifiers, we exhibit other choices of coefficients that improve the results on large gaps between the zeros of the Riemann zeta-function. Precisely, assuming the Generalized Riemann Hypothesis (GRH), we show that there exist infinitely many consecutive gaps greater than 3.033 times the average spacing.
We prove that there exist infinitely many consecutive zeros of the Riemann zeta-function on the critical line whose gaps are greater than $3.18$ times the average spacing. Using a modification of our method, we also show that there are even larger gaps between the multiple zeros of the zeta function on the critical line (if such zeros exist).
Assuming the Riemann Hypothesis, it is known that there are infinitely many consecutive pairs of zeros of the Riemann zeta-function within 0.515396 times the average spacing. This is obtained using the method of Montgomery and Odlyzko. We prove that this method can never find infinitely many pairs of consecutive zeros within 0.5042 times the average spacing.
We examine the behaviour of the zeros of the real and imaginary parts of $ξ(s)$ on the vertical line $\Re s = 1/2+λ$, for $λ\neq 0$. This can be rephrased in terms of studying the zeros of families of entire functions $A(s) = {1/2} (ξ(s+λ) + ξ(s - λ))$ and $B(s) = \frac{1}{2i} (ξ(s+λ) - ξ(s - λ))$. We will prove some unconditional analogues of results appearing in \cite{Lag}, specifically that the normalized spacings of the zeros of these functions converges to a limiting distribution consisting of equal spacings of length 1, in contrast to the expected GUE distribution for the same zeros at $…
If $f$ is a polynomial with all of its roots on the real line, then the roots of the derivative $f'$ are more evenly spaced than the roots of $f$. The same holds for a real entire function of order~1 with all its zeros on a line. In particular, we show that if $f$ is entire of order~1 and has sufficient regularity in its zero spacing, then under repeated differentiation the function approaches (a change of variables from) the cosine function. We also study polynomials with all their zeros on a circle, and we find a close analogy between the two situations. This sheds light on the spacing betwe…
We investigate the proportion of the nontrivial roots of the equation $ζ(s)=a$, which lie on the line $\Re s=1/2$ for $a \in \mathbb C$ not equal to zero. We show that at most one-half of these points lie on the line $\Re s=1/2$. Moreover, assuming a spacing condition on the ordinates of zeros of the Riemann zeta-function, we prove that zero percent of the nontrivial solutions to $ζ(s)=a$ lie on the line $\Re s=1/2$ for any nonzero complex number $a$.
Feng and Wu introduced a new general coefficient sequence into Montgomery and Odlyzko's method for exhibiting irregularity in the gaps between consecutive zeros of $ζ(s)$ assuming the Riemann Hypothesis. They used a special case of their sequence to improve upon earlier results on the gaps. In this paper we consider an equivalent form of the general sequence of Feng and Wu, and introduce a somewhat less general sequence $\{a_n\}$ for which we write the Montgomery-Odlyzko expressions explicitly. As an application, we give the following slight improvement of Feng and Wu's result: infinitely ofte…
In this paper, first by employing inequalities derived from the Opial inequality due to David Boyd with best constant, we will establish new unconditional lower bounds for the gaps between the zeros of the Riemann zeta function. Second, on the hypothesis that the moments of the Hardy Z-function and its derivatives are correctly predicted, we establish some explicit formulae for the lower bounds of the gaps between the zeros and use them to establish some new conditional bounds. In particular it is proved that the consecutive nontrivial zeros often differ by at least 6.1392 (conditionally) time…
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 spacing. To obtain this result, we generalize a set of weights that were developed by Xiaosheng Wu, who used them to find a positive proportion of large and small gaps between zeros of the Riemann zeta-function.
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 zeros closer together than 0.37 times their average spacing. More generally, we investigate zero spacings within the family of twists by Dirichlet characters of a fixed L-function and give precise bounds for small gaps which depend only on the degree of the L-function.
A one-parameter family of point processes describing the distribution of the critical points of the characteristic polynomial of large random Hermitian matrices on the scale of mean spacing is investigated. Conditionally on the Riemann hypothesis and the multiple correlation conjecture, we show that one of these limiting processes also describes the distribution of the critical points of the Riemann $ξ$-function on the critical line. We prove that each of these processes boasts stronger level repulsion than the sine process describing the limiting statistics of the eigenvalues: the probability…
Assuming the Riemann hypothesis and Montgomery's Pair Correlation Conjecture, we investigate the distribution of the sequences $(\log|ζ(ρ+z)|)$ and $(\argζ(ρ+z)).$ Here $ρ=\frac12+iγ$ runs over the nontrivial zeros of the zeta-function, $0<γ\leq T,$ $T$ is a large real number, and $z=u+iv$ is a nonzero complex number of modulus $\ll 1/\log T.$ Our approach proceeds via a study of the integral moments of these sequences. If we let $z$ tend to $0$ and further assume that all the zeros $ρ$ are simple, we can replace the pair correlation conjecture with a weaker spacing hypothesis on the zeros and…
In this paper, we prove the lower bound of the unconditional large gap is 3.5555 which improves the obtained value 3.079 in the literature. Next, on the hypothesis that the moments of the Hardy Z-function and its derivatives are correctly predicted we establish a new explicit formula of the gaps and use it to establish some lower bounds for k=3,4,...,15. In particular it is proved that lower bound when k=15 is 9.6435 which means that the consecutive nontrivial zeros often differ by at least 9.6435 times the average spacing. The main results are proved by employed an integral inequality with a …
We postulate the existence of a self-adjoint operator associated to a system with countably infinite number of degrees of freedom whose spectrum is the sequence of the nontrivial zeros of the Riemann zeta function. We assume that it describes a massive scalar field coupled to a background field in a $(d+1)$-dimensional flat space-time. The scalar field is confined to the interval $[0,a]$ in one dimension and is not restricted in the other dimensions. The renormalized zero-point energy of this system is presented using techniques of dimensional and analytic regularization. In even dimensional s…
We investigate the distribution of the fractional parts of ag, where a is a fixed non-zero real number and g runs over the imaginary parts of the non-trivial zeros of the Riemann zeta function. The revision includes several minor corrections.
We consider a model of the Riemann zeta function on the critical axis and study its maximum over intervals of length $(\log T)^θ$, where $θ$ is either fixed or tends to zero at a suitable rate. It is shown that the deterministic level of the maximum interpolates smoothly between the ones of log-correlated variables and of i.i.d. random variables, exhibiting a smooth transition 'from $\frac34$ to $\frac14$' in the second order. This provides a natural context where extreme value statistics of log-correlated variables with time-dependent variance and rate occur. A key ingredient of the proof is …
V. I. Vasyunin has introduced special sequences of step functions related to the strong Nyman-Beurling criterion that converge pointwise to 1 in $[1,\infty)$. We show here that the first and simplest such sequence considered by Vasyunin diverges in $L_1((1,\infty),x^{-2}dx)$, which of course precludes the $L_2((1,\infty),x^{-2}dx)$-convergence needed for the Riemann hypothesis. Whether all sequences considered by this author also diverge remains an interesting open question.
