Related papers: The Riemann Zeta-Function and the Sine Kernel

In this paper we study two multicritical correlation kernels and prove that they converge to the Pearcey kernel in a certain double scaling limit. The first kernel appears in a model of non-intersecting Brownian motions at a tacnode. The…

We obtain conditional upper bounds for negative discrete moments of the derivative of the Riemann zeta-function averaged over a subfamily of zeros of the zeta function which is expected to have full density inside the set of all zeros. For…

We consider iterated integrals of $\log\zeta(s)$ on certain vertical and horizontal lines. Here, the function $\zeta(s)$ is the Riemann zeta-function. It is a well known open problem whether or not the values of the Riemann zeta-function on…

We prove a lower bound on the maximum of the Riemann zeta function in a typical short interval on the critical line. Together with the upper bound from the previous work of the authors, this implies tightness of $$ \max_{|h|\leq…

A variant for the Hilbert and Polya spectral interpretation of the Riemann zeta function is proposed. Instead of looking for a self-adjoint linear operator H, whose spectrum coincides with the Riemann zeta zeros, we look for the complex…

Keating and Snaith modeled the Riemann zeta-function \zeta(s) by characteristic polynomials of random NxN unitary matrices, and used this to conjecture the asymptotic main term for the 2k-th moment of \zeta(1/2+it) when k>-1/2. However, an…

As a generalization of the Dedekind zeta function, Weng defined the high rank zeta functions and proved that they have standard properties of zeta functions, namely, meromorphic continuation, functional equation, and having only two simple…

Already in 1734 Euler found a short explicit formula for the value of Riemann zeta function Zeta(s) when the argument s equals a positive integer 2n where n=1,2,3,. No such formula exists for odd positive integer arguments of Zeta. The…

Assuming the existence of a sequence of exceptional discriminants of quadratic fields, we show that a hundred percent of zeros of the Riemann zeta function are on the critical line in specific segments. This is a special case of a more…

I present two independent proofs of the Riemann Hypothesis considered by many the greatest unsolved problem in mathematics. I find that the admissible domain of complex zeros of the Riemann Zeta Function is the critical line. The methods…

We examine published arguments which suggest that the Riemann Hypothesis may not be true. In each case we provide evidence to explain why the claimed argument does not provide a good reason to doubt the Riemann Hypothesis. The evidence we…

The Riemann hypothesis states that all nontrivial zeros of the zeta function lie on the critical line $\Re(s)=1/2$. Hilbert and P\'olya suggested a possible approach to prove it, based on spectral theory. Within this context, some authors…

We prove an equivalent of the Riemann hypothesis in terms of the functional equation (in its asymmetrical form) and the $a$-points of the zeta-function, i.e., the roots of the equation $\zeta(s)=a$, where $a$ is an arbitrary fixed complex…

A new definition for the Dirichlet beta function for positive integer arguments is discovered and presented for the first time. This redefinition of the Dirichlet beta function, based on the polygamma function for some special values,…

This article proves the Riemann hypothesis, which states that all non-trivial zeros of the zeta function have a real part equal to 1/2. We inspect in detail the integral form of the (symmetrized) completed zeta function, which is a product…

We analyse a collection of twisted mixed moments of the Riemann zeta function and establish the validity of asymptotic formulae comprising on some instances secondary terms of the shape $P(\log T) T^{C}$ for a suitable constant $C<1$ and a…

In this unpublished note, we sketch an idea of using a three-piece mollifier to slightly improve the known percentages of zeros and simple zeros of the Riemann zeta-function on the critical line. This uses the recent result of Bettin, Bui,…

Applying Littlewood's lemma in connection to Riemann's Hypothesis and exploiting the symmetry of Riemann's $xi$ function we show that almost all nontrivial Riemann's Zeta zeros are on the critical line.

We prove Riemann hypothesis. Method is to show the convexity of function which has zeros on open critical strip the same as zeta function.

We give an overview of the intimate connections between natural direct and inverse spectral problems for fractal strings, on the one hand, and the Riemann zeta function and the Riemann hypothesis, on the other hand (in joint works of the…