Related papers: Zeros of Jensen polynomials and asymptotics for th…

We show that there are an infinite number of Riemann zeros on the critical line, enumerated by the positive integers $n=1,2,\dotsc$, whose ordinates can be obtained as the solution of a new transcendental equation that depends only on $n$.…

Riemann numerically approximated at least three zeta zeros. According to Edwards, Riemann even took steps to verify that the lowest zero he computed was indeed the first zeta zero. This approach to verification is developed, improved, and…

We derive Mehler--Heine type asymptotic formulas for Charlier and Meixner polynomials, and also for their associated families. These formulas provide good approximations for the polynomials in the neighborhood of $x=0,$ and determine the…

The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line $\Re(s)=1/2$. Hilbert and P\'olya suggested that one possible way to prove the Riemann hypothesis is to interpret the nontrivial zeros in…

Under the Riemann Hypothesis, we prove for any natural number $r$ there exist infinitely many large natural numbers $n$ such that $(\gamma_{n+r}-\gamma_n)/(2\pi /\log \gamma_n) > r + \Theta\sqrt{r}$ and $(\gamma_{n+r}-\gamma_n)/(2\pi /\log…

An asymptotic formula is presented for the number of planar lattice convex polygonal lines joining the origin to a distant point of the diagonal. The formula involves the non-trivial zeros of the zeta function and leads to a necessary and…

Fix $\delta\in(0,1]$, $\sigma_0\in[0,1)$ and a real-valued function $\varepsilon(x)$ for which $\limsup_{x\to\infty}\varepsilon(x)\le 0$. For every set of primes ${\mathcal P}$ whose counting function $\pi_{\mathcal P}(x)$ satisfies an…

Suppose that the Riemann hypothesis is false and $\rho_{*} = 1/2 + \eta_{*} + i \gamma_{*}$, $\eta_{*} > 0$, is a nontrivial zero of the Riemann $\zeta$-function off the critical line. Under the negation of the Riemann hypothesis for the…

A Hamiltonian operator $\hat H$ is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical…

Assuming the Riemann Hypothesis (RH), Montgomery proved a theorem concerning pair correlation of zeros of the Riemann zeta-function. One consequence of this theorem is that, assuming RH, at least $67.9\%$ of the nontrivial zeros are simple.…

Several arguments against the truth of the Riemann hypothesis are extensively discussed. These include the Lehmer phenomenon, the Davenport-Heilbronn zeta-function, large and mean values of $|\zeta(1/2+it)|$ on the critical line, and zeros…

We show that certain sums of products of Hermite-Biehler entire functions have only real zeros, extending results of Cardon. As applications of this theorem we construct sums of exponential functions having only real zeros, we construct…

Recently, we have established the generalized Li's criterion equivalent to the Riemann hypothesis, viz. demonstrated that the sums over all non-trivial Riemann function zeroes k_n,b=Sum_rho(1-(1-((rho+b)/(rho-b-1))**n) for any real b not…

For $n \times n$ matrices $A$ and $B$ define $$\eta(A,B)=\sum_{S}\det(A[S])\det(B[S']),$$ where the summation is over all subsets of $\{1,..., n\}$, $S'$ is the complement of $S$, and $A[S]$ is the principal submatrix of $A$ with rows and…

It is shown that, for open sets in classical potential theory and - more generally - for elliptic harmonic spaces, the set of Jensen measures for a point is a simple union of closed faces of a compact convex set which has been thoroughly…

We use expansions with functions related to some special functions such as Hermite or Laguerre to get some conjectural expansions of the Riemann Zeta function in the critical strip involving a set of polynomials which have their zeros on…

The transformations of the sum identities for generalized harmonic and oscillatory numbers, obtained earlier in our recent report [1], enable us to derive the new identities expressed in terms of the corresponding square roots of x. At…

We consider transcendental entire solutions of linear $q$-difference equations with polynomial coefficients and determine the asymptotic behavior of their Taylor coefficients. We use this to show that under a suitable hypothesis on the…

In this work we consider an equation for the Riemann zeta-function in the critical half-strip. With the help of this equation we prove that finding non-trivial zeros of the Riemann zeta-function outside the critical line would be equivalent…

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