Related papers: Exploring Riemann's Functional Equation

Let $\Xi(t)$ be a function relating to the Riemann zeta function $\zeta (s)$ with $s = \frac{1} {2} + it$. In this paper, we construct a function $v$ containing $t$ and $\Xi(t)$, and prove that $v$ satisfies a nonadjoint boundary value…

This paper investigates the analytic properties of the Liouville function's Dirichlet series that obtains from the function F(s)= zeta(2s)/zeta(s), where s is a complex variable and zeta(s) is the Riemann zeta function. The paper employs a…

We show that if the derivative of the Riemann zeta function has sufficiently many zeros close to the critical line, then the zeta function has many closely spaced zeros. This gives a condition on the zeros of the derivative of the zeta…

We investigate the intersections of the curve $\mathbb{R}\ni t\mapsto \zeta({1\over 2}+it)$ with the real axis. We show unconditionally that the zeta-function takes arbitrarily large positive and negative values on the critical line.

We introduce a new method to detect the zeros of the Riemann zeta function which is sensitive to the vertical distribution of the zeros. This allows us to prove there are few `half-isolated' zeros. By combining this with classical methods,…

We settle a conjecture of Farmer and Ki in a stronger form. Roughly speaking we show that there is a positive proportion of small gaps between consecutive zeros of the zeta-function $\zeta(s)$ if and only if there is a positive proportion…

We study the value-distribution of the Riemann zeta-function and related functions on and near the critical line. Amongst others, we focus on the following: The critical line is a natural boundary of the Voronin-type universality property…

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…

We have studied some properties of the special Gram points of the Riemann zeta function which lie on contour lines ${\bf Im}(\zeta ( s )) = 0$ which do not contain zeroes of $\zeta ( s )$. We find that certain functions of these points,…

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.…

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 $n$.…

In this paper, we show that Riemann hypothesis (concerning zeros of the zeta function in the critical strip) is equivalent to the analytic continuation of Euler products obtained by restricting the Euler zeta product to suitable subsets…

This paper shows that, in the critical strip, the Riemann zeta function $\zeta(s)$ have the same set of zeros as $F(s):=\int_0^\infty t^{s-1}(e^t+1)^{-1}dt$, and then discusses the behavior of $F(s)$.

The Riemann Hypothesis (RH), one of the most profound unsolved problems in mathematics, concerns the nontrivial zeros of the Riemann zeta function. Establishing connections between the RH and physical phenomena could offer new perspectives…

A strategy for proving (not a proof of, as was the first over-optimistic belief) the Riemann hypothesis is suggested. The vanishing of Riemann Zeta reduces to an orthogonality condition for the eigenfunctions of a non-Hermitian operator D^+…

Montgomery in 1973 introduced the pair correlation method to study the vertical distribution of Riemann zeta-function zeros. This work assumed the Riemann Hypothesis (RH). One striking application was a short proof that at least 2/3 of…

We present the first applications of the recently established by us (arXiv:1304.7895; Ukrainian Math. J. - 2014. -66. - P. 371-383) generalized Li's criterion equivalent to the Riemann Hypothesis. This criterion is the statement that the…

We prove a general result on representing the Riemann zeta function as a convergent infinite series in a complex vertical strip containing the critical line. We use this result to re-derive known expansions as well as to discover new series…

Some computations made about the Riemann Hypothesis and in particular, the verification that zeroes of zeta belong on the critical line and the extension of zero-free region are useful to get better effective estimates of number theory…

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.