Related papers: Generalized Bombieri - Lagarias' theorem and gener…

Riemann conjectured that all the zeros of the Riemann $\Xi$-function are real, which is now known as the Riemann Hypothesis (RH). In this article we introduce the study of the zeros of the truncated sums $\Xi_N(z)$ in Riemann's uniformly…

Gonek, Graham, and Lee have shown recently that the Riemann Hypothesis (RH) can be reformulated in terms of certain asymptotic estimates for twisted sums with von Mangoldt function $\Lambda$. Building on their ideas, for each…

In this paper, we establish a new mean value theorem of Bombieri-Vinogradov type over Piatetski-Shapiro sequence. Namely, it is proved that for any given constant $A>0$ and any sufficiently small $\varepsilon>0$, there holds…

We derive an asymptotic formula for the sum $$ H = \sum_{0<\gamma_k\leqslant T,\, 1\leqslant k\leqslant m}h(a_1\gamma_1+a_2\gamma_2+\cdots + a_m\gamma_m), $$ where $a_1, a_2, \ldots, a_m$ are integers whose sum equals zero, $\gamma_1,…

We study the Lagarias inequality, an elementary criterion equivalent to the Riemann Hypothesis. Using a continuous extension of the harmonic numbers, we show that the sequence $B_n=\frac{H_n+e^{H_n}\log(H_n)}{n}$ is strictly increasing for…

For any real $\beta_0\in[\tfrac12,1)$, let ${\rm GRH}[\beta_0]$ be the assertion that for every Dirichlet character $\chi$ and all zeros $\rho=\beta+i\gamma$ of $L(s,\chi)$, one has $\beta\le\beta_0$ (in particular, ${\rm GRH}[\frac12]$ is…

Recently, Grynkiewicz et al. [{\it Israel J. Math.} {\bf 193} (2013), 359--398], using tools from additive combinatorics and group theory, proved necessary and sufficient conditions under which the linear congruence $a_1x_1+\cdots…

For sums $S_n=\sum_{k=1}^n X_k$, $n\ge 1$ of independent random variables $ X_k $ taking values in $\Z$ we prove, as a consequence of a more general result, that if (i) For some function $1\le \phi(t)\uparrow \infty $ as $t\to \infty$, and…

For each positive integer $n$, function $f$, and point $c$, the GGR Theorem states that $f$ is $n$ times Peano differentiable at $c$ if and only if $f$ is $n-1$ times Peano differentiable at $c$ and the following $n$-th generalized…

We present a complete algebraic classification of the Lie symmetries for generalized Zoomeron equations. For the generalized 1+1 and 2+1 Zoomeron equations we solve the Lie symmetry conditions in order to constrain the free functions of the…

From Bombieri's mean value theorem one can deduce the prime number theorem being equivalent to the Riemann hypothesis and the least prime P(q) satisfying P(q)= O(q^2 [ln q]^32) in any arithmetic progressions with common difference q.

We consider Mertens' function M(x,q,a) in arithmetic progression, Assuming the generalized Riemann hypothesis (GRH), we show an upper bound that is uniform for all moduli which are not too large. For the proof, a former method of K.…

Let $\Lambda$ be the von Mangoldt function and $(r_G(n) = \sum_{m_1 + m_2 = n} \Lambda(m_1) \Lambda(m_2))$ be the counting function for the Goldbach numbers. Let $N \geq 2$ be an integer. We prove that $$\begin{align} &\sum_{n \le N} r_G(n)…

Exact and asymptotic formulae are displayed for the coefficients $\lambda_n$ used in Li's criterion for the Riemann Hypothesis. For $n \to \infty$ we obtain that if (and only if) the Hypothesis is true, $\lambda_n \sim n(A \log n +B)$ (with…

We study the algebraic properties of Generalized Laguerre polynomials for negative integral values of a given parameter which is $L_{n}^{(-1-n-r)}(x)= \sum\limits_{j=0}^{n} \binom{n-j+r}{n-j} \frac{x^{j}}{j!}$ for integers $r\geq 0, n\geq…

We study the algebraic properties of Generalized Laguerre Polynomials for negative integral values of the parameter. For integers $r,n\geq 0$, we conjecture that $L_n^{(-1-n-r)}(x) = \sum_{j=0}^n \binom{n-j+r}{n-j}x^j/j!$ is a…

If a sequence $(a_n)$ of non-negative real numbers has ``best possible'' distribution in arithmetic progressions, Bombieri showed that one can deduce an asymptotic formula for the sum $\sum_{n\le x} a_n \Lambda_k(n)$ for $k\ge 2$. By…

We prove the following result: Let $N \geq 2$ and assume the Riemann Hypothesis (RH) holds. Then \[ \sum_{n=1}^{N} R(n) =\frac{N^{2}}{2} -2 \sum_{\rho} \frac{N^{\rho + 1}}{\rho (\rho + 1)} + O(N \log^{3}N), \] where $\rho=1/2+i\gamma$ runs…

We present a different proof of the following identity due to Munarini, which generalizes a curious binomial identity of Simons. \begin{align*} \sum_{k=0}^{n}\binom{\alpha}{n-k}\binom{\beta+k}{k}x^k…

J. Marcinkiewicz and A. Zygmund proved in 1936 that, for all functions $f$ and points $x$, the existence of the $n$th Peano derivative $f_{(n)}(x)$ is equivalent to the existence of both $f_{(n-1)}(x)$ and the $n$th generalized Riemann…