Related papers: Admissibility of H\"ormander--Bernhardsson extrema…

The Keiper--Li sequence $\{ \lambda_n \}$ is most sensitive to the Riemann Hypothesis asymptotically ($n \to \infty$), but highly elusive both analytically and numerically. We deform it to fully explicit sequences, simpler to analyze and to…

We present another view dealing with the Arnold-Givental conjecture on a real symplectic manifold $(M, \omega, \tau)$ with nonempty and compact real part $L={\rm Fix}(\tau)$. For given $\Lambda\in (0, +\infty]$ and $m\in\N\cup\{0\}$ we show…

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…

Make an exponential transformation in the integral formulation of Riemann's zeta-function zeta(s) for Re(s) > 0. Separately, in addition make the substitution s -> 1 - s and then transform back to s again using the functional equation.…

We consider a model of 2D gravity with the coefficient of the Einstein-Hilbert action having an imaginary part $\pi/2$. This is equivalent to introduce a $\Theta$-vacuum structure in the genus expansion whose effect is to convert the…

We consider the asymptotic expansion of the functional series \[S_{\mu}^\pm(a;\lambda)=\sum_{n=0}^\infty \frac{(\pm 1)^n e^{-\lambda n}}{(n^2+a^2)^\mu}\] for $\lambda>0$ and $\mu\geq0$ as $|a|\to \infty$ in the sector $|\arg\,a|<\pi/2$. The…

We consider a pure death process $(Z(t), t\ge0)$ with death rates $\lambda_n$ satisfying the condition $\sum_{n=2}^\infty \lambda_n^{-1}<\infty$ of coming from infinity, $Z(0)=\infty$, down to an absorbing state $n=1$. We establish limit…

We prove formulas for the Bernoulli numbers by using the Newton-Girard identities to evaluate the Riemann zeta function at positive even integers. To do this, we define a sequence of positive integers, a sequence of polynomials, and a…

We prove some identities, which involve the non-trivial zeros of the Riemann zeta function. From them we derive some convergent asymptotic expansions related to the work by Cram\'er, and also new representations for some arithmetical…

The linear $\delta$ expansion is used to obtain corrections up to O$(\delta^2)$ to the self-energy for a complex scalar field theory with a $\lambda (\phi^{\star}\phi)^2$ interaction at high temperature and non-zero charge density. The…

We obtain a uniform ergodic theorem for the sequence $\frac1{s(n)} \sum_{k=0}^n(\varDelta s)(n-k)\,T^k$, where $\varDelta$ is the inverse of the endomorphism on the vector space of scalar sequences which maps each sequence into the sequence…

Let $\lambda =\left( \lambda_{1},\lambda_{2},...,\lambda_{r}\right) $ be an integer partition, and $\left[p_{\lambda }\right] $ the $q$-analog of the symmetric power function $%p_{\lambda }$. This $q$-analogue has been defined as a special…

We extend the classical van der Corput inequality to the real line. As a consequence, we obtain a simple proof of the Wiener-Wintner theorem for the $\mathbb{R}$-action which assert that for any family of maps $(T_t)_{t \in \mathbb{R}}$…

This paper studies zeta functions of the form $\sum_{n=1}^{\infty} \chi(n) n^{-s}$, with $\chi$ a completely multiplicative function taking only unimodular values. We denote by $\sigma(\chi)$ the infimum of those $\alpha$ such that the…

This paper deals with an attempt of proof of the Riemann Hypothesis (RH). Let $T>10^{10}$ arbitrarily large. Let the region $\Omega_T=\Big\{z=x+i y\ \Big|\ \frac{1}{2}<x<1, \ 0<y<T\Big\}.$ There is a finite number $N_T$ of roots of…

Following the seminal works of Asorey-Ibort-Marmo and Mu\~{n}oz-Casta\~{n}eda-Asorey about selfadjoint extensions and quantum fields in bounded domains, we compute all the heat kernel coefficients for any strongly consistent selfadjoint…

Some recent results concerning the Sherrington-Kirkpatrick model are reported. For $T$ near the critical temperature $T_c$, the replica free energy of the Sherrington-Kirkpatrick model is taken as the starting point of an expansion in…

For a class of Laplace exponents we derive the heat trace asymptotics of the generator of the corresponding subordinate Brownian motion on Euclidean space. The terms in the asymptotic expansion are found to depend both on the geometry of…

For a class of generalized holomorphic Eisenstein series, we establish complete asymptotic expansions (Theorems~1~and~2), which together with the explicit expression of the latter remainder (Theorem~3), naturally transfer to several new…

We establish for $2 \le k \le n-1$ the strict concavity of the function $f_k(\lambda)=\log(\sigma_k(\lambda))$ on a subset of the positive cone $\Gamma_n=\{\lambda=(\lambda_{1}, \lambda_{2}, \cdots,\lambda_{n})\in \mathbb{R}^n;…