Related papers: A Riemann sum upper bound in the Riemann-Lebesque …

In previous work, the first author obtained conjecturally sharp upper bounds for the joint moments of the $(2k-2h)^{\text{th}}$ power of the Riemann zeta function with the $2h^{\text{th}}$ power of its derivative on the critical line in the…

We prove the inequality sum_{k=1}^infty (-1)^{k+1} r^k cos(k*phi) (k+2)^{-1} < sum_{k=1}^infty(-1)^{k+1} r^k (k+2)^{-1} for 0 < r <= 1 and 0 < phi < pi. For the case r = 1 we give two proofs. The first one is by means of a general numerical…

We improve on previous upper bounds for the $q$th norm of the partial sums of the Riemann zeta function on the half line when $0<q\leqslant 1$. In particular, we show that the 1-norm is bounded above by $(\log N)^{1/4}(\log\log N)^{1/4}$.

Let $\pi S(t)$ denote the argument of the Riemann zeta-function at the point $\frac12+it$. Assuming the Riemann Hypothesis, we sharpen the constant in the best currently known bounds for $S(t)$ and for the change of $S(t)$ in intervals. We…

We prove, in respect of an arbitrary Hecke congruence subgroup \Gamma =\Gamma_0(q_0) of the group SL(2,Z[i]), some new upper bounds (or `spectral large sieve inequalities') for sums involving Fourier coefficients of \Gamma -automorphic cusp…

This manuscript includes some classical results we select apart from the new results we've found on the Analysis of Boolean Functions and Fourier-Entropy-Influence conjecture. We try to ensure the self-completeness of this work so that…

This paper shows how the Lebesgue integral can be obtained as a Riemann sum and provides an extension of the Morse Covering Theorem to open sets. Let $X$ be a finite dimensional normed space; let $\mu$ be a Radon measure on $X$ and let…

Assuming the Generalized Riemann Hypothesis, we provide uniform upper bounds with explicit main terms for moduli of $\left(\cL'/\cL\right)(s)$ and $\log{\cL(s)}$ for $1/2+\delta\leq\sigma<1$, fixed $\delta\in(0,1/2)$ and for functions in…

Assuming the generalized Riemann hypothesis and a bound for the negative discrete moments of the Riemann zeta function (resp. Dirichlet $L$-functions), we prove the existence of a logarithmic limiting distribution for the normalized partial…

For $L(\cdot,\pi)$ in a large class of $L$-functions, assuming the generalized Riemann hypothesis, we show an explicit bound for the function $S_1(t,\pi)=\frac{1}{\pi}\int_{1/2}^\infty\log|L(\sigma+it,\pi)|\,d\sigma$, expressed in terms of…

An elementary proof of a quantitative version of the Riemann-Lebesgue lemma for functions supported on the half line is given. Applications to differential models with memory are discussed.

The Cram\'er-Granville conjecture is an upper bound on prime gaps, $g_n = p_{n+1} - p_n < \cCramer \, \log^2 p_n$ for some constant $\cCramer \geq 1$. Using a formula of Selberg, we first prove the weaker summed version: $\sum_{n=1}^N g_n <…

We obtain reasonably tight upper and lower bounds on the sum $\sum_{n \leqslant x} \varphi \left( \left\lfloor{x/n}\right\rfloor\right)$, involving the Euler functions $\varphi$ and the integer parts $\left\lfloor{x/n}\right\rfloor$ of the…

This work proves pointwise convergence of the truncated Fourier double integral of non-Lebesgue integrable bounded variation functions. This leads to the Dirichlet-Jordan theorem proof for non-Lebesgue integrable functions, which has not…

Let $\bf f$ be a primitive Hilbert cusp form of weight $k$ and level $\mathfrak{n}$ with Fourier coefficients $c_{\bf f}(\mathfrak{m})$. We prove a non-trivial upper bound for almost all Fourier coefficients $c_{\bf f}(\mathfrak{m})$ of…

A Fourier restriction estimate is obtained for a broad class of conic surfaces by adding a weight to the usual underlying measure. The new restriction estimate exhibits a certain affine-invariance and implies the sharp $L^p-L^q$ restriction…

Let $\lambda$ denote the Liouville function. We prove that $$\sum_{X \leq x < 2X} \sup_{\alpha \in \mathbb{R}/\mathbb{Z}} \bigg\lvert\!\sum_{x \leq n < x+H} \lambda(n) e(n\alpha)\bigg\rvert = o(HX)$$ as $X\to \infty$, in the regime $H =…

Let $R_k(x)$ denote the error incurred by approximating the number of $k$-free integers less than $x$ by $x/\zeta(k)$. It is well known that $R_k(x)=\Omega(x^{\frac{1}{2k}})$, and widely conjectured that…

We establish a lower bound and an upper bound to the sum of the Fractional-Logarithmic Laplacian. A main challenge in such a study comes from the fact that this operator has a Fourier symbol that is not globally monotone in its radial…

We substantially apply the Li criterion for the Riemann hypothesis to hold. Based upon a series representation for the sequence \{\lambda_k\}, which are certain logarithmic derivatives of the Riemann xi function evaluated at unity, we…