Related papers: Decoupling, exponential sums and the Riemann zeta …

We give an informal introduction to the most basic techniques used to evaluate moments on the critical line of the Riemann zeta-function and to find asymptotics for sums of arithmetic functions.

Let $t$ be random and uniformly distributed in the interval $[T,2T]$, and consider the quantity $N(t+1/\log T) - N(t)$, a count of zeros of the Riemann zeta function in a box of height $1/\log T$. Conditioned on the Riemann hypothesis, we…

The purpose of the present paper is to provide a general overview of a variety of results related to a category of cotangent sums which have been proven to be associated to the so-called Nyman-Beurling criterion for the Riemann Hypothesis.…

The general relation between the standard expansion coefficients and the beta function for the QCD coupling is exactly derived in a mathematically strict way. It is accordingly found that an infinite number of logarithmic terms are lost in…

We derive a central limit theorem for sums of a function of independent sums of independent and identically distributed random variables. In particular we show that previously known result from Rempa\la and Weso\lowski (Statist. Probab.…

In this paper we use operation of crossbreeding on the set of six transmutations of corresponding asymptotic complete hybrid formulas from our previous paper. We obtain in result the set of fifteen exact meta-functional equations. Every of…

We establish pointwise convergence for nonconventional ergodic averages taken along $\lfloor p^c\rfloor$, where $p$ is a prime number and $c\in(1,4/3)$ on $L^r$, $r\in(1,\infty)$. In fact, we consider averages along more general sequences…

In this paper we obtain some essential generalizations of certain Ramachandra's inequality, i. e. we obtain new lower estimates for the energies of some complicated signals generated by the Riemann zeta-function on the critical line.

We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}\tau_k(n)e(n\alpha)\bigg|d\alpha$ where $\tau_k(n) = \sum_{d_1\dots d_k = n} 1$. This follows from a quick application of the circle method. Along the way, we find minor arc…

For $N \in \mathbb{N}$ consider the $N$-th section of the approximate functional equation $$ \zeta_N(s)= \sum_{n =1 }^N B_n(s),$$ where $$ B_n(s)= \frac{1}{2} \left [ n^{-s} + \chi(s) \cdot n^{s-1} \right ].$$ Our aim in this work is to…

In this paper, some new results are reported for the study of Riemann zeta function $\zeta(s)$ in the critical strip $0<Re(s)<1$, such as $\zeta(s)$ expressed in a generalized Euler product only involving prime numbers. Particularly, some…

In this paper we extend a result on the asymptotics of moments of certain cotangent sums associated to the Estermann and Riemann zeta functions established in a previous paper for integer exponents to arbitrary positive real exponents.

Let $\alpha \in (1/2,1)$ be fixed. We prove that $$ \max_{0 \leq t \leq T} |\zeta(\alpha+it)| \geq \exp\left(\frac{c_\alpha (\log T)^{1-\alpha}}{(\log \log T)^\alpha}\right) $$ for all sufficiently large $T$, where we can choose $c_\alpha =…

Explicit estimates for the Riemann zeta-function on the $1$-line are derived using various methods, in particular van der Corput lemmas of high order and a theorem of Borel and Carath\'{e}odory.

We introduce iterated beta integrals, a new class of iterated integrals on the universal abelian covering of the punctured projective line that unifies hyperlogarithms and classical beta integrals while preserving their fundamental…

The function $S_n (t) = \pi \left( \frac{3}{2} - {frac} \left( \frac{\vartheta(t)}{\pi} \right) + \left( \lfloor \frac{t \ln \left( \frac{t}{2 \pi e}\right)}{2 \pi} + \frac{7}{8} \rfloor - n \right) \right)$ is conjectured to be equal to $S…

As well known, the important hypothesis formulated by B.G. RIEMANN in 1859 states that all non-trivial zeroes of the Zeta function $Z(s)=\sum_{n=1}^{\infty } n^{-s}$ should fall on the Critical Line (C.L.) $Re(s)=\frac{1}{2}$.\\ Although…

Assuming the Riemann Hypothesis, Goldston, Gonek and Montgomery \cite{GGM} studied the second moment of the log-derivative of $\zeta$, shifted away from the half line by $a/\log T$, and its connection with the pair correlation conjecture.…

Building on work in \cite{AB24} on the Riemann zeta function at height $T$ off the critical line, we prove an unconditional lower bound on the critical line for real large deviations of the order $V\sim\alpha\log\log T$ for any $\alpha>0.$…

We give explicit formulae for all of the terms in the asymptotic expansion of the mean fourth power of the Riemann zeta-function on the critical line.