Related papers: An explicit mean-value estimate for the PNT in int…

We study the arithmetic (real) function f=g*1, with g "essentially bounded" and supported over the integers of [1,Q]. In particular, we obtain non-trivial bounds, through f "correlations", for the "Selberg integral" and the "symmetry…

The aim of this paper is to give a direct interpretation of the validity of the Riemann hypothesis up to a certain height $T$ in terms of the prime-counting function $\pi(x)$. This is done by proving the well-known explicit Schoenfeld bound…

95 years ago Hoheisel proved the existence of primes in the sub-linear interval \[ \left[x, x+x^{1-{1\over 33000}}\right] \qquad \hbox{for $x$ sufficiently large}. \] This was improved by Heilbronn, proving existence of primes in the…

This paper gives some results for the logarithm of the Riemann zeta-function and its iterated integrals. We obtain a certain explicit approximation formula for these functions. The formula has some applications, which are related with the…

We improve the universality theorem of the Riemann zeta-function in short intervals by establishing universality for significantly shorter intervals $[T,T+H]$. Assuming the Riemann Hypothesis, we prove that universality in such short…

We prove a kind of "almost all symmetry" result for the primes, i.e. we give non-trivial bounds for the "symmetry integral", say $I_{\Lambda}(N,h)$, of the von Mangoldt function $\Lambda(n)$ ($:= \log p$ for prime-powers $n=p^r$, 0…

We give two improved explicit versions of the prime number theorem for primes in arithmetic progression: the first isolating the contribution of the Siegel zero and the second completely explicit, where the improvement is for medium-sized…

Numerical study of the distribution of the Riemann zeros differences following the work [1] shows the significance of the function for which the prime sum expression is proposed. Computational results related to this definition explored…

In this note, we generalise two results on prime numbers in short intervals. The first result is Ingham's theorem which connects the zero-density estimates with short intervals where the prime number theorem holds, and the second result is…

Using Watt's mean value theorem and a delicate sieve decomposition, the author shows that the interval $[x, x+x^{\frac{1}{2}+\varepsilon}]$ contains an integer with a prime factor larger than $x^{\frac{35}{36}-\varepsilon}$ for sufficiently…

The Goldbach conjecture that, every even integer is the sum of two primes, has been open since 1742. This paper details a road map to a proof of Goldbachs conjecture based on a function that estimates the number of Goldbach pairs. It is…

We show that, on the Riemann hypothesis, $\limsup_{X\to\infty}I(X)/X^{2} \leq 0.8603$, where $I(X) = \int_X^{2X} (\psi(x)-x)^2\,dx.$ This proves (and improves on) a claim by Pintz from 1982. We also show unconditionally that…

Let $N$ be a sufficiently large, odd integer. We prove an asymptotic formula for the number of representations of $N$ as the sum of three primes, one of which is smaller than a given $U$. By inserting the currently best zero-density…

Let $0<\gamma_1\leq \gamma_2\leq \ldots$ denote the positive ordinates of the non-trivial zeros of the Riemann zeta-function. A result first announced by Selberg states that there exist absolute constants $\Theta, \vartheta>0$ such that for…

This paper presents some considerations about the Goldbach's conjecture (GC). The work is based on elementary results of the number theory and it provides a constructive method that permits, given an even integer, to find at least a pair of…

We prove that for every nonnegative integer $m$ there exists an $\varepsilon>0$ such that if $\lambda\in (0,\varepsilon]$ and $x$ is sufficiently large in terms of $m$, then the number of positive integers $n\leq x$ for which the interval…

The Schinzel hypothesis is a famous conjectural statement about primes in value sets of polynomials, which generalizes the Dirichlet theorem about primes in an arithmetic progression. We consider the situation that the ring of integers is…

Let $\Lambda\left(n\right)$ be the Von Mangoldt function, let \[ r_{G}\left(n\right)=\underset{{\scriptstyle m_{1}+m_{2}=n}}{\sum_{m_{1},m_{2}\leq n}}\Lambda\left(m_{1}\right)\Lambda\left(m_{2}\right), \] \[…

Conditionally on the Riemann Hypothesis we obtain bounds of the correct order of magnitude for the 2k-th moment of the Riemann zeta-function for all positive real k < 2.181. This provides for the first time an upper bound of the correct…

We approach a new proof of the strong Goldbach's conjecture for sufficiently large even integers by applying the Dirichlet's series. Using the Perron formula and the Residue Theorem in complex variable integration, one could show that any…