225 research outputs found
Prime numbers in logarithmic intervals
No full textLet be a large parameter. We will first give a new estimate for the integral moments of primes in short intervals of the type , where is a prime number and h=\odi{X}. Then we will apply this to prove that for every there exists a positive proportion of primes such that the interval contains at least a prime number. As a consequence we improve Cheer and Goldston's result on the size of real numbers with the property that there is a positive proportion of integers such that the interval contains no primes. We also prove other results concerning the moments of the gaps between consecutive primes and about the positive proportion of integers such that the interval contains at least a prime number. The last application of these techniques are two theorems (the first one unconditional and the second one in which we assume the validity of the Riemann Hypothesis and of a form of the Montgomery pair correlation conjecture) on the positive proportion of primes such that the interval contains no prime
On an average ternary problem with prime powers
No full textWe continue our work on averages for ternary additive problems with powers of prime numbers in Languasco and Zaccagnini (J Number Theory 159:45–58, 2016; Rocky Mountain J Math arXiv:1806.04934, 2018) and Cantarini et al. (Proc Amer Math Soc arXiv:1805.09008, 2018)
A note on Mertens' formula for arithmetic progressions
No full textWe study the Mertens product over primes in arithmetic progressions, and find a uniform version of previous results on the asymptotic formula, improving at the same time the size of the error term, and giving an alternative, simpler value for the constant appearing in the main term
A fast algorithm to compute the Ramanujan-Deninger Gamma function and some number-theoretic applications
No full textWe introduce a fast algorithm to compute the Ramanujan-Deninger gamma function and its logarithmic derivative at positive values. Such an algorithm allows us to greatly extend the numerical investigations about the Euler-Kronecker constants , and , where is an odd prime, runs over the primitive Dirichlet characters , is the principal Dirichlet character and is the Dirichlet -function associated to .
Using such algorithms we obtained that
and
thus getting a new negative value for .
Moreover we also computed , and for every odd prime , 10^6< q le 10^7, thus extending the results in Languasco (2019). As a consequence we obtain that both and are positive for every odd prime up to and that
rac{17}{20} log log q< M_q < rac{5}{4} log log q
for every odd prime 1531 < qle 10^7. In fact the lower bound holds true for q>13.
The programs used and the results here described are collected at the following address
\url{http://www.math.unipd.it/~languasc/Scomp-appl.html}
A singular series average and Goldbach numbers in short intervals
No full textLet \Sing(n) = 2
\prod\limits_{p>2}\left(1-\frac{1}{(p-1)^2}\right) \prod\limits_{\substack
{p\mid n\\ p>2}} \left(\frac{p-1}{p-2}\right) if is even and \Sing(n)
=0 if is odd, be the singular series of the Goldbach problem. Let
be a fixed real number. We prove that \sum_{n\leq X}
\Sing(n)^\nu = c_1X +c_2(\log X)^\nu + O((\log X)^{\nu-1/3}), where
and the implicit constant depend on . As a consequence, we
improve the known results on the positive proportion of Goldbach numbers in
short intervals
A Cesáro average for generalised Hardy-Littlewood numbers
No full textWe continue our recent work on additive problems with prime summands: we already studied the emph{average} number of representations of an integer as a sum of two primes, and also considered individual integers. Furthermore, we dealt with representations of integers as sums of powers of prime numbers. In this paper, we study a Ces`aro weighted partial emph{explicit} formula for generalised Hardy-Littlewood numbers (integers that can be written as a sum of a prime power and a square) thus extending and improving our earlier results
On a Diophantine problem with two primes and s powers of two
No full textWe refine a recent result of Parsell on the values of the form , where are prime numbers, are positive integers, is negative and irrational and , \lambda_2/\mu_2 \in \Q
A conditional result on Goldbach numbers in short intervals
No full textAssume the Riemann
Hypothesis and a weaker form of Montgomery's pair correlation conjecture,
i.e., for every where , , run over the imaginary part of the
non-trivial zeros of the Riemann zeta-function, holds uniformly for
, where . Then, for all sufficiently
large and , we have that the interval
contains a even integer which is a sum of two primes
A Diophantine problem with a prime and three squares of primes
No full textWe prove that if λ 1, λ 2, λ 3 and λ 4 are non-zero real numbers, not all of the same sign, λ 1/λ 2 is irrational, and π{variant} is any real number then, for any ε>0, the inequality |λ1p1+λ2p22+λ3p32+λ4p42+π{variant}|≤(maxjpj)-1/18+ε has infinitely many solutions in prime variables p 1,..., p 4. © 2012 Elsevier Inc
A Cesàro average for Hardy-Littlewood numbers
No full textLet be the von Mangoldt function and
be the counting function for the Hardy-Littlewood numbers. Let
be a sufficiently large integer.
We prove that
%
\begin{align*}
\sum_{n \le N} r_{\textit{HL}}(n) \frac{(1 - n/N)^k}{\Gamma(k + 1)}
&=
\frac{\pi^{1 / 2}}2 \frac{N^{3 / 2}}{\Gamma(k + 5 / 2)}
-
\frac 12 \frac{N}{\Gamma(k + 2)}
-
\frac{\pi^{1 / 2}}2
\sum_{\rho}
\frac{\Gamma(\rho)}{\Gamma(k + 3 / 2 + \rho)} N^{1 / 2 + \rho} \\
&+
\frac 12
\sum_{\rho}
\frac{\Gamma(\rho)}{\Gamma(k + 1 + \rho)} N^{\rho}
+
\frac{N^{3 / 4 - k / 2}}{\pi^{k + 1}}
\sum_{\ell \ge 1}
\frac{J_{k + 3 / 2} (2 \pi \ell N^{1 / 2})}{\ell^{k + 3 / 2}} \\
&-
\frac{N^{1 / 4 - k / 2}}{\pi^k}
\sum_{\rho} \Gamma(\rho) \frac{N^{\rho / 2}}{\pi^\rho}
\sum_{\ell \ge 1}
\frac{J_{k + 1 / 2 + \rho} (2 \pi \ell N^{1 / 2})}
{\ell^{k + 1 / 2 + \rho}}
+
\Odip{k}{1}.
\end{align*}
%
for , where runs over
the non-trivial zeros of the Riemann zeta-function
and denotes the Bessel function of complex order
and real argument
- …
