Related papers: Explicit formulae for primes in arithmetic progres…

Let $E_x(q,a)$ be the error term when counting primes in arithmetic progressions and let $M(Q)=\sum_{q\leq Q}\phi(q)\sum_{a=1}^qE_x(q,a)^3$. We show that $M(Q)<<Q^3(x/Q)^{7/5}$ for large $Q$ close to $x$ (in the usual BDH sense) thereby…

Let $q\ge 3$ be a non-exceptional modulus $q\ge3$, and let $a$ be a positive integer coprime with $q$. For any $\epsilon>0$, there exists $\alpha>0$ (computable), such that for all $x\ge \alpha (\log q)^2$, the interval $\left[…

We establish completely log-free bounds for exponential sums over the primes and the M\"{o}bius function. Let $0<\eta \leq 1/10$, and suppose $\alpha = a/q + \delta/x$, with $(a,q)=1$ and $|\delta| \leq x^{1/5 + \eta}/q$, and set $\delta_0…

Let $\chi$ be a Dirichlet character mod $D$ with $L(s,\chi)$ its associated $L$-function, and let $\psi(x,q,a)$ be Chebyshev's prime-counting function for primes congruent to $a$ modulo $q$. We show that under the assumption of an…

If a set S of pairwise coprime moduli q, less than x^(9/40), is considered, one obtains the expected behavior for primes up to x in arithmetic progressions mod q, except for a subset of S whose cardinality is bounded by a power of log x.

Let \beta be a real number. Then for almost all irrational \alpha>0 (in the sense of Lebesgue measure) \limsup_{x\to\infty}\pi_{\alpha,\beta}^*(x)(\log x)^2/x>=1, where \pi_{\alpha,\beta}^*(x)={p<=x: both p and [\alpha p+\beta] are primes}.

Let $a>1$. Denote by $l_a(p)$ the multiplicative order of $a$ modulo $p$. We look for an estimate of sum of $\frac{l_a(p)}{p-1}$ over primes $p\leq x$ on average. When we average over $a\leq N$, we observe a statistic of $C\mathrm{Li}(x)$.…

Let $A\subset\left\{ 1,\dots,N\right\} $ be a set of prime numbers containing no non-trivial arithmetic progressions. Suppose that $A$ has relative density $\alpha=|A|/\pi(N)$, where $\pi(N)$ denotes the number of primes in the set $\left\{…

We find a lower bound for the number of Chen primes in the arithmetic progression $a \bmod q$, where $(a,q)=(a+2,q)=1$. Our estimate is uniform for $q \leq \log^M x$, where $M>0$ is fixed.

For $(M,a)=1$, put \begin{equation*} G(X;M,a)=\sup_{p^\prime_n\leq X}(p^\prime_{n+1}-p^\prime_n), \end{equation*} where $p^\prime_n$ denotes the $n$-th prime that is congruent to $a\pmod{M}$. We show that for any positive $C$, provided $X$…

We prove new mean value theorems for primes in arithmetic progressions to moduli larger than $x^{1/2}$, extending the Bombieri-Vinogradov theorem to moduli of size $x^{1/2+\delta}$ which have conveniently sized divisors. The main feature of…

In this paper we show that for some constant $c>0$ and for any $A>0$ there exist some $x(A)>0$ such that, If $q\leq (\log x)^{A}$ then we have \begin{align} \Psi_z(x;\mathcal{N}_q(a,b),q) &= \frac{\Theta (z)}{2\phi(q)}x +…

We fix a non-zero integer $a$ and consider arithmetic progressions $a \bmod q$, with $q$ varying over a given range. We show that for certain specific values of $a$, the arithmetic progressions $a \bmod q$ contain, on average, significantly…

Let $a>1$ be an integer. Denote by $l_a(p)$ the multiplicative order of $a$ modulo primes $p$. We prove that if $\frac{x}{\log x\log\log x}=o(y)$, then $$\frac 1 y \sum_{a\leq y}\sum_{p\leq x}\frac{1}{l_a(p)}=\log x + C\log\log…

Let $q\geq 1$ be any integer and let $ \epsilon \in [\frac{1}{11}, \frac{1}{2})$ be a given real number. In this short note, we prove that for all primes $p$ satisfying $$ p\equiv 1\pmod{q}, \quad \log\log p > \frac{\log…

We continue to study the distribution of prime numbers $p$, satisfying the condition $\{ p^{\alpha} \} \in I \subset [0; 1)$, in arithmetic progressions. In the paper, we prove an analogue of Bombieri-Vinogradov theorem for $0 < \alpha <…

An improved estimate is given for $|\theta(x) -x|$, where $\theta(x) = \sum_{p\leq x} \log p$. Three applications are given: the first to arithmetic progressions that have points in common, the second to primes in short intervals, and the…

Let $x,h$ and $Q$ be three parameters. We show that, for most moduli $q\le Q$ and for most positive real numbers $y\le x$, every reduced arithmetic progression $a\mod q$ has approximately the expected number of primes $p$ from the interval…

We give explicit numerical values with 100 decimal digits for the constant in the Mertens product over primes in the arithmetic progressions $a \bmod q$, for $q \in \{3$, ..., $100\}$ and $(a, q) = 1$.

Assuming a uniform $q$-variant of the prime $k$-tuple conjecture, we compute moments of the number of primes in arithmetic progressions to a large modulus $q$ as the residue classes vary. Consequently, depending on the size of $\varphi(q)$,…