Related papers: Explicit formulae for primes in arithmetic progres…

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…

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…

Part-and-parcel of the study of "multiplicative number theory" is the study of the distribution of multiplicative functions in arithmetic progressions. Although appropriate analogies to the Bombieri-Vingradov Theorem have been proved for…

We prove large sieve inequalities with multivariate polynomial moduli and deduce a general Bombieri--Vinogradov type theorem for a class of polynomial moduli having a sufficient number of variables compared to its degree. This sharpens…

We obtain the analog of the Bombieri-Vinogradov theorem for square moduli up to any power of x less than 1/2.

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.

We shall give an explicit formula for $\psi(x, q, a)$ with an error term of the form $C/\log^\alpha x$ under the condition that $q<\log^{\alpha_1} x$ is nonexceptional, for various values of $\alpha$ and $\alpha_1$. We shall also give an…

We prove a version of the Bombieri--Vinogradov Theorem with certain products of Gaussian primes as moduli, making use of their special form as polynomial expressions in several variables. Adapting Vaughan's proof of the classical…

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 <…

In this paper, we establish theorems of Bombieri-Vinogradov type and Barban-Davenport-Halberstam type for sparse sets of moduli. As an application, we prove that there exist infinitely many primes of the form $p=am^2+1$ such that $a\leq…

In this paper, we establish a Bombieri-Vinogradov type result for prime numbers of the form $p=x^2+y^2+1$. The proof is based on the enveloping sieve.

We generalize the classical Bombieri-Vinogradov theorem to a short interval, non-abelian setting. This leads to variants of the prime number theorem for short intervals where the primes lie in arithmetic progressions that are "twisted" by a…

In this paper we establish a generalization of Bombieri-Vinogradov theorem for primes represented by a fixed positive definite binary quadratic form. Then we apply this theorem to generalize a result of Vatwani on bounded gap between…

In this article, we extend our recent work on a Bombieri-Vinogradov-type theorem for sparse sets of prime powers $p^N\le x^{1/4-\varepsilon}$ with $p\le (\log x)^C$ to sparse sets of moduli $s\le x^{1/3-\varepsilon}$ with radical rad$(s)\le…

In 2020, Roger Baker \cite{Bak} proved a result on the exceptional set of moduli in the prime number theorem for arithmetic progressions of the following kind. Let $\mathcal{S}$ be a set of pairwise coprime moduli $q\le x^{9/40}$. Then the…

In this paper, we establish a theorem of Bombieri -- Vinogradov type for exponential sums over Piatetski-Shapiro primes $p= [n^{1/\gamma}]$ with $\frac{865}{886}<\gamma < 1$.

Let $\alpha > 0$ be any fixed non-integer, $I$ be any subinterval of $[0; 1)$. In the paper, we prove an analogue of Bombieri-Vinogradov theorem for the set of primes $p$ satisfying the condition $\{ p^{\alpha} \} \in I$. This strengthens…

We prove a new equidistribution estimate for the divisor function in arithmetic progression to moduli that have two small factors. We give two applications. First, we show an asymptotic formula for the divisor function over arithmetic…

Let $p>0$ be a prime, $k$ a field of characteristic $p$ and $G$ and elementary abelian $p$-group of order $q = p^n$. Let $W$ be an indecomposable $kG$-module of dimension 2 and define $V_i=S^{i-1}(W^*)$ for each $i=1 \ldots q$. We show that…

We study the average distribution of primes of size $x$ in arithmetic progressions to moduli larger than $x^{\frac{1}{2}}$. Using arithmetic information from the works of many authors together with different variants of the original…