相关论文: On the second moment for primes in an arithmetic p…

A 1976 result from Norton may be used to give an asymptotic (but not explicit) description of the constant in Mertens' second theorem for primes in arithmetic progressions. Assuming the Generalized Riemann Hypothesis, we give an effective…

In this note, assuming a variant of the Generalized Riemann Hypothesis, which does not exclude the existence of real zeros, we prove an asymptotic formula for the mean value of the representation function for the sum of two primes in…

We obtain asymptotic formulae for the second discrete moments of the Riemann zeta function over arithmetic progressions $\frac{1}{2} + i(a n + b)$. It reveals noticeable relation between the discrete moments and the continuous moment of the…

We obtain an asymptotic formula for the second discrete moment of the Riemann zeta function over the arithmetic progression $\frac{1}{2} + in$. It shows that the first main term is equal to that of the continuous mean value.

Assuming the generalized Riemann hypothesis, we give asymptotic bounds on the size of intervals that contain primes from a given arithmetic progression using the approach developed by Carneiro, Milinovich and Soundararajan [Comment. Math.…

We prove explicit versions of Cram\'er's theorem for primes in arithmetic progressions, on the assumption of the generalized Riemann hypothesis.

We prove some distribution results for the $k$-fold divisor function in arithmetic progressions to moduli that exceed the square-root of length $X$ of the sum, with appropriate constrains and averaging on the moduli, saving a power of $X$…

We establish unconditional $\Omega$-results for all weighted even moments of primes in arithmetic progressions. We also study the moments of these moments and establish lower bounds under GRH. Finally, under GRH and LI we prove an…

We assume the Riemann Hypothesis and an quantitative form of the Twin Prime Conjecture, and obtain an asymptotic formula for the second moment of $S(T)$ with better error term.

Assuming the Riemann hypothesis, we prove the latest explicit version of the prime number theorem for short intervals. Using this result, and assuming the generalised Riemann hypothesis for Dirichlet $L$-functions is true, we then establish…

We prove an explicit error term for the $\psi(x,\chi)$ function assuming the Generalized Riemann Hypothesis. Using this estimate, we prove a conditional explicit bound for the number of primes in arithmetic progressions.

In various papers the authors have derived asymptotics for moments of certain cotangent sums related to the Riemann Hypothesis. S. Bettin has given an upper bound for the error term in these asymptotic results. In the present paper the…

We prove that the primes below $x$ are, on average, equidistributed in arithmetic progressions to smooth moduli of size up to $x^{1/2+1/40-\epsilon}$. The exponent of distribution $\tfrac{1}{2} + \tfrac{1}{40}$ improves on a result of…

We prove the analog of Cram\'er's short intervals theorem for primes in arithmetic progressions and prime ideals, under the relevant Riemann Hypothesis. Both results are uniform in the data of the underlying structure. Our approach is based…

We furnish an explicit bound for the prime number theorem in short intervals on the assumption of the Riemann hypothesis.

Assuming the generalized Lindel\"of hypothesis for Dirichlet $L$-functions, we establish that the least prime $p\equiv a\pmod{q}$ satisfies $p\ll_{\varepsilon} q^{2+\varepsilon}$. This achieves a bound that nearly matches the classical…

In the present paper, we adopt a pretentious approach and prove a strongly uniform estimate for the sums of the von Mangoldt function $\Lambda$ on arithmetic progressions. This estimate is analogous to an estimate that Linnik established in…

We show that the twisted second moments of the Riemann zeta function averaged over the arithmetic progression $1/2 + i(an + b)$ with $a > 0$, $b$ real, exhibits a remarkable correspondance with the analogous continuous average and derive…

We describe an algorithm for obtaining explicit expressions for lower terms for the conjectured full asymptotics of the moments of the Riemann zeta function, and give two distinct methods for obtaining numerical values of these…

We investigate the approximation to the number of primes in arithmetic progressions given by Vaughan. Instead of averaging the expected error term over all residue classes to modules in a given range, here we only consider subsets of…