Related papers: The prime number theorem for primes in arithmetic …

Assuming the Generalised Riemann Hypothesis, we prove a sharp upper bound on moments of shifted Dirichlet $L$-functions. We use this to obtain conditional upper bounds on high moments of theta functions. Both of these results strengthen…

We study the conjecture that $\sum_{n\leq x} \chi(n)=o(x)$ for any primitive Dirichlet character $\chi \pmod q$ with $x\geq q^\epsilon$, which is known to be true if the Riemann Hypothesis holds for $L(s,\chi)$. We show that it holds under…

We prove an asymptotic formula for the eighth moment of Dirichlet $L$-functions averaged over primitive characters $\chi$ modulo $q$, over all moduli $q\leq Q$ and with a short average on the critical line. Previously the same result was…

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…

In this article, we derive a Euler prime product formula for the magnitude of the Riemann zeta function $\zeta(s)$ valid for $\Re(s)>1$, as well as similar formulas for $\zeta(s)$ valid for an even and odd $k$th positive integer argument.…

We establish the existence of infinitely many \emph{polynomial} progressions in the primes; more precisely, given any integer-valued polynomials $P_1, >..., P_k \in \Z[\m]$ in one unknown $\m$ with $P_1(0) = ... = P_k(0) = 0$ and any $\eps…

We consider Dirichlet $L$-functions $L(s, \chi)$ where $\chi$ is a real, non-principal character modulo $q$. Using Pintz's refinement of Page's theorem, we prove that for $q\geq 3$ the function $L(s, \chi)$ has at most one real zero $\beta$…

In previous work it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its $L$-function is valid to the right of the…

Rubinstein and Sarnak have shown, conditional on the Riemann hypothesis (RH) and the linear independence hypothesis (LI) on the non-real zeros of $\zeta(s)$, that the set of real numbers $x\ge2$ for which $\pi(x)>$ li$(x)$ has a logarithmic…

We consider Dirichlet $L$-functions $L(s, \chi)$ where $\chi$ is a non-principal quadratic character to the modulus $q$. We make explicit a result due to Pintz and Stephens by showing that $|L(1, \chi)|\leq \frac{1}{2}\log q$ for all $q\geq…

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…

In this paper we establish explicit upper and lower bounds for the ratio of the arithmetic and geometric means of the prime numbers, which improve the current best estimates. Further, we prove several conjectures related to this ration…

We show that for every fixed $A>0$ and $\theta>0$ there is a $\vartheta=\vartheta(A,\theta)>0$ with the following property. Let $n$ be odd and sufficiently large, and let $Q_{1}=Q_{2}:=n^{\h}(\log n)^{-\vartheta}$ and $Q_{3}:=(\log…

The author sharpens a result of Jia (1996), showing that the interval $[n, n+n^{\frac{1}{21.5}+\varepsilon}]$ contains prime numbers for almost all $n$. Watt's mean value bound, a delicate sieve decomposition and more accurate estimates for…

For $a/q\in\mathbb{Q}$ the Estermann function is defined as $D(s,a/q):=\sum_{n\geq1}d(n)n^{-s}\operatorname{e}(n\frac aq)$ if $\Re(s)>1$ and by meromorphic continuation otherwise. For $q$ prime, we compute the moments of $D(s,a/q)$ at the…

Using the method of multiple Dirichlet series, we develop L-functions ratios conjecture with one shift in both the numerator and denominator in certain ranges for quadratic families of Dirichlet and Hecke L-functions of primerelated moduli…

The pentagonal number theorem is extended to the sequence of the number of integer partitions with all parts equal. The new pentagonal number theorem implies that the distribution of the primes is just a specific detail of the application…

We establish estimates for the number of ways to represent any reduced residue class as a product of a prime and an integer free of small prime factors. Our best results is conditional on the Generalised Riemann hypothesis (GRH). As a…

We prove a new mean-value theorem for Dirichlet polynomials with coefficients given by the von Mangoldt function. We then use our theorem to derive new estimates for certain exponential sums over primes. The latter have applications to…

Assuming a conjecture on distinct zeros of Dirichlet L-functions we get asymptotic results on the average number of representations of an integer as the sum of two primes in arithmetic progression. On the other hand the existence of good…