Related papers: On Linnik's constant

Define $G(x;q)$ to be the variance of primes $p\le x$ in the arithmetic progressions modulo $q$, weighted by $\log p$. Hooley conjectured that as soon as $q$ tends to infinity and $x\ge q$, we have the upper bound $G(x;q) \ll x \log q$. In…

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 $\beta$…

We show that Grothendieck's real constant $K_{G}$ satisfies $K_G\geq c+10^{-26}$, improving on the lower bound of $c=1.676956674215576\ldots$ of Davie and Reeds from 1984 and 1991, respectively.

Let $I(n,l)$ denote the maximum possible number of incidences between $n$ points and $l$ lines. It is well known that $I(n,l) = \Theta(n^{2/3}l^{2/3} + n + l)$. Let $c_{\mathrm{SzTr}}$ denote the lower bound on the constant of…

In 1845, Bertrand conjectured that twice any prime strictly exceeds the next prime. Tchebichef proved Bertrand's postulate in 1850. In 1934, Ishikawa proved a stronger result: the sum of any two consecutive primes strictly exceeds the next…

Let $q\ge3$ be an integer, $\chi$ denote a Dirichlet character modulo $q$, for any real number $a\ge 0$, we define the generalized Dirichlet $L$-functions $$ L(s,\chi,a)=\sum_{n=1}^{\infty}\frac{\chi(n)}{(n+a)^s}, $$ where $s=\sigma+it$…

Inspired by a result of Soundararajan, assuming the Riemann hypothesis (RH), we prove a new inequality for the logarithm of the modulus of the Riemann zeta-function on the critical line in terms of a Dirichlet polynomial over primes and…

In this paper, we study derivatives of powers of Euclidean norm. We prove their H\"older continuity and establish explicit expressions for the corresponding constants. We show that these constants are optimal for odd derivatives and at most…

We prove asymptotics for mollified first and second moments of subfamilies of Dirichlet $L$-functions given by shrinking angular restrictions on the root number. Using these moments, we prove that for even primitive characters with prime…

In 1976 Lieb and Thirring established upper bounds on sums of powers of the negative eigenvalues of a Schr\"odinger operator in terms of semiclassical phase-space integrals. Over the last 45 years the optimal constants in these…

The problem of finding upper bounds for L-functions at the edge of the critical strip has a long and interesting history. Here, the situation for classical L-functions such as Dirichlet L-functions is relatively well understood. The reason…

Assuming the generalized Riemann hypothesis, we rediscover and sharpen some of the best known results regarding the distribution of low-lying zeros of Dirichlet $L$-functions. This builds upon earlier work of Omar, which relies on the…

We verify that $\liminf_{q\to\infty} q\cdot |q|_p\cdot ||qx||<\epsilon$ for all real $x$, small primes $p$ and relatively small $\epsilon$. This result supports the famous $p$-adic Littlewood conjecture which states that the above lower…

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…

We establish an asymptotic formula with a power-saving error term for the twisted mixed moment of Dirichlet $L$-functions and automorphic $L$-functions twisted by all primitive characters modulo $q$, valid for all admissible moduli. As a…

According to a popular belief, the decimal digits of mathematical constants such as {\pi} behave like statistically independent random variables, each taking the values 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 with equal probability of 1/10. If…

We consider small solutions of quadratic congruences of the form $x_1^2+\alpha_2x_2^2+\alpha_3x_3^2\equiv 0 \bmod{q}$, where $q=p^m$ is an odd prime power. Here, $\alpha_2$ is arbitrary but fixed and $\alpha_3$ is variable, and we assume…

In a landmark paper on arithmetical properties of Lambert series, Erd\H{o}s proved that $\sum_{n=1}^{\infty} \frac{1}{2^{n} - 1}$ is irrational. This value $E$ is now referred to as the Erd\H{o}s-Borwein constant. Crandall, in 2012, studied…

First we reprove, using representation theory and the relative trace formula of Jacquet, an average value result of Duke for modular L-series at the critical center. We also establish a refinement. To be precise, the L-value which appears…

In this paper we use techniques first introduced by Florea to improve the asymptotic formula for the first moment of the quadratic Dirichlet L-functions over the rational function field, running over all monic, square-free polynomials of…