Related papers: Explicit bounds for small prime nonresidues

For any large prime $q$, $1 \leq x \leq q$ and any real $0 \leq k \leq 1$, we prove an upper bound for the following $2k$-th moment $$\displaystyle \sum_{\substack{\chi \bmod q}} \Big| \sum_{n\leq x} \chi(n)\lambda(n)\Big|^{2k},$$ where…

We obtain a lower bound for \[ \#\{x/2< p_{n}\leq x:\ p_n \equiv\ldots\equiv p_{n+m}\equiv a\text{ (mod $q$)},\ p_{n+m} - p_{n}\leq y\}, \] where $p_{n}$ is the $n^{\text{th}}$ prime.

Issai Schur once asked if it was possible to determine a bound, preferably using elementary methods, such that for all prime numbers p greater than the bound, the greatest number of consecutive quadratic non-residues modulo p is always less…

Let $g \geq 3$ be fixed and odd, and for large $q$ let $\chi$ be a primitive Dirichlet character modulo $q$ of order $g$. Conditionally on GRH we improve the existing upper bounds in the P\'{o}lya-Vinogradov inequality for $\chi$, showing…

Let $t \in \mathbb{N}$, $\eta >0$. Suppose that $x$ is a sufficiently large real number and $q$ is a natural number with $q \leq x^{5/12-\eta}$, $q$ not a multiple of the conductor of the exceptional character $\chi^*$ (if it exists).…

We use character sum estimates to give a bound on the least square-full primitive root modulo a prime. Specifically, we show that there is a square-full primitive root mod $p$ less than $p^{2/3 + 3/(4 \sqrt{e})+ \epsilon}$, and we give some…

Let $\chi$ be a real and non-principal Dirichlet character, $L(s,\chi)$ its Dirichlet $L$-function and let $p$ be a generic prime number. We prove the following result: If for some $0\leq \sigma<1$ the partial sums $\sum_{p\leq…

Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Let $b\in\mathbb Z$ and $\varepsilon\in\{\pm 1\}$. We mainly prove that $$\left|\left\{N_p(a,b):\ 1<a<p\ \text{and}\ \left(\frac…

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 $\beta$…

Let $p_n$ denote the $n$-th prime. For any $m\geq 1$, there exist infinitely many $n$ such that $p_{n}-p_{n-m}\leq C_m$ for some large constant $C_m>0$, and $$p_{n+1}-p_n\geq \frac{c_m\log n\log\log n\log\log\log\log n}{\log\log\log n}, $$…

Let $G$ be a finite (not necessarily abelian) group and let $p=p(G)$ be the smallest prime number dividing $|G|$. We prove that $d(G)\leq \frac{|G|}{p}+9p^2-10p$, where $d(G)$ denotes the small Davenport constant of $G$ which is defined as…

Let $\chi$ be a non-principal Dirichlet character of modulus $q$ with associated \textit{L}-function $L(s,\chi)$. We prove that $$|L(1,\chi)|\le\left(\frac{1}{2}+O\Big(\frac{\log\log q}{\log q}\Big)\right)\frac{\varphi(q)}{q}\log q\,,$$…

This paper studies explicit and theoretical bounds for several interesting quantities in number theory, conditionally on the Generalized Riemann Hypothesis. Specifically, we improve the existing explicit bounds for the least quadratic…

Let $\chi$ be a primitive character modulo a prime $q$, and let $\delta > 0$. It has previously been observed that if $\chi$ has large order $d \geq d_0(\delta)$ then $\chi(n) \neq 1$ for some $n \leq q^{\delta}$, in analogy with…

Let $G$ be a finite nilpotent group, $\chi$ and $\psi$ be irreducible complex characters of $G$ of prime degree. Assume that $\chi(1)=p$. Then either the product $\chi\psi$ is a multiple of an irreducible character or $\chi\psi$ is the…

Let $p$ be a prime, $\varepsilon>0$ and $0<L+1<L+N < p$. We prove that if $p^{1/2+\varepsilon}< N <p^{1-\varepsilon}$, then $$ \#\{n!\!\!\! \pmod p;\,\, L+1\le n\le L+N\} > c (N\log N)^{1/2},\,\, c=c(\varepsilon)>0. $$ We use this bound to…

Fix $a \in \mathbb{Z}$, $a\notin \{0,\pm 1\}$. A simple argument shows that for each $\epsilon > 0$, and almost all (asymptotically 100% of) primes $p$, the multiplicative order of $a$ modulo $p$ exceeds $p^{\frac12-\epsilon}$. It is an…

For a given $x$ we consider the minimum of $\sum_{n\le x} \chi(n)/n$ as $\chi$ ranges over all quadratic Dirichlet characters. For all large $x$, this minimum is negative and we give upper and lower bounds for it.

In this paper we give a refinement of the bound of D. A. Burgess for multiplicative character sums modulo a prime number $q$. This continues a series of previous logarithmic improvements, which are mostly due to H. Iwaniec and E. Kowalski.…

We prove several results which imply the following consequences. For any $\varepsilon>0$ and any sufficiently large prime $p$, if $\cI_1,\ldots, \cI_{13}$ are intervals of cardinalities $|\cI_j|>p^{1/4+\varepsilon}$ and $abc\not\equiv…