William D. Banks

University of Missouri | Mizzou · Department of Mathematics

Let $y\ne 0$ and $C>0$. Under the Riemann Hypothesis, there is a number $T_*>0$ $($depending on $y$ and $C)$ such that for every $T\ge T_*$, both \[ \zeta(\tfrac12+i\gamma)=0 \quad\text{and}\quad\zeta(\tfrac12+i(\gamma+y))\ne 0 \] hold for at least one $\gamma$ in the interval $[T,T(1+\epsilon)]$, where $\epsilon:=T^{-C/\log\log T}$.

The methods of Korobov and Vinogradov produce a zero-free region for the Riemann zeta function [Formula: see text] of the form [Formula: see text] For many decades, the general shape of the zero-free region has not changed (although explicit known values for [Formula: see text] have improved over the years). In this paper, we show that if the zero-...

We show that the Generalized Riemann Hypothesis for Dirichlet L-functions is a consequence of certain conjectural properties of the zeros of the Riemann zeta function. Conversely, we prove that the zeros of ζ(s)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \use...

For each primitive Dirichlet character $\chi$, a hypothesis ${\rm GRH}^\dagger[\chi]$ is formulated in terms of zeros of the associated $L$-function $L(s,\chi)$. It is shown that for any such character, ${\rm GRH}^\dagger[\chi]$ is equivalent to the Generalized Riemann Hypothesis.

The type $\tau$($\alpha$) of an irrational number $\alpha$ measures the extent to which rational numbers can closely approximate $\alpha$. More precisely, $\tau$($\alpha$) is the infimum over those t$\in$R for which |$\alpha$--h/k|<k^{--t--1} has at most finitely many solutions h,k$\in$Z, k>0. In this paper, we regard the type as a function $\tau$:...

We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval [1,x]\documentcl...

We show that the Generalized Riemann Hypothesis for all Dirichlet L-functions is a consequence of certain conjectural properties of the zeros of the Riemann zeta function. Conversely, we prove that the zeros of $\zeta(s)$ satisfy those properties under GRH.

For any real \(\beta _0\in [\tfrac{1}{2},1)\), let \(\textrm{GRH}[\beta _0]\) be the assertion that for every Dirichlet character \(\chi \) and all zeros \(\rho =\beta +i\gamma \) of \(L(s,\chi )\), one has \(\beta \leqslant \beta _0\) (in particular, \(\textrm{GRH}[\frac{1}{2}]\) is the Generalized Riemann Hypothesis). In this paper, we show that...

Let $\chi_1$ and $\chi_2$ be distinct primitive Dirichlet characters whose moduli are $q_1$ and $q_2$, respectively, and put $q:={\rm lcm}[q_1,q_2]\ge 3$. We show that for some absolute constants $C_1,C_2>0$, the $L$-functions $L(s,\chi_1)$ and $L(s,\chi_2)$ do not have the same zeros (counted with multiplicity) in the region $$ \big\{s=\sigma+it\i...

Gonek, Graham, and Lee have shown recently that the Riemann Hypothesis can be reformulated in terms of certain asymptotic estimates for twisted sums with von Mangoldt function $\Lambda$. Building on their ideas, for each $k\in\mathbb{N}$, we study twisted sums with the generalized von Mangoldt function$$\Lambda_k(n):=\sum_{d\,\mid\,n}\mu(d)\Big(\lo...

For any real $\beta_0\in[\tfrac12,1)$, let ${\rm GRH}[\beta_0]$ be the assertion that for every Dirichlet character $\chi$ and all zeros $\rho=\beta+i\gamma$ of $L(s,\chi)$, one has $\beta\le\beta_0$ (in particular, ${\rm GRH}[\frac12]$ is the Generalized Riemann Hypothesis). In this paper, we show that the validity of ${\rm GRH}[\frac{9}{10}]$ dep...

Motivated by a question of V. Bergelson and F. K. Richter (2017), we obtain asymptotic formulas for the number of relatively prime tuples composed of positive integers $n\le N$ and integer parts of polynomials evaluated at $n$. The error terms in our formulas depend on the Diophantine properties of the leading coefficients of these polynomials.

Fix coprime natural numbers a,q. Assuming the Prime k-tuple Conjecture, we show that there exist arbitrarily long arithmetic progressions of Carmichael numbers, each of which lies in the reduced residue class a mod q and is a product of three distinct prime numbers.

Fix coprime natural numbers $a,q$. Assuming the Prime $k$-tuple Conjecture, we show that there exist arbitrarily long arithmetic progressions of Carmichael numbers, each of which lies in the reduced residue class $a$ mod $q$ and is a product of three distinct prime numbers.

Building on recent work of A. Harper (2012), and using various results of M. C. Chang (2014) and H. Iwaniec (1974) on the zero-free regions of $L$-functions $L(s,\chi)$ for characters $\chi$ with a smooth modulus $q$, we establish a conjecture of K. Soundararajan (2008) on the distribution of smooth numbers over reduced residue classes for such mod...

Given a prime p, an integer H∈[1,p), and an arbitrary set M⊆Fp∗, where Fp is the finite field with p elements, let J(H,M) denote the number of solutions to the congruence xm≡ynmodpfor which x,y∈[1,H] and m,n∈M. In this paper, we bound J(H,M) in terms of p, H, and the cardinality of M. In a wide range of parameters, this bound is optimal. We give tw...

We combine a classical idea of Postnikov (1956) with the method of Korobov (1974) for estimating double Weyl sums, deriving new bounds on short character sums when the modulus q has a small core Πp|qp. Using this estimate, we improve certain bounds of Gallagher (1972) and Iwaniec (1974) for the corresponding L-functions. In turn, this allows us to...

In an earlier paper we considered the distribution of integers $n$ for which Euler’s totient function at $n$ has all small prime factors. Here we obtain an improvement that is likely to be best possible.

We introduce a new probabilistic model of the primes consisting of integers that survive the sieving process when a random residue class is selected for every prime modulus below a specific bound. From a rigorous analysis of this model, we obtain heuristic upper and lower bounds for the size of the largest prime gap in the interval $[1,x]$. Our res...

Given a prime $p$, an integer $H\in[1,p)$, and an arbitrary set $\cal M\subseteq \mathbb F_p^*$, where $\mathbb F_p$ is the finite field with $p$ elements, let $J(H,\cal M)$ denote the number of solutions to the congruence $$ xm\equiv yn\bmod p $$ for which $x,y\in[1,H]$ and $m,n\in\cal M$. In this paper, we bound $J(H,\cal M)$ in terms of $p$, $H$...

We fix a gap in our proof of an upper bound for the number of positive integers $n\le x$ for which the Euler function $\varphi(n)$ has all prime factors at most $y$. While doing this we obtain a stronger, likely best-possible result.

In their recent work, the authors (2016) have combined classical ideas of A. G. Postnikov (1956) and N. M. Korobov (1974) to derive improved bounds on short character sums for certain nonprincipal characters with powerful moduli. In the present paper, these results are used to bound sums of the Mobius function twisted by characters of the same type...

Fix $\delta\in(0,1]$, $\sigma_0\in[0,1)$ and a real-valued function $\varepsilon(x)$ for which $\limsup_{x\to\infty}\varepsilon(x)\le 0$. For every set of primes ${\mathcal P}$ whose counting function $\pi_{\mathcal P}(x)$ satisfies an estimate of the form $$\pi_{\mathcal P}(x)=\delta\,\pi(x)+O\bigl(x^{\sigma_0+\varepsilon(x)}\bigr),$$ we define a...

International audience We give a new proof that the Riemann zeta function is nonzero in the half-plane {s ∈ C : σ > 1}. A novel feature of this proof is that it makes no use of the Euler product for ζ(s).

Let g>1 be an integer and f(X)â Z[X] a polynomial of positive degree with no multiple roots, and put u(n)=f(gn). In this note, we study the sequence of quadratic fields Q(u(n)) as n varies over the consecutive integers M+1,...,M+N. Fields of this type include Shanks fields and their generalizations. Using the square sieve together with new bounds o...

Let $\alpha>1$ be an irrational number of finite type $\tau$. In this paper, we introduce and study a zeta function $Z_\alpha^\sharp(r,q;s)$ that is closely related to the Lipschitz-Lerch zeta function and is naturally associated with the Beatty sequence ${\mathcal B}(\alpha):=(\lfloor\alpha m\rfloor)_{m\in{\mathbb N}}$. If $r$ is an element of the...

Fix irrational numbers $\alpha,\hat\alpha>1$ of finite type and real numbers $\beta,\hat\beta\ge 0$, and let $B$ and $\hat B$ be the Beatty sequences $$ B:=(\lfloor\alpha m+\beta\rfloor)_{m\ge 1}\quad\text{and}\quad\hat B:=(\lfloor\hat\alpha m+\hat\beta\rfloor)_{m\ge 1}. $$ In this note, we study the distribution of pairs $(p,p^\sharp)$ of consecut...

Fix irrational numbers $\alpha,\hat\alpha>1$ of finite type and real numbers $\beta,\hat\beta\ge 0$, and let $B$ and $\hat B$ be the Beatty sequences $$ B:=(\lfloor\alpha m+\beta\rfloor)_{m\ge 1}\quad\text{and}\quad\hat B:=(\lfloor\hat\alpha m+\hat\beta\rfloor)_{m\ge 1}. $$ In this note, we study the distribution of pairs $(p,p^\sharp)$ of consecut...

We combine a classical idea of Postnikov (1956) with the method of Korobov (1974) for estimating double Weyl sums, deriving new bounds on short character sums when the modulus $q$ has a small core $\prod_{p\mid q}p$. Using this estimate, we improve certain bounds of Gallagher (1972) and Iwaniec (1974) for the corresponding $L$-functions. In turn, t...

For any odd prime number $p$, let $(\cdot|p)$ be the Legendre symbol, and let $n_1(p)<n_2(p)<\cdots$ be the sequence of positive nonresidues modulo $p$, i.e., $(n_k|p)=-1$ for each $k$. In 1957, Burgess showed that the upper bound $n_1(p)\ll_\epsilon p^{(4\sqrt{e})^{-1}+\epsilon}$ holds for any fixed $\epsilon>0$. In this paper, we prove that the s...

Let with and , where is the set of prime numbers, and is the floor function. We show that for every such there are infinitely many members of having at most prime factors, giving explicit estimates for when is near one and also when is large.

It is shown that the set of decimal palindromes is an additive basis for the natural numbers. Specifically, we prove that every natural number can be expressed as the sum of forty-nine (possibly zero) decimal palindromes.

Let $${\mathbb P}^c=(\lfloor p^c\rfloor)_{p\in{\mathbb P}} \qquad (c>1,\ c\not\in {\mathbb N}), $$ where ${\mathbb P}$ is the set of prime numbers, and $\lfloor\cdot\rfloor$ is the floor function. For $c<149/87$, we obtain the expected asymptotic formula for the number of squarefree integers in ${\mathbb P}^c$ up to $x$. We show further that for ev...

Using the sieve, we show that there are infinitely many Carmichael numbers whose prime factors all have the form $p = 1 + a^2 + b^2$ with $a,b \in{\mathbb Z}$.

For c > 1, c ε ℤ and χ a primitive character (mod q), we estimate the sum Σx<n≤x+yχ([nc]). Our results complement earlier work of Banks and Shparlinski. We also give a new bound for the least n for which [nc] is a quadratic non-residue (mod q), in the case of prime q, for 1 < c < 371/309. © 2015 2015. Published by Oxford University Press. All right...

We establish several related results on Carmichael, Sierpiński and Riesel numbers. First, we prove that almost all odd natural numbers k have the property that 2nk + 1 is not a Carmichael number for any n ∈ N; this implies the existence of a set K of positive lower density such that for any k ∈ K the number 2nk + 1 is neither prime nor Carmichael f...

Let $d$ be a probability distribution. Under certain mild conditions we show that $$ \lim_{x\to\infty}x\sum_{n=1}^\infty \frac{d^{*n}(x)}{n}=1,\qquad\text{where}\quad d^{*n}:=\underbrace{\,d*d*\cdots*d\,}_{n\text{ times}}. $$ For a compactly supported distribution $d$, we show that if $c>0$ is a given constant and the function $f(k):=\widehat d(k)-...

Using a recent improvement by Bettin and Chandee to a bound of Duke, Friedlander and Iwaniec~(1997) on double exponential sums with Kloosterman fractions, we establish a uniformity of distribution result for the fractional parts of Dedekind sums $s(m,n)$ with $m$ and $n$ running over rather general sets. Our result extends earlier work of Myerson (...

Let �|·|� be the floor function. In this paper, we show that for any fixed c ∈ (�1, 77/76) � there are infinitely many primes of the form p = |�n^c|�, where n is a natural number with at most eight prime factors (counted with multiplicity).

A set of natural numbers is primitive if no element of the set divides another. Erdős conjectured that if S is any primitive set, then (Formula Presented) where P denotes the set of primes. In this paper, we make progress towards this conjecture by restricting the setting to smaller sets of primes. Let P denote any subset of ℙ, and let ℕ(P) denote...

Let $p_n$ denote the $n$th smallest prime number, and let $\boldsymbol{L}$ denote the set of limit points of the sequence $ \left\{(p_{n+1} - p_n)/\log p_n\right\}_{n = 1}^{\infty} $ of normalized differences between consecutive primes. We show that for $k = 50$ and for any sequence of $k$ nonnegative real numbers $\beta_1 \le \beta_2 \le \cdots \l...

We investigate in various ways the representation of a large natural number N as a sum of s positive k-th powers of numbers from a fixed Beatty sequence. Inter alia, a very general form of the local to global principle is established in additive number theory. Although the proof is very short, it depends on a deep theorem of M. Kneser. © Société Ar...

For any finite Galois extension K of Q and any conjugacy class C in Gal(K/Q), we show that there exist infinitely many Carmichael numbers composed solely of primes for which the associated class of Frobenius automorphisms is C. This result implies that for every natural number n there are infinitely many Carmichael numbers of the form a 2 + nb 2 wi...

In a stunning new advance towards the Prime k-tuple Conjecture, Maynard and Tao have shown that if k is sufficiently large in terms of m, then for any admissible k-tuple H(x)={g_jx+h_j : j = 1,...,k} of linear forms in Z[x], the set H(n)={g_jn+h_j: j = 1,...,k} contains at least m primes for infinitely many n. In this short note, we show that if k...

For the Gauss sums which are defined by S_n(a,q) := \sum_{x (mod q)} e(ax^n/q), Stechkin (1975) conjectured that the quantity A := \sup_{n,q\ge 2} \max_{\gcd(a,q)=1} |S_n(a,q)|/q^(1-1/n) is finite. Shparlinski (1991) proved that A is finite, but in the absence of effective bounds on the sums S_n(a,q) the precise determination of A has remained intr...

A set of natural numbers is primitive if no element of the set divides another. Erd\H{o}s conjectured that if S is any primitive set, then \sum_{n\in S} 1/(n log n) \le \sum_{n\in \P} 1/(p log p), where \P denotes the set of primes. In this paper, we make progress towards this conjecture by restricting the setting to smaller sets of primes. Let P d...

We show that the Riemann zeta function \zeta\ has only countably many self-intersections on the critical line, i.e., for all but countably many z in C the equation \zeta(1/2+it)=z has at most one solution t in R. More generally, we prove that if F is analytic in a complex neighborhood of R and locally injective on R, then either the set {(a,b) in R...

Let φ be the Euler function. Fix l Î \mathbb N{\ell\in{\mathbb N}} , and let \fancyscript P{{\fancyscript P}} be an arbitrary set of primes of positive lower natural density. Using a variant of the Alford–Granville–Pomerance construction, we show that there are infinitely many Carmichael numbers N with a totient of the form j(N)=ml [(m)\tilde]{...

Let $\zeta (s)$ be the Riemann zeta function. In this paper, we study repeated values of $\zeta (s)$ on the critical line, and we give evidence to support our conjecture that for every nonzero complex number $z$, the equation $\zeta (1/2 + i t) = z$ has at most two solutions $t \in R$. We prove a number of related results, some of which are uncondi...

We consider various arithmetic questions for the Piatetski-Shapiro sequences $\fl{n^c}$ ($n=1,2,3,...$) with $c>1$, $c\not\in\N$. We exhibit a positive function $\theta(c)$ with the property that the largest prime factor of $\fl{n^c}$ exceeds $n^{\theta(c)-\eps}$ infinitely often. For $c\in(1,\tfrac{149}{87})$ we show that the counting function of...

We give new bounds on sums of the form ∑ n≤NΛ(n)exp (2πiagn/m) and ∑ n≤NΛ(n)χ(gn+a), where Λ is the von Mangoldt function, m is a natural number, a and g are integers coprime to m, and χ is a multiplicative character modulo m. In particular, our results yield bounds on the sums ∑ p≤Nexp (2πiaMp/m) and ∑ p≤Nχ(Mp) with Mersenne numbers Mp=2p−1, where...

Let α, β ∈ R be fixed with α > 1, and suppose that α is irrational and of finite type. We show that there are infinitely many Carmichael numbers composed solely of primes from the non-homogeneous Beatty sequence B_α,β = (\lfloor nα + β \rfloor)^∞_{ n=1} . We conjecture that the same result holds true when α is an irrational number of infinite type.

A natural number n is called a k-almost prime if n has precisely k prime factors, counted with multiplicity. For any fixed k≥2, let ℱ k (X) be the number of k-th powers m k ≤X such that φ(n)=m k for some squarefree k-almost prime n, where φ(·) is the Euler function. We show that the lower bound ℱ k (X)≫X 1/k /(logX) 2k holds, where the implied cons...

We estimate the sizes of the sumset A + A and the productset A $\cdot$ A in the special case that A = S (x, y), the set of positive integers n less than or equal to x, free of prime factors exceeding y. Comment: 12 pages

We bound short sums of the form ån £ X(c1*c2)(n){\sum_{n\le X}(\chi_1{*}\chi_2)(n)}, where χ 1*χ 2 is the convolution of two primitive Dirichlet characters χ 1 and χ 2 with conductors q 1 and q 2, respectively. Mathematics Subject Classification (2000)Primary: 11L07-Secondary: 11N37-11M06

We study the collection of group structures that can be realized as a group of rational points on an elliptic curve over a finite field (such groups are well known to be of rank at most two). We also study various subsets of this collection which correspond to curves over prime fields or to curves with a prescribed torsion. Some of our results are...

For a nontrivial multiplicative character χ modulo p, we bound character sums taken on the integer parts of a real-valued, twice-differentiable function f whose second derivative decays at an appropriate rate. For the special case that f(x) = xη with some positive real number η, our bounds extend recent results of several authors.

We dedicate this paper to our friend Alf van der Poorten Abstract Assuming a weak version of a conjecture of Heath-Brown on the least prime in a residue class, we show that for any coprime integers a and m 1, there are infinitely many Carmichael numbers in the arithmetic progression a mod m.

Let φ denote the Euler function. In this paper, we show that for all large χ there are more than χ0.33 Carmichael numbers n ≤ χ with the property that φ(n) is a perfect square. We also obtain similar results for higher powers.

We show that there is an infinite set of natural numbers with the property that is square-free for every finite subset .

We study sums of the form S α (f,x)=∑ n≤x,n∈ℬ α f(n), where f is an arbitrary arithmetic function satisfying a mild growth condition, and ℬ α =(⌊αk) k∈ℕ ⌋ is the homogeneous Beatty sequence corresponding to a real number α>1. We show that for almost all α>1 the asymptotic formula S α (f,x)∼α -1 ∑ n≤x f(n)(x→∞) holds, and we give a strong bound on t...

In this note, we determine the order of growth of the number of positive integers n ≤ x such that λ(n) is a sum of two square numbers, where λ(n) is the Carmichael function.

Giuga has conjectured that if the sum of the $(n-1)$-st powers of the residues modulo $n$ is $-1pmod n$, then $n$ is 1 or prime. It is known that any counterexample is a Carmichael number. Lehmer has asked if $varphi(n)$ divides $n-1$, with $varphi$ being Euler&apos;s function, must it be true that $n$ is 1 or prime. No examples are known, but a co...

We give bounds on the number of integers 1≤n≤N such that p∣s(n), where p is a prime and s(n) is the sum of aliquot divisors function given by s(n)=σ(n)-n, where σ(n) is the sum of divisors function. Using this result, we obtain nontrivial bounds in certain ranges for rational exponential sums of the form S p (a,N)=∑ n≤N exp(2πias(n)/p),gcd(a,p)=1·

For any natural number $n$, let $X'_n$ be the set of primitive Dirichlet characters modulo $n$. We show that if the Riemann hypothesis is true, then the inequality $|X'_{2n_k}|\le C_2 e^{-\gamma} \phi(2n_k)/\log\log(2n_k)$ holds for all $k\ge 1$, where $n_k$ is the product of the first $k$ primes, $\gamma$ is the Euler-Mascheroni constant, $C_2$ is...

D. H. Lehmer asked whether there are any composite integers for which φ(n)∣n-1, where φ is the Euler function. In this paper, we show that the number of such integers n≥x is o(x 1/2 ) as x→∞.

We define Wieferich numbers to be those odd integers w≥3 that satisfy the congruence 2 φ(w)≡1 (mod  w 2). It is clear that the distribution of Wieferich numbers is closely related to the distribution of Wieferich primes, and we give some quantitative forms of this statement. We establish several unconditional asymptotic results about Wieferich numb...

In 1984, G. Robin proved that the Riemann hypothesis is true if and only if the Robin inequality $\sigma(n)<e^\gamma n\log\log n$ holds for every integer $n>5040$, where $\sigma(n)$ is the sum of divisors function, and $\gamma$ is the Euler-Mascheroni constant. We exhibit a broad class of subsets $\cS$ of the natural numbers such that the Robin ine...

In this note, we show that the number of composite integers n ≤ x such that ϕ(n)|n − 1 is at most O(x 1/2(log log x)1/2), thus improving earlier results by Pomerance and by Shan.

Let $\varphi(\cdot)$ be the Euler function and let $\sigma(\cdot)$ be the sum-of-divisors function. In this note, we bound the number of positive integers $n\le x$ with the property that $s(n)=\sigma(n)-n$ divides $\varphi(n)$.

A study of certain Hamiltonian systems has lead Y. Long to conjecture the existence of infinitely many primes of the form $p=2[\alpha n]+1$, where $1<\alpha<2$ is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progres...

We study the values of arithmetic functions taken on the elements of a non-homogeneous Beatty sequence ⌊αn+β⌋, n=1,2,…, where α,β∈R, and α>0 is irrational. For example, we show that∑n⩽Nω(⌊αn+β⌋)∼NloglogNand∑n⩽N(−1)Ω(⌊αn+β⌋)=o(N), where Ω(k) and ω(k) denote the number of prime divisors of an integer k≠0 counted with and without multiplicities, respe...

We study the behavior of the arithmetic functions defined by

We study the problem of representing integers as sums of prime numbers from a fixed Beatty sequence $B_{\alpha,\beta}$, where $\alpha>1$ is irrational and of finite type.

Let P denote the set of prime numbers, and let P (n) denote the largest prime factor of an integer n > 1. We show that, for every real number 32/17 < η < (4 + 3√2)/4, there exists a constant c(η) > 1 such that for every integer a ≠ 0, the set {p ∈ P : p = P(q - a) for some prime q with pη < q < c(η) pη} has relative asymptotic density one in the se...

In this note, we study those positive integers n with the property that the sum of the distinct prime factors of n divides the n-th Mersenne number.

For every integer n∈ℕ, let R(n) be the integer obtained by reversing the order of the base-g digits of n and call R(n) the reversal of n (with respect to g). In this paper, we introduce and study a sequence C ˜(g)={R(n)} n=1 ∞ which is closely related to the van der Corput sequence. We establish a few fundamental divisibility properties of reversal...

We obtain asymptotic formulae for the number of primes p ≤ x for which the reduction modulo p of the elliptic curve $$ E_{a,b} :Y^2 = X^3 + aX + b $$ satisfies certain “natural” properties, on average over integers a and b such that |a| ⩽ A and |b| ⩽ B, where A and B are small relative to x. More precisely, we investigate behavior with respect to t...

We establish an asymptotic formula for the number of positive integers n⩽x for which φ(n) is free of kth powers.

We give upper bounds for the number of solutions to con- gruences with the Euler function '(n) and with the Carmichael function �(n). We also give nontrivial bounds for certain exponential sums in- volving '(n). Analogous results can also be obtained for the sum of divisors function and similar arithmetic functions.

We obtain asymptotic formulae for the number of primes $p\le x$ for which the reduction modulo $p$ of the elliptic curve $$ \E_{a,b} : Y^2 = X^3 + aX + b $$ satisfies certain ``natural'' properties, on average over integers $a$ and $b$ with $|a|\le A$ and $|b| \le B$, where $A$ and $B$ are small relative to $x$. Specifically, we investigate behavio...

We estimate multiplicative character sums taken on the values of a non-homogeneous Beatty sequence $\{[\alpha n + \beta] : n =1,2,... \}$, where $\alpha,\beta\in\R$, and $\alpha$ is irrational. Our bounds are nontrivial over the same short intervals for which the classical character sum estimates of Burgess have been established.

We show that for any fixed $\eps>0$, there are numbers $\delta>0$ and $p_0\ge 2$ with the following property: for every prime $p\ge p_0$ and every integer $N$ such that $p^{1/(4\sqrt{e})+\eps}\le N\le p$, the sequence $1,2,...,N$ contains at least $\delta N$ quadratic non-residues modulo $p$. We use this result to obtain strong upper bounds on the...

We estimate multiplicative character sums taken on the values of a non- homogeneous Beatty sequence {� αn + β� : n =1 , 2 ,... } ,w hereα, β ∈ R ,a ndα is irrational. In particular, our bounds imply that for fixed α, β and a small real number ε> 0, if p is sufficiently large and p1/3+ε ≤ N ≤ p1/2+ε, then among the first N elements of the Beatty seq...

We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence B-alpha,B-beta = {[alpha n + beta] : n = 1, 2,3,...}, where alpha,beta is an element of R, and alpha is irrational. In particular, our bounds imply that for every fixed epsilon > 0, if p is sufficiently large and p(1/2+epsilon) <= N <= p, then among the...

We give nontrivial bounds in various ranges for exponential sums of the formunder(∑, n ∈ S (x, y)) exp (2 π i a θ{symbol}n / m) and under(∑, n ∈ St (x, y)) exp (2 π i a θ{symbol}n / m)where m ≥ 2, φ is an element of order t in the multiplicative group Z*m, gcd(a,m) = 1, S(x,y) is the set of y-smooth integers n≤x, and St(x,y) is the subset of S(x,y)...

Let a be an integer and q a prime number. In this paper, we find an asymptotic formula for the number of positive integers n < x with the property that no divisor d > 1 of n lies in the arithmetic progression a modulo q.

We establish upper bounds for multiplicative character sums with the function σg (n) which computes the sum of the digits of n in a fixed base g ≥ 2. Our results may be viewed as analogues of some previously known results for exponential sums with sum of g-ary digits function.

For a positive integer n, we let '(n) and (n) denote the Euler function and the Carmichael function, respectively. We dene (n) as the ratio '(n)= (n) and study various arithmetic properties of (n).

Let $p$ be a prime and $\vartheta$ an integer of order $t$ in the multiplicative group modulo $p$. In this paper, we continue the study of the distribution of Diffie–Hellman triples $(\vartheta^x, \vartheta^y, \vartheta^{xy})$ by considering the closely related problem of estimating exponential sums formed from linear combinations of the entries in...