Related papers: On Kloosterman sums with multiplicative coefficien…

Kloosterman sums play a special role in analytic number theory, for expressing the integer Fourier coefficients of modular forms as an infinite sum of Bessel functions, also known as Rademacher formula. The generalization to vector-valued…

We consider the numerical approximation of $f({\cal A})b$ where $b\in{\mathbb R}^{N}$ and $\cal A$ is the sum of Kronecker products, that is ${\cal A}=M_2 \otimes I + I \otimes M_1\in{\mathbb R}^{N\times N}$. Here $f$ is a regular function…

We prove that if $f(n)$ is a Steinhaus or Rademacher random multiplicative function, there almost surely exist arbitrarily large values of $x$ for which $|\sum_{n \leq x} f(n)| \geq \sqrt{x} (\log\log x)^{1/4+o(1)}$. This is the first such…

We investigate the solubility of the congruence xy=1 (mod p), where p is a prime and x,y are restricted to lie in suitable short intervals. Our work relies on a mean value theorem for incomplete Kloosterman sums.

Recently there has been a large number of works on bilinear sums with Kloosterman sums and on sums of Kloosterman sums twisted by arithmetic functions. Motivated by these, we consider several related new questions about sums of Kloosterman…

Let $f(n)$ be a multiplicative function with $|f(n)|\leq 1, q$ be a prime number and $a$ be an integer with $(a, q)=1, \chi$ be a non-principal Dirichlet character modulo $q$. Let $\varepsilon$ be a sufficiently small positive constant, $A$…

We provide examples of multiplicative functions $f$ supported on the $k$-free integers such that at primes $f(p)=\pm 1$ and such that the partial sums of $f$ up to $x$ are $o(x^{1/k})$. Further, if we assume the Generalized Riemann…

For a fixed prime $p$, the maximum coefficient (in absolute value) $M(p)$ of the cyclotomic polynomial $\Phi_{pqr}(x)$, where $r$ and $q$ are free primes satisfying $r>q>p$ exists. Sister Beiter conjectured in 1968 that $M(p)\le(p+1)/2$. In…

Sums of Kloosterman sums have deep connections with the theory of modular forms, and their estimation has many important consequences. Kuznetsov used his famous trace formula and got a power-saving estimate with respect to $x$ with implied…

In this paper, we derive the following asymptotic formula $$ \mathop{{\sum}'}_{n\leqslant x}\dfrac{r(n)}{r(n+1)} = {x}{(\ln x)^{-3/4}}(c+o(1)),\ \ x \to +\infty,$$ where $r(n)$ is the number of representations of $n$ as a sum of two…

We introduce a simple sieve-theoretic approach to studying partial sums of multiplicative functions which are close to their mean value. This enables us to obtain various new results as well as strengthen existing results with new proofs.…

We consider the distribution of the polygonal paths joining partial sums of classical Kloosterman sums, as their parameter varies modulo a prime tending to infinity. Using independence of Kloosterman sheaves, we prove convergence in the…

G. Ricotta and E. Royer (2018) have recently proved that the polygonal paths joining the partial sums of the normalized classical Kloosterman sums $S(a,b;p^n)/p^(n/2) converge in law in the Banach space of complex-valued continuous function…

We derive explicit formulas for some Kloosterman sums on $\Gamma_0(N)$, and for the Fourier coefficients of Eisenstein series attached to arbitrary cusps, around a general Atkin-Lehner cusp.

The polynomial coefficient $\binom {n,q}{k}$ is defined to be the coefficient of $x^{k}$ in the expansion of $(1+x+x^2+... +x^{q-1})^n$. In this note we give an asymptotic estimate for $\binom {n,q}{cn}$ as $n$ tends to infinity, where $c$…

Inspired by the work of Bourgain and Garaev (2013), we provide new bounds for certain weighted bilinear Kloosterman sums in polynomial rings over a finite field. As an application, we build upon and extend some results of Sawin and…

We say a power series $a_0+a_1q+a_2q^2+\cdots$ is \emph{multiplicative} if $n\mapsto a_n/a_1$ for positive integers $n$ is a multiplicative function. Given the Eisenstein series $E_{2k}(q)$, we consider formal multiplicative power series…

We prove that when $f$ is a Rademacher random multiplicative function for any $\epsilon>0$, then $\sum_{n \leqslant x}\frac{f(n)}{\sqrt{n}} \ll (\log\log(x))^{3/4+\epsilon}$ for almost all $f$. We also show that there exist arbitrarily…

We give results characterising ternary Kloosterman sums modulo 9 and 27. This leads to a complete characterisation of values that ternary Kloosterman sums assume modulo 18 and 54. The proofs uses Stickelberger's theorem, the Gross-Koblitz…

A $\textit{square-full}$ number is a positive integer for which all its prime divisors divide itself at least twice. The counting function of square-full integers of the form $f(n)$ for $n\leqslant N$ is denoted by…