Related papers: Waring problem with the Ramanujan $\tau$-function

Necessary and sufficient conditions for the existence of an integer solution of the diophantine equation $m/n=1/x(\lambda)+1/y(\lambda)+1/z(\lambda)$ with $n=b+a\lambda$ are explicitly given for a,b coprime and a not a multiple of m . The…

We determine asymptotically the maximal order of log d(d(n)), where d(n) is the number of positive divisors of n. This solves a problem first put forth by Ramanujan in 1915.

Towards the end of his life Ramanujan wrote a manuscript on properties of the partition and tau functions, some parts of which remained unpublished until very recently. Nevertheless, this manuscript gave rise to a lot of subsequent work. In…

Let f(n)=1 if n=1, 2^(2^(n-2)) if n \in {2,3,4,5}, (2+2^(2^(n-4)))^(2^(n-4)) if n \in {6,7,8,...}. We conjecture that if a system T \subseteq {x_i+1=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}} has only finitely many solutions in positive…

The sufficient conditions for insolvability of the Diophantine equation $\sum_{i=1}^{m}x_i^{n}=bc^{n}$ ($n, m \geq 2$, $b, c\in \mathbb{N}$) in nonnegative integers are obtained for the case where the canonical decomposition of the number…

Let $\mu_1, \ldots, \mu_s$ be real numbers, with $\mu_1$ irrational. We investigate sums of shifted $k$th powers $\mathfrak{F}(x_1, \ldots, x_s) = (x_1 - \mu_1)^k + \ldots + (x_s - \mu_s)^k$. For $k \ge 4$, we bound the number of variables…

Naciri proved that for any integer $k\geq2$, the Brocard--Ramanujan equation $n!+1=x^2$ has only finitely many integer solutions, assuming $x\pm1$ is a $k$-free integer or a prime power. In the present paper we prove similar statements for…

Let $f(z)=\sum_{n=1}^\infty a(n)q^n\in S^{\text{new}}_ k (\Gamma_0(N))$ be a newform with squarefree level $N$ that does not have complex multiplication. For a prime $p$, define $\theta_p\in[0,\pi]$ to be the angle for which $a(p)=2p^{( k…

In this paper, we consider the Diophantine equation $\lambda_1U_{n_1}+\ldots+\lambda_kU_{n_k}=wp_1^{z_1} \cdots p_s^{z_s},$ where $\{U_n\}_{n\geq 0}$ is a fixed non-degenerate linear recurrence sequence of order greater than or equal to 2;…

In this article, we present relations for the Euler totient function $\varphi(n)$ and the number of divisors $\tau(n)$ in terms of finite sums of integer parts of rational numbers or greatest common divisors of pairs of integers. Some of…

In this paper, we solve the simultaneous Diophantine equations m.(x_1^k+....+x_{t_1}^k)=n.(y_1^k+....+y_{t_2}^k); k=1,3, where t_1, t_2>3, and m, n are fixed arbitrary and relatively prime positive integers. This is done by choosing two…

The authors assume the Deep Riemann Hypothesis to prove that a weighted sum of Ramanujan's $\tau$-function has a bias to being positive. This phenomenon is an analogue of Chebyshev's bias.

In this paper we introduce a new diophantine equation with prime numbers. Let $[\, \cdot\,]$ be the floor function. We prove that when $1<c<\frac{23}{21}$ and $\theta>1$ is a fixed, then every sufficiently large positive integer $N$ can be…

We describe mean value estimates for exponential sums of degree exceeding 2 that approach those conjectured to be best possible. The vehicle for this recent progress is the efficient congruencing method, which iteratively exploits the…

Let $1<k<7/6$, $\lambda_1,\lambda_2,\lambda_3$ and $\lambda_4$ be non-zero real numbers, not all of the same sign such that $\lambda_1/\lambda_2$ is irrational and let $\omega$ be a real number. We prove that the inequality…

We establish that almost every positive integer $n$ is the sum of four cubes, two of which are at most $n^{\theta}$, as long as $\theta\geq192/869$. An asymptotic formula for the number of such representations is established when…

Let E_n={x_i=1, x_i+x_j=x_k, x_i \cdot x_j=x_k: i,j,k \in {1,...,n}}. There is an algorithm that for every computable function f:N->N returns a positive integer m(f), for which a second algorithm accepts on the input f and any integer…

Let $1<c<832/825$. For large real numbers $N>0$ and a small constant $\vartheta>0$, the inequality \begin{equation*} |p_1^c+p_2^c+p_3^c+p_4^c-N|<\vartheta \end{equation*} has a solution in prime numbers $p_1,\,p_2,\,p_3,\,p_4$ such that,…

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.

We give upper bounds on the size of the gap between the constant term and the next non-zero Fourier coefficient of an entire modular form of given weight for \Gamma_0(2). Numerical evidence indicates that a sharper bound holds for the…