Related papers: The Explicit Sato-Tate Conjecture For Primes In Ar…

It is shown that the density of the values set {Tau(n): n <= x} of the nth coefficients Tau(n) of the discriminant function Delta(z), a cusp form of level N = 1 and weight k = 12, has the lower bound #{Tau(n): n <= x} >> x/log x. The…

Fix a non-CM elliptic curve $E/\mathbb{Q}$, and let $a_E(p) = p + 1 - \#E(\mathbb{F}_p)$ denote the trace of Frobenius at $p$. The Sato-Tate conjecture gives the limiting distribution $\mu_{ST}$ of $a_E(p)/(2\sqrt{p})$ within $[-1, 1]$. We…

The Dedekind eta function $\eta(\tau)$ is defined by \[\eta(\tau)=e^{\pi i\tau/12}\prod_{n=1}^{\infty}\left(1-e^{2\pi i n\tau}\right),\quad\text{when}\;\text{Im}\,\tau>0.\] It plays an important role in number theory, especially in the…

By using the elementary symmetric polynomials and some results of number theory, we solve the well known problem of Lehmer on Euler's totient function. As application, we obtain a new characterization of prime numbers.

We formulate and prove the analogue of the Ramanujan Conjectures for modular forms of half-integral weight subject to some ramification restriction in the setting of a polynomial ring over a finite field. This is applied to give an…

Given a multiplicative function $f$ which is periodic over the primes, we obtain a full asymptotic expansion for the shifted convolution sum $\sum_{|h|<n\leq x} f(n) \tau(n-h)$, where $\tau$ denotes the divisor function and…

Let $f \in S_{2r}(\Gamma_0(N))$ be a normalized newform of weight $2r$ which is good at $p$. Let $K$ be an imaginary quadratic field of class number one in which every prime divisor of $pN$ splits. Let $\chi$ be an anticyclotomic Hecke…

In this paper, for a positive integer $n\ge 1$, we look at the size and prime factors of the iterates of the Ramanujan $\tau$ function applied to $n$.

$\tau$-tilting theory can be thought of as a generalization of the classical tilting theory which allows mutations at any indecomposable summand of a support $\tau$-tilting pair. Indeed, for any algebra $\Lambda$ its tilting modules…

We study generalizations of some results of Jean-Louis Nicolas regarding the relation between small values of Euler's function $\varphi(n)$ and the Riemann Hypothesis. Among other things, we prove that for $1\leq q\leq 10$ and for $q=12,…

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erd\H{o}s proved in 1935 that the weighted sum $\sum1/(n \log n)$ for $n$ ranging over a primitive set $A$ is universally bounded over all choices for…

We conjecture average counting functions for prime $k$-tuples based on a gamma distribution hypothesis for prime powers. The conjecture is closely related to the Hardy-Littlewood conjecture for $k$-tuples but yields better estimates.…

We make an analytical proof for Lehmer's totient conjecture. Lehmer conjectured that there is no solution for the congruence equation $n-1\equiv 0~(mod~\phi(n))$ with composite integers,$n$, where $\phi(n)$ denotes Euler's totient function.…

De Bruijn and Newman introduced a deformation of the Riemann zeta function $\zeta(s)$, and found a real constant $\Lambda$ which encodes the movement of the zeros of $\zeta(s)$ under the deformation. The Riemann hypothesis (RH) is…

A representation of divisor function $\tau(n)\equiv \sigma_{0}(n)$ by means of logarithmic residue of a function of complex variable is suggested. This representation may be useful theoretical instrument for further investigations of…

Let $n$ and $t$ be positive integers with $t\geq 2$. Let $R_t(n)$ be the number of $t$-regular partitions of $n$. A class of functions, denoted $\tau_k(n)$, is defined as follows:…

Let $A/\mathbb{Q}$ be an elliptic curve with split multiplicative reduction at a prime $p$. We prove (an analogue of) a conjecture of Perrin-Riou, relating $p$-adic Beilinson$-$Kato elements to Heegner points in $A(\mathbb{Q})$, and a large…

We study which integers are admissible as Fourier coefficients of even integer weight newforms. In the specific case of the tau-function, we show that for all odd primes $\ell < 100$ and all integers $m \geq 1$, we have $$ \tau(n) \neq \pm…

Towards confirming Sun's conjecture on the strict log-concavity of combinatorial sequence involving the n$th$ Bernoulli number, Chen, Guo and Wang proposed a conjecture about the log-concavity of the function…

The Tu--Deng Conjecture is concerned with the sum of digits $w(n)$ of $n$ in base~$2$ (the Hamming weight of the binary expansion of $n$) and states the following: assume that $k$ is a positive integer and $1\leq t<2^k-1$. Then \[\Bigl…