Odd prime values of the Ramanujan tau function

  title={Odd prime values of the Ramanujan tau function},
  author={Nik Lygeros and Olivier Rozier},
  journal={The Ramanujan Journal},
We study the odd prime values of the Ramanujan tau function, which form a thin set of large primes. To this end, we define LR(p,n):=τ(pn−1) and we show that the odd prime values are of the form LR(p,q) where p,q are odd primes. Then we exhibit arithmetical properties and congruences of the LR numbers using more general results on Lucas sequences. Finally, we propose estimations and discuss numerical results on pairs (p,q) for which LR(p,q) is prime. 

Some remarks on small values of $$\tau (n)$$

A natural variant of Lehmer’s conjecture that the Ramanujan τ -function never vanishes asks whether, for any given integer α, there exist any n ∈ Z such that τ(n) = α. A series of recent papers

Small Primes And The Tau Function

This note shows that the prime values of the Ramanujan tau function $\tau(n)=\pm p$ misses every prime $p\leq 8.0\times 10^{25}$.

Approximate Atkin-Serre Conjecture

Abstract : Let λ(n) be the nth coefficient of a modular form f(z) = ∑ n≥1 λ(n)q n of weight k ≥ 4, let pm be a prime power, and let ε > 0 be a small number. A pair of completely different

Spectral Methods And Prime Numbers Counting Problems

A recent heuristic argument based on basic concepts in spectral analysis showed that the twin prime conjecture and a few other related primes counting problems are valid. A rigorous version of the

Fourier Coefficients Of Some Cusp Forms

The possible values of the nth Fourier coefficients a(n) of some cusp forms f(z) of weight k => 12 are studied in this article. In particular, the values of the tau function are investigated in some

Variants of Lehmer's speculation for newforms

In the spirit of Lehmer's unresolved speculation on the nonvanishing of Ramanujan's tau-function, it is natural to ask whether a fixed integer is a value of $\tau(n)$ or is a Fourier coefficient

Approximate Akin-Serre Conjecture

where s ∈ C is a complex number in the upper half plane, q = ei2πs, are the topics of many studies. Basic information on modular forms, classified by various parameters such as level N ≥ 1, weight k

Odd values of the Ramanujan tau function

We prove a number of results regarding odd values of the Ramanujan $$\tau $$ τ -function. For example, we prove the existence of an effectively computable positive constant $$\kappa $$ κ such that if

An isomorphism between the convolution product and the componentwise sum connected to the D’Arcais numbers and the Ramanujan tau function

Given a commutative ring R with identity, let $$H_R$$HR be the set of sequences of elements in R. We investigate a novel isomorphism between $$(H_R, +)$$(HR,+) and $$(\tilde{H}_R,*)$$(H~R,∗), where

Polynomial interpolation of modular forms for Hecke groups

For $m = 3, 4, ...$, let $\lambda_m = 2 \cos \pi/m$, let $G(\lambda_m)$ be a Hecke group, and let $J_m (m = 3, 4, ...$) be a triangle function for $G(\lambda_m)$ such that, when normalized



The Primality of Ramanujan's Tau-Function

Although a number of remarkable properties of r(n) have been established, some of which are cited below, there remains a number of unsolved questions about ,r(n); for example: What is the exact order

On ℓ-adic representations and congruences for coefficients of modular forms (II)

The work I shall describe in these lectures has two themes, a classical one going back to Ramanujan [8] and a modern one initiated by Serre [9] and Deligne [3]. To describe the classical theme, let

A proof of Bertrand's postulate

We discuss the formalization, in the Matita Interactive Theorem Prover, of some results by Chebyshev concerning the distribution of prime numbers, subsuming, as a corollary, Bertrand's postulate.

On Certain Arithmetical Functions Due to

1. Introduction. G. Humbert has discussed, in a series of brief notes, 1 a certain class of entire functions with interesting arithmetical properties. These functions are defined, in an essentially

Divisors of Mersenne numbers

We add to the heuristic and empirical evidence for a conjecture of Gillies about the distribution of the prime divisors of Mersenne numbers. We list some large prime divisors of Mersenne numbers Mp

Existence of Primitive Divisors of Lucas and Lehmer Numbers

We prove that for n > 30, every n-th Lucas and Lehmer number has a primitive divisor. This allows us to list all Lucas and Lehmer numbers without a primitive divisor.

Implementing the asymptotically fast version of the elliptic curve primality proving algorithm

  • F. Morain
  • Computer Science, Mathematics
    Math. Comput.
  • 2007
The elliptic curve primality proving algorithm is one of the current fastest practical algorithms for proving the primality of large numbers, and an asymptotically fast version, attributed to J. O. Shallit, is described.

The new book of prime number records

1 How Many Prime Numbers Are There?.- I. Euclid's Proof.- II. Goldbach Did It Too!.- III. Euler's Proof.- IV. Thue's Proof.- V. Three Forgotten Proofs.- A. Perott's Proof.- B. Auric's Proof.- C.

A New Solution to the Equation (p) ≡ 0 (mod p)

The known solutions to the equation �(p) ≡ 0 (mod p) were p = 2, 3, 5, 7, and 2411. Here we present our method to compute the next solution, which is p = 7758337633. There are no other solutions up

Divisibilité de certaines fonctions arithmétiques

© Séminaire Delange-Pisot-Poitou. Théorie des nombres (Secrétariat mathématique, Paris), 1974-1975, tous droits réservés. L’accès aux archives de la collection « Séminaire Delange-Pisot-Poitou.