The Ramanujan tau function, studied by , is the function
defined by the following identity:
:
where with ,
is the
Euler function
In mathematics, the Euler function is given by
:\phi(q)=\prod_^\infty (1-q^k),\quad , q, A000203
On account of the identity \sum_ d = \sum_ \frac, this may also be written as
:\ln(\phi(q)) = -\sum_^\infty \frac \sum_ d.
Also if a,b\in\mathbb^ ...
, is the
Dedekind eta function
In mathematics, the Dedekind eta function, named after Richard Dedekind, is a modular form of weight 1/2 and is a function defined on the upper half-plane of complex numbers, where the imaginary part is positive. It also occurs in bosonic string t ...
, and the function is a
holomorphic
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex derivati ...
cusp form In number theory, a branch of mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion.
Introduction
A cusp form is distinguished in the case of modular forms for the modular gro ...
of weight 12 and level 1, known as the
discriminant modular form (some authors, notably
Apostol
Apostol may refer to:
People
Apostol is an East European name and name element derived from Ancient Greek ἀπόστολος "apostle", and therefore found mainly in Christian societies and cultures.
Given name
* Apostol Mărgărit (1832–1903 ...
, write
instead of
). It appears in connection to an "error term" involved in counting the number of ways of expressing an integer as a sum of 24 squares. A formula due to
Ian G. Macdonald was given in .
Values
The first few values of the tau function are given in the following table :
Ramanujan's conjectures
observed, but did not prove, the following three properties of :
* if (meaning that is a
multiplicative function
In number theory, a multiplicative function is an arithmetic function ''f''(''n'') of a positive integer ''n'' with the property that ''f''(1) = 1 and
f(ab) = f(a)f(b) whenever ''a'' and ''b'' are coprime.
An arithmetic function ''f''(''n'') is ...
)
* for prime and .
* for all
primes
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...
.
The first two properties were proved by and the third one, called the
Ramanujan conjecture, was proved by
Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Pr ...
in 1974 as a consequence of his proof of the
Weil conjectures
In mathematics, the Weil conjectures were highly influential proposals by . They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.
Th ...
(specifically, he deduced it by applying them to a Kuga-Sato variety).
Congruences for the tau function
For and , define as the sum of the th powers of the divisors of . The tau function satisfies several congruence relations; many of them can be expressed in terms of . Here are some:
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#][Due to D. H. Lehmer]
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For prime, we have][
]
-
-
-
Explicit formula
In 1975 Douglas Niebur proved an explicit formula for the Ramanujan tau function:
:
This also shows that the tau function is always an integer.
Conjectures on ''τ''(''n'')
Suppose that is a weight- integer newform and the Fourier coefficients are integers. Consider the problem:
: Given that does not have complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...
, do almost all primes have the property that ?
Indeed, most primes should have this property, and hence they are called ''ordinary''. Despite the big advances by Deligne and Serre on Galois representations, which determine for coprime to , it is unclear how to compute . The only theorem in this regard is Elkies' famous result for modular elliptic curves, which guarantees that there are infinitely many primes such that , which thus are congruent to 0 modulo . There are no known examples of non-CM with weight greater than 2 for which for infinitely many primes (although it should be true for almost all ). There are also no known examples with for infinitely many . Some researchers had begun to doubt whether for infinitely many . As evidence, many provided Ramanujan's (case of weight 12). The only solutions up to 1010 to the equation are 2, 3, 5, 7, 2411, and .
conjectured that for all , an assertion sometimes known as Lehmer's conjecture. Lehmer verified the conjecture for up to (Apostol 1997, p. 22). The following table summarizes progress on finding successively larger values of for which this condition holds for all .
Ramanujan's ''L''-function
Ramanujan's ''L''-function is defined by
:
if and by analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new ...
otherwise. It satisfies the functional equation
:
and has the Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard Eul ...
:
Ramanujan conjectured that all nontrivial zeros of have real part equal to .
Notes
References
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*
*
*{{Citation
, last=Wilton
, first=J. R.
, title=Congruence properties of Ramanujan's function τ(''n'')
, year=1930
, journal=Proceedings of the London Mathematical Society
, volume=31
, pages=1–10
, doi=10.1112/plms/s2-31.1.1
Modular forms
Multiplicative functions
Srinivasa Ramanujan
Zeta and L-functions