Giuseppe Melfi

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Giuseppe Melfi
Melfi2004.jpg
Born (1967-06-11) 11 June 1967 (age 56)
NationalityFlag of Italy.svg  Italy
Flag of Switzerland (Pantone).svg   Switzerland
Known for Practical numbers
Ramanujan-type identities
AwardsPremio Ulisse (2010) [1]
Scientific career
Fields Mathematics
Institutions University of Neuchâtel
University of Applied Sciences Western Switzerland
University of Teacher Education BEJUNE

Giuseppe Melfi (June 11, 1967) is an Italo-Swiss mathematician who works on practical numbers and modular forms.

Contents

Career

He gained his PhD in mathematics in 1997 at the University of Pisa. After some time spent at the University of Lausanne during 1997-2000, Melfi was appointed at the University of Neuchâtel, as well as at the University of Applied Sciences Western Switzerland and at the local University of Teacher Education.

Work

His major contributions are in the field of practical numbers. This prime-like sequence of numbers is known for having an asymptotic behavior and other distribution properties similar to the sequence of primes. Melfi proved two conjectures both raised in 1984 [2] one of which is the corresponding of the Goldbach conjecture for practical numbers: every even number is a sum of two practical numbers. He also proved that there exist infinitely many triples of practical numbers of the form .

Another notable contribution has been in an application of the theory of modular forms, where he found new Ramanujan-type identities for the sum-of-divisor functions. His seven new identities extended the ten other identities found by Ramanujan in 1913. [3] In particular he found the remarkable identity

where is the sum of the divisors of and is the sum of the third powers of the divisors of .

Among other problems in elementary number theory, he is the author of a theorem that allowed him to get a 5328-digit number that has been for a while the largest known primitive weird number.

In applied mathematics his research interests include probability and simulation.

Selected research publications

See also

Related Research Articles

In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function f(n) whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their definition the requirement that an arithmetical function "expresses some arithmetical property of n". There is a larger class of number-theoretic functions that do not fit this definition, for example, the prime-counting functions. This article provides links to functions of both classes.

<span class="mw-page-title-main">Amicable numbers</span> Pair of integers related by their divisors

Amicable numbers are two different natural numbers related in such a way that the sum of the proper divisors of each is equal to the other number. That is, s(a)=b and s(b)=a, where s(n)=σ(n)-n is equal to the sum of positive divisors of n except n itself (see also divisor function).

<span class="mw-page-title-main">Carmichael number</span> Composite number in number theory

In number theory, a Carmichael number is a composite number which in modular arithmetic satisfies the congruence relation:

<span class="mw-page-title-main">Partition function (number theory)</span> The number of partitions of an integer

In number theory, the partition functionp(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has the five partitions 1 + 1 + 1 + 1, 1 + 1 + 2, 1 + 3, 2 + 2, and 4.

<span class="mw-page-title-main">Weird number</span> Number which is abundant but not semiperfect

In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors of the number is greater than the number, but no subset of those divisors sums to the number itself.

<span class="mw-page-title-main">Lagrange's four-square theorem</span> Every natural number can be represented as the sum of four integer squares

Lagrange's four-square theorem, also known as Bachet's conjecture, states that every natural number can be represented as a sum of four non-negative integer squares. That is, the squares form an additive basis of order four.

<span class="mw-page-title-main">Divisor function</span> Arithmetic function related to the divisors of an integer

In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer. It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum.

In mathematics, a harmonic divisor number or Ore number is a positive integer whose divisors have a harmonic mean that is an integer. The first few harmonic divisor numbers are

In mathematics, an untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer. That is, these numbers are not in the image of the aliquot sum function. Their study goes back at least to Abu Mansur al-Baghdadi, who observed that both 2 and 5 are untouchable.

<span class="mw-page-title-main">Practical number</span> Number such that it and all smaller numbers may be represented as sums of its distinct divisors

In number theory, a practical number or panarithmic number is a positive integer such that all smaller positive integers can be represented as sums of distinct divisors of . For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2.

<span class="mw-page-title-main">Ramanujan tau function</span>

The Ramanujan tau function, studied by Ramanujan, is the function defined by the following identity:

Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generalized in the theory of automorphic forms.

In mathematics, a superabundant number is a certain kind of natural number. A natural number n is called superabundant precisely when, for all m < n:

<span class="mw-page-title-main">Colossally abundant number</span> Type of natural number

In number theory, a colossally abundant number is a natural number that, in a particular, rigorous sense, has many divisors. Particularly, it is defined by a ratio between the sum of an integer's divisors and that integer raised to a power higher than one. For any such exponent, whichever integer has the highest ratio is a colossally abundant number. It is a stronger restriction than that of a superabundant number, but not strictly stronger than that of an abundant number.

A Fibonacci prime is a Fibonacci number that is prime, a type of integer sequence prime.

In mathematics, a Kloosterman sum is a particular kind of exponential sum. They are named for the Dutch mathematician Hendrik Kloosterman, who introduced them in 1926 when he adapted the Hardy–Littlewood circle method to tackle a problem involving positive definite diagonal quadratic forms in four as opposed to five or more variables, which he had dealt with in his dissertation in 1924.

In mathematics, a natural number a is a unitary divisor of a number b if a is a divisor of b and if a and are coprime, having no common factor other than 1. Equivalently, a divisor a of b is a unitary divisor if and only if every prime factor of a has the same multiplicity in a as it has in b.

In number theory, Ramanujan's sum, usually denoted cq(n), is a function of two positive integer variables q and n defined by the formula

In number theory, the aliquot sums(n) of a positive integer n is the sum of all proper divisors of n, that is, all divisors of n other than n itself. That is,

References

  1. "Consegnati i premi "Ulisse", La Sicilia, 15th August 2010, p. 38" . Retrieved 2021-08-10.
  2. Margenstern, M., Résultats et conjectures sur les nombres pratiques, C, R. Acad. Sci. Sér. 1 299, No. 18 (1984), 895-898.
  3. Ramanujan, S., On certain arithmetical functions, Transactions of the Cambridge Philosophical Society, 22 (9), 1916, p. 159-184.