Giuseppe Melfi (June 11, 1967) is an Italo-

Swiss Swiss may refer to: * the adjectival form of Switzerland * Swiss people Places * Swiss, Missouri * Swiss, North Carolina *Swiss, West Virginia * Swiss, Wisconsin Other uses *Swiss-system tournament, in various games and sports *Swiss Internation ...

mathematician who works on

practical number In number theory, a practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. For example, 12 is a practical number because all the numbers from 1 ...

s and

modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...

Career

He gained his PhD in mathematics in 1997 at the

University of Pisa The University of Pisa ( it, Università di Pisa, UniPi), officially founded in 1343, is one of the oldest universities in Europe. History The Origins The University of Pisa was officially founded in 1343, although various scholars place ...

. After some time spent at the

University of Lausanne The University of Lausanne (UNIL; french: links=no, Université de Lausanne) in Lausanne, Switzerland was founded in 1537 as a school of Protestant theology, before being made a university in 1890. The university is the second oldest in Switzer ...

during 1997-2000, Melfi was appointed at the

University of Neuchâtel The University of Neuchâtel (UniNE) is a French-speaking university based in Neuchâtel, Switzerland. The university has four faculties (schools) and more than a dozen institutes, including arts and human sciences, natural sciences, law and eco ...

, as well as at the

University of Applied Sciences Western Switzerland The University of Applied Sciences and Arts of Western Switzerland (french: HES-SO Haute école spécialisée de Suisse occidentale) is situated in Western Switzerland. It is formally accredited by the Swiss Accreditation Council. The universit ...

and at the local University of Teacher Education.

Work

His major contributions are in the field of

s. This prime-like sequence of numbers is known for having an asymptotic behavior and other distribution properties similar to the sequence of

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 ...

. Melfi proved two conjectures both raised in 1984 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

m-2,m,m+2

. Another notable contribution has been in an application of the theory of

modular forms In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the group action of the modular group, and also satisfying a growth condition. The theory of ...

, 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.Ramanujan, S., ''On certain arithmetical functions'', Transactions of the Cambridge Philosophical Society, 22 (9), 1916, p. 159-184. In particular he found the remarkable identity :

\sum_ \sigma(k)\sigma(n-k)=\frac19\sigma_3(n) \qquad \mboxn\equiv2\bmod3

where

\sigma(n)

is the sum of the divisors of

n

and

\sigma_3(n)

is the sum of the third powers of the divisors of

n

. Among other problems in elementary

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, he is the author of a theorem that allowed him to get a 5328-digit number that is at present the largest known primitive

weird number 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 (divisors including 1 but not itself) of the number is greater than the number, but no subset of those diviso ...

. In applied mathematics his research interests include

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

and

simulation A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or proc ...

Selected research publications

* *. * *

References

External links

Giuseppe Melfi's home page

The proof of conjectures on practical numbers
an
the joint work with Paul Erdős
on

Zentralblatt zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastructur ...

.
Tables of practical numbers
compiled by Giuseppe Melfi
Academic research query for "Giuseppe Melfi"
{{DEFAULTSORT:Melfi, Giuseppe 1967 births 20th-century Italian mathematicians 21st-century Italian mathematicians Living people Number theorists Mathematicians from Sicily Academic staff of the University of Neuchâtel

Career

Work

Selected research publications

See also

References

External links