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Template:Infobox scientist Giuseppe Melfi (born June 11, 1967) is an Italo-Swiss mathematician. He achieved his PhD in mathematics in 1997 at the University of Pisa. After some years spent at the University of Lausanne, he works now at the University of Neuchâtel, as well as at the University of Applied Sciences Western Switzerland and at the local University of Teacher Education.

His work

His major contributions are in the theory 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 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, where he found new Ramanujan-type identities for the sum-of-divisor functions. In particular he found the remarkable identity

$\sum _{k=0}^{n}\sigma _{1}(3k+1)\sigma _{1}(3n-3k+1)={\frac {1}{9}}\sigma _{3}(3n+2),$

where $\sigma _{k}(n)$ is the sum of the $k$-th powers of divisors of $n$.

Other topics of research include sum-free sequences in a joint work with Erdős and Deshouillers and other problems in elementary number theory.

In applied mathematics his research interests include probability and simulation.

Selected research publications

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See also

• Applications of randomness

External links

• Giuseppe Melfi's home page
• The proof of conjectures on practical numbers and the joint work with Paul Erdős on Zentralblatt.
• Tables of practical numbers compiled by Giuseppe Melfi
• Academic research query for "Giuseppe Melfi"

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