Giuseppe Melfi

Giuseppe Melfi
Born	11 June 1967 Uznach, Switzerland
Nationality	Italy   Switzerland
Fields	Mathematics
Institutions	University of Neuchâtel University of Applied Sciences Western Switzerland University of Teacher Education BEJUNE
Known for	Practical numbers Ramanujan-type identities
Notable awards	Premio Ulisse (2010)

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

$\sum_{k=0}^n\sigma_1(3k+1)\sigma_1(3n-3k+1)=\frac19\sigma_3(3n+2),$

where $\sigma_k(n)$ is the sum of the $k$-th powers of divisors of $n$.

In a joint work with Erdős and Deshouillers he proved the existence of sum-free sequences having a polynomial growing: so he has an Erdős number = 1 being one of the 511 direct collaborators to the Hungarian mathematician disappeared in 1996.^[1]

Among other problems in elementary number theory, he is the author of a theorem that allowed him getting a 5328-digit number that is at present the largest known primitive weird number.

In applied mathematics his research interests include probability and simulation.

Selected research publications

• Giuseppe Melfi and Yadolah Dodge (2008). Premiers pas en simulation. Springer-Verlag. ISBN 978-2-287-79493-3.
• Deshouillers, Jean-Marc; Erdős, Pál; Melfi, Giuseppe (1999), "On a question about sum-free sequences", Discrete Mathematics 200: 49–54, doi:10.1016/s0012-365x(98)00322-7, MR 1692278.
• Melfi, Giuseppe (2015). "On the conditional infiniteness of primitive weird numbers". Journal of Number Theory (Elsevier) 147: 508–514. doi:10.1016/j.jnt.2014.07.024.
• Melfi, Giuseppe (1996), "On two conjectures about practical numbers", Journal of Number Theory 56 (1): 205–210, doi:10.1006/jnth.1996.0012, MR 1370203

See also

• Applications of randomness

References

1. Erdős Number Project, list of 511 direct collaborators with Erdős Number=1

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"