<%@ Page Language="C#" MasterPageFile="~/Main.master" AutoEventWireup="true" Title="Volume 13 Issue 3" %>
A New Lower Bound on the Number of Odd Values of the Ordinary Partition Function
Dennis Eichhorn
Department of Mathematics, University of California, Irvine, CA 92697, USA
deichhor@math.uci.edu
Annals of Combinatorics 13 (3) pp.297-303 September, 2009
AMS Subject Classification: 05A17, 11P81, 11P82
Abstract:
The parity of $p(n)$, the ordinary partition function, has been studied for at least a century, yet it still remains something of a mystery. Although much work has been done, the known lower bounds for the number of even and odd values of $p(n)$ for $n \leq N$ still appear to have a great deal of room for improvement. In this paper, we use classical methods to give a new lower bound for the number of odd values of $p(n)$.
Keywords: parity, partitions, odd values, p(n)

References:

1. Ahlgren, S.: Distribution of parity of the partition function in arithmetic progressions. Indag. Math. (N.S.) 10, 173-181 (1999)

2. Berndt, B.C., Yee, A.J., Zaharescu, A.: On the parity of partition functions. Internat. J. Math. 14, 437-459 (2003)

3. Berndt, B.C., Yee, A.J., Zaharescu, A.: New theorems on the parity of partition functions. J. Reine Angew. Math. 566, 91-109 (2004)

4. Bombieri, E., Davenport, H.: Small differences between prime numbers. Proc. Roy. Soc. Ser. A 293, 1-18 (1966)

5. Garvan, F., Kim, D., Stanton, D.: Cranks and t-cores. Invent. Math. 101, 1-17 (1990)

6. Hirschhorn, M.D., Sellers, J.A.: Two congruences involving 4-cores. Electron. J. Combin. 3, #10 (1996)

7. Hirschhorn, M.D., Subbarao, M.V.: On the parity of p(n). Acta Arith. 50, 355-356 (1988)

8. Kolberg, O.: Note on the parity of the partition function. Math. Scand. 7, 377-378 (1959)

9. Ono, K.: Parity of the partition function in arithmetic progressions. J. Reine Angew. Math. 472, 1-15 (1996)

10. Nicolas, J.-L., Ruzsa, I.Z., S´ark¨ozy, A.: On the parity of additive representation functions. J. Number Theory 73, 292-317 (1998)

11. Parkin, T.R., Shanks, D.: On the distribution of parity in the partition function. Math. Comp. 21, 466-480 (1967)