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Mock modular forms and singular combinatorial series

Amanda Folsom, Susie Kimport (2013)

Acta Arithmetica

A celebrated result of Bringmann and Ono shows that the combinatorial rank generating function exhibits automorphic properties after being completed by the addition of a non-holomorphic integral. Since then, automorphic properties of various related combinatorial families have been studied. Here, extending work of Andrews and Bringmann, we study general infinite families of combinatorial q-series pertaining to k-marked Durfee symbols, in which we allow additional singularities. We show that these...

Mod 2 normal numbers and skew products

Geon Ho Choe, Toshihiro Hamachi, Hitoshi Nakada (2004)

Studia Mathematica

Let E be an interval in the unit interval [0,1). For each x ∈ [0,1) define dₙ(x) ∈ 0,1 by d ₙ ( x ) : = ∑ i = 1 n 1 E ( 2 i - 1 x ) ( m o d 2 ) , where t is the fractional part of t. Then x is called a normal number mod 2 with respect to E if N - 1 ∑ n = 1 N d ₙ ( x ) converges to 1/2. It is shown that for any interval E ≠(1/6, 5/6) a.e. x is a normal number mod 2 with respect to E. For E = (1/6, 5/6) it is proved that N - 1 ∑ n = 1 N d ₙ ( x ) converges a.e. and the limit equals 1/3 or 2/3 depending on x.

Mod p structure of alternating and non-alternating multiple harmonic sums

Jianqiang Zhao (2011)

Journal de Théorie des Nombres de Bordeaux

The well-known Wolstenholme’s Theorem says that for every prime p > 3 the ( p - 1 ) -st partial sum of the harmonic series is congruent to 0 modulo p 2 . If one replaces the harmonic series by ∑ k ≥ 1 1 / n k for k even, then the modulus has to be changed from p 2 to just p . One may consider generalizations of this to multiple harmonic sums (MHS) and alternating multiple harmonic sums (AMHS) which are partial sums of multiple zeta value series and the alternating Euler sums, respectively. A lot of results along this direction...

Modifications of the Eratosthenes sieve

Jerzy Browkin, Hui-Qin Cao (2014)

Colloquium Mathematicae

We discuss some cancellation algorithms such that the first non-cancelled number is a prime number p or a number of some specific type. We investigate which numbers in the interval (p,2p) are non-cancelled.

Modified proof of a local analogue of the Grothendieck conjecture

Victor Abrashkin (2010)

Journal de Théorie des Nombres de Bordeaux

A local analogue of the Grothendieck Conjecture is an equivalence between the category of complete discrete valuation fields K with finite residue fields of characteristic p ≠ 0 and the category of absolute Galois groups of fields K together with their ramification filtrations. The case of characteristic 0 fields K was studied by Mochizuki several years ago. Then the author of this paper proved it by a different method in the case p > 2 (but with no restrictions on the characteristic of K ). In this paper...

Modular case of Levinson's theorem

Damien Bernard (2015)

Acta Arithmetica

We evaluate the integral mollified second moment of L-functions of primitive cusp forms and we obtain, for such L-functions, an explicit positive proportion of zeros which lie on the critical line.

Currently displaying 241 – 260 of 440