Journal•ISSN: 1793-7310
International Journal of Number Theory
World Scientific
About: International Journal of Number Theory is an academic journal published by World Scientific. The journal publishes majorly in the area(s): Modular form & Prime (order theory). It has an ISSN identifier of 1793-7310. Over the lifetime, 1940 publications have been published receiving 13032 citations. The journal is also known as: IJNT.
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TL;DR: In this article, it was shown that if and only if ρ pδ(e)|A| is an interval, there is a relation to estimates on incomplete Kloosterman sums.
Abstract: In this paper we establish new estimates on sum-product sets and certain exponential sums in finite fields of prime order. Our first result is an extension of the sum-product theorem from [8] when sets of different sizes are involed. It is shown that if and pe pδ (e)|A|. Next we exploit the Szemeredi–Trotter theorem in finite fields (also obtained in [8]) to derive several new facts on expanders and extractors. It is shown for instance that the function f(x,y) = x(x+y) from to satisfies |F(A,B)| > pβ for some β = β (α) > α whenever and |A| ~ |B|~ pα, 0 pρ where ρ pδ(e)|A|. This is the finite fields version of a problem considered in [11]. If A is an interval, there is a relation to estimates on incomplete Kloosterman sums. In the Appendix, we obtain an apparently new bound on bilinear Kloosterman sums over relatively short intervals (without the restrictions of Karatsuba's result [14]) which is of relevance to problems involving the divisor function (see [1]) and the distribution (mod p) of certain rational functions on the primes (cf. [12]).
329 citations
TL;DR: In this paper, the p-divisibility of MHS is studied and some generalizations of the classical Wolstenholme's Theorem to both homogeneous and non-homogeneous sums are provided.
Abstract: In this paper, we will study the p-divisibility of multiple harmonic sums (MHS) which are partial sums of multiple zeta value series. In particular, we provide some generalizations of the classical Wolstenholme's Theorem to both homogeneous and non-homogeneous sums. We make a few conjectures at the end of the paper and provide some very convincing evidence.
122 citations
TL;DR: In this article, an analog of Ramanujan's "Most Beautiful Identity" is presented, which is closely related to the beautiful results involving the cubic continued fraction in this paper.
Abstract: In this paper, we prove an analog of Ramanujan's "Most Beautiful Identity". This analog is closely related to Ramanujan's beautiful results involving the cubic continued fraction.
118 citations
TL;DR: In this article, the authors consider holomorphic vector-valued modular forms F of integral weight k on the full modular group Γ = SL(2, ℤ) corresponding to representations of Γ of arbitrary finite dimension p. Assuming that the component functions of F are linearly independent, the inequality k ≥ 1 - p always holds, and that equality holds only in the trivial case when p = 1 and k = 0.
Abstract: We consider holomorphic vector-valued modular forms F of integral weight k on the full modular group Γ = SL(2, ℤ) corresponding to representations of Γ of arbitrary finite dimension p. Assuming that the component functions of F are linearly independent, we prove that the inequality k ≥ 1 - p always holds, and that equality holds only in the trivial case when p = 1 and k = 0. For any p ≥ 2, we show how to construct large numbers of representations of Γ for which k = 2 - p. The key idea is to consider representations of Γ on spaces of solutions of certain linear differential equations whose coefficients are modular forms.
79 citations
TL;DR: In this paper, the authors established congruences involving central binomial coefficients as well as Catalan numbers and showed that for every n = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 34, 34
Abstract: In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let p be a prime and let a be any positive integer. We determine for d = 0, …, pa and for δ = 0, 1. We also show that for every n = 0, 1, 2, …, where Cm is the Catalan number , and is the Legendre symbol.
78 citations