On a conjecture of Corradi and Katai
We consider Goldbach type sums corresponding to the Liouville function and prove the existence of sufficient cancellations. We also consider applications to sign patterns in the Liouville function.
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arXiv subjects
Krishnarjun Krishnamoorthy
Publications and source records attributed to Krishnarjun Krishnamoorthy.
Prime detecting quasi-modular forms in higher level
In a previous work, the authors resolved a conjecture about the structure of prime-detecting quasi-modular forms by studying sign changes occurring in quasi-modular cusp forms. In this paper, we extend the considerations to prime-detecting quasi-modular forms of higher level, in particular describing the structure of the space of quasi-modular forms that detect primes in various arithmetic progressions. We also provide an ``analytic'' proof of the level one case.
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Diophantine tuples and Integral Ideals of $\mathbb{Q}(\sqrt{d})$
Suppose $n$ is the fundamental discriminant associated with a quadratic extension of $\mathbb{Q}$. We show that for every Diophantine $m$-tuple $ \{t_1, t_2, \ldots, t_m\} $ with the property $ D(n) $, there exists integral ideals $ \mathfrak{t}_1, \mathfrak{t}_2, \ldots, \mathfrak{t}_m $ of $ \mathbb{Q}(\sqrt{n}) $ and $c\in \{1,2\}$ such that $ t_i= c\mathcal{N}(\mathfrak{t}_i) $ for $ i=1,2, \ldots, m $. Here, $ \mathcal{N}(\cdot) $ denotes the norm map from $\mathbb{Q}(\sqrt{n})$ to $\mathbb{Q}$. Moreover, we explicitly construct the above ideals for Diophantine pairs $\{a_1, a_2\}$ whenever $\gcd(a_1, a_2) = 1$.
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On a conjecture about prime-detecting quasimodular forms
Motivated by weighted partition of $n$ that vanish if and only if $n$ is a prime, Craig, van Ittersum, and Ono conjecture a classification of quasimodular forms which detect primes in the sense that the $n$-th Fourier coefficient vanishes if and only if $n$ is a prime. In this paper, we prove this conjecture by showing that Fourier coefficients of quasimodular cusp forms exhibit infinitely many sign changes.
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On variants of Chowla's conjecture
We study the shifted convolution sums associated to completely multiplicative functions taking values in $\{\pm 1\}$ and give combinatorical proofs of two recent results in the direction of Chowla's conjecture. We also determine the corresponding "spectrum".
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Determination of Hilbert modular forms using squarefree coefficients
Let $F$ (over $\mathbb{Q}$) be a totally real number field of narrow class number $1$. We generalize a result of Kohnen on the determination of half integral weight modular forms by their Fourier coefficients supported on squarefree (algebraic) integers. We also give a soft proof that infinitely many Fourier coefficients supported on squarefree integers are non-vanishing.
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Moments of non-normal number fields -- II
Suppose $K$ is a number field and $a_K(m)$ is the number of integral ideals of norm equal to $m$ in $K$, then for any integer $l$, we asymptotically evaluate the sum \[ \sum_{m\leqslant T} a_K^l(m) \] as $T\to\infty$. We also consider the moments of the corresponding Dedekind zeta function. We prove lower bounds of expected order of magnitude and slightly improve the known upper bound for the second moment in the non-Galois case.
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A Note on Holomorphic Quantum Unique Ergodicity
In this paper we give a new proof of the Quantum Unique Ergodicity conjecture for holomorphic integral weight modular forms on the upper half plane. The proof requires only partial results towards the Ramanujan conjecture and the shifted convolution problem. Furthermore the proof is applicable to a wider class of cusp forms other than Hecke eigenforms. We also prove some interesting corollaries, particularly towards the Lehmer's conjecture on the non vanishing of the Fourier coefficients.
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On some symmetries of the base $ n $ expansion of $ 1/m $ : Comments on Artin's Primitive root conjecture
Suppose $ m,n\geq 2 $ are co prime integers. We prove certain new symmetries of the base $ n $ representation of $ 1/m $, and in particular characterize the subgroup generated by $ n $ inside $ (\mathbb{Z}/m\mathbb{Z})^\times $. As an application we give a sufficient condition for a prime $ p $ such that a non square number $ n $ is a primitive root modulo $ p $.
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On some symmetries of the base $ n $ expansion of $ 1/m $ : The Class Number connection
Suppose that $ m\equiv 1\mod 4 $ is a prime and that $ n\equiv 3\mod 4 $ is a primitive root modulo $ m $. In this paper we obtain a relation between the class number of the imaginary quadratic field $ \Q(\sqrt{-nm}) $ and the digits of the base $ n $ expansion of $ 1/m $. Secondly, if $ m\equiv 3\mod 4 $, we study some convoluted sums involving the base $ n $ digits of $ 1/m $ and arrive at certain congruence relations involving the class number of $ \Q(\sqrt{-m}) $ modulo certain primes $ p $ which properly divide $ n+1 $.
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