arXiv subjects

Sanoli Gun

Publications and source records attributed to Sanoli Gun.

31 records · Page 2Linked to original sources

arXiv2018

The first simultaneous sign change and non-vanishing of Hecke eigenvalues of newforms

Let $f$ and $g$ be two distinct newforms which are normalized Hecke eigenforms of weights $k_1, k_2 \ge 2$ and levels $N_1, N_2 \ge 1$ respectively. Also let $a_f(n)$ and $a_g(n)$ be the $n$-th Fourier-coefficients of $f$ and $g$ respectively. In this article, we investigate the first sign change of the sequence $\{a_f(p^α)a_g(p^α) \}_{p^α \in \N, α\le 2}$, where $p$ is a prime number. We further study the non-vanishing of the sequence $\{a_f(n)a_g(n) \}_{n \in \N}$ and derive bounds for first non-vanishing term in this sequence. We also show, using ideas of Kowalski-Robert-Wu and Murty-Murty, that there exists a set of primes $S$ of natural density one such that for any prime $p \in S$, the sequence $\{a_f(p^n)a_g(p^m) \}_{n,m \in \N}$ has no zero elements. This improves a recent work of Kumari and Ram Murty. Finally, using $\B$-free numbers, we investigate simultaneous non-vanishing of coefficients of $m$-th symmetric power $L$-functions of non-CM forms in short intervals.

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arXiv2018

On Hecke eigenvalues of Siegel modular forms in the Maass space

In this article, we prove an omega-result for the Hecke eigenvalues $λ_F(n)$ of Maass forms $F$ which are Hecke eigenforms in the space of Siegel modular forms of weight $k$, genus two for the Siegel modular group $Sp_2(\Z)$. In particular, we prove $$ λ_F(n)= Ω(n^{k-1}\text{exp} (c \frac{\sqrt{\log n}}{\log\log n})), $$ when $c>0$ is an absolute constant. This improves the earlier result $$ λ_F(n)= Ω(n^{k-1} (\frac{\sqrt{\log n}}{\log\log n})) $$ of Das and the third author. We also show that for any $n \ge 3$, one has $$ λ_F(n) \leq n^{k-1}\text{exp} \left(c_1\sqrt{\frac{\log n}{\log\log n}}\right), $$ where $c_1>0$ is an absolute constant. This improves an earlier result of Pitale and Schmidt. Further, we investigate the limit points of the sequence $\{\frac{λ_F(n)}{n^{k-1}}\}_{n \in \N}$ and show that it has infinitely many limit points. Finally, we show that $λ_F(n) >0$ for all $n$, a result earlier proved by Breulmann by a different technique.

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arXiv2017

Multiple Lerch zeta functions and an idea of Ramanujan

In this article, we derive meromorphic continuation of multiple Lerch zeta functions by generalising an elegant identity of Ramanujan. Further, we describe the set of all possible singularities of these functions. Finally, for the multiple Hurwitz zeta functions, we list the exact set of singularities.

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arXiv2016

Linear and algebraic independence of Generalized Euler-Briggs constants

Possible transcendental nature of Euler's constant $γ$ has been the focus of study for sometime now. One possible approach is to consider $γ$ not in isolation, but as an element of the infinite family of generalised Euler-Briggs constants. In a recent work \cite{GSS}, it is shown that the infinite list of generalized Euler-Briggs constants can have at most one algebraic number. In this paper, we study the dimension of spaces generated by these generalized Euler-Briggs constants over number fields. More precisely, we obtain non-trivial lower bounds (see \thmref{pre} and \thmref{linear-ind}) on the dimension of these spaces and consequently establish the infinite dimensionality of the space spanned. Further, we study linear and algebraic independence of these constants over the field of all algebraic numbers.

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arXiv2016

A number field extension of a question of Milnor

Milnor formulated a conjecture about rational linear independence of some special Hurwitz zeta values. The second and third authors along with Ram Murty studied this conjecture and suggested an extension of Milnor's conjecture. In this note, we investigate the number field generalisation of this extended Milnor conjecture. We indicate the motivation for considering this number field case by noting that such a phenomenon is true in an analogous context. We also study some new spaces related to normalised Hurwitz zeta values.

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arXiv2014

On the zeros of generalized Hurwitz zeta functions

In this note, we prove the existence of infinitely many zeros of certain generalized Hurwitz zeta functions in the domain of absolute convergence. This is a generalization of a classical problem of Davenport, Heilbronn and Cassels about the zeros of the Hurwitz zeta function.

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arXiv2014

On the zeros of weakly holomorphic modular forms

In this article, we study the nature of zeros of weakly holomorphic modular forms. In particular, we prove results about transcendental zeros of modular forms of higher levels and for certain Fricke groups which extend a work of Kohnen. Furthermore, we investigate the algebraic independence of values of weakly holomorphic modular forms.

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arXiv2014

A variant of Lehmer's conjecture, II: The CM-case

Let $f$ be a normalized Hecke eigenform with rational integer Fourier coefficients. It is an interesting question to know how often an integer $n$ has a factor common with the $n$-th Fourier coefficient of $f$. The second author \cite{kumar3} showed that this happens very often. In this paper, we give an asymptotic formula for the number of integers $n$ for which $(n, a(n))=1$, where $a(n)$ is the $n$-th Fourier coefficient of a normalized Hecke eigenform $f$ of weight $2$ with rational integer Fourier coefficients and has complex multiplication.

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arXiv2014

Divisors of Fourier coefficients of modular forms

Let $d(n)$ denote the number of divisors of $n$. In this paper, we study the average value of $d(a(p))$, where $p$ is a prime and $a(p)$ is the $p$-th Fourier coefficient of a normalized Hecke eigenform of weight $k \ge 2$ for $Γ_0(N)$ having rational integer Fourier coefficients.

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arXiv2014

An explicit bound for the first sign change of the Fourier coefficients

We give an explicit upper bound for the first sign change of the Fourier coefficients of an arbitrary non-zero Siegel cusp form $F$ of even integral weight on the Siegel modular group of arbitrary genus $ g\geq 2 $.

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arXiv2013

A Note on Fourier-Jacobi coefficients of Siegel modular forms

Let F be a Siegel cusp form of weight k and genus n>1 with Fourier-Jacobi coefficients f_m. In this article, we estimate the growth of the Petersson norms of f_m, where m runs over an arithmetic progression. This result sharpens a recent result of Kohnen in [5].

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arXiv2013

The Digamma function, Euler-Lehmer constants and their $p$-adic counterparts

The goal of this article is twofold. We first extend a result of Murty and Saradha \cite{MS} related to the digamma function at rational arguments. Further, we extend another result of the same authors \cite{MS1} about the nature of $p$-adic Euler-Lehmer constants.

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arXiv2012

The circle method and non lacunarity of Modular Functions

Serre proved that any holomorphic cusp form of weight one for $Γ_1(N)$ is lacunary while a holomorphic modular form for $Γ_1(N)$ of higher integer weight is lacunary if and only if it is a linear combination of cusp forms of CM-type (see Serre, subsections 7.6 and 7.7). In this paper, we show that when a non-zero modular function of arbitrary real weight for any finite index subgroup of the modular group ${\SL}_2(\Z)$ is lacunary, it is necessarily holomorphic on the upper-half plane, finite at the cusps and has non-negative weight.

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