Narrow Results

Let $f(z)$ be the normalized primitive holomorphic Hecke eigenforms of even integral weight $k$ for the full modular group SL$(2,\mathbb{Z})$. In this paper, we investigate the second moment of Fourier coefficients of symmetric power $L$-function $L(s,{\mathrm{\text{sym}}}^{j}f)$ for $j\ge 2$ and improve on previous error estimates. We also consider the second moment of ${\lambda }_f(n^2)$.

Let p be a prime and let f be any cusp form of level l ∈ {2,3,5,7,13} whose weight satisfy a certain congruence modulo (p-1). Then we exhibit explicit linear combinations of the coefficients of f that must be divisible by p. For a normalized Hecke eigenform, this translates (under mild restrictions) into the pth coefficient itself being divisible by a prime ideal above p in the ring generated by the coefficients of f. This provides many instances of the so-called non-ordinary primes. We also discuss linear relations satisfied universally on the space of modular forms of these levels. These results extend recent work of Choie, Kohnen and Ono in the level 1 case.

We determine formulae for the numbers of representations of a positive integer by certain octonary quadratic forms whose coefficients are 1, 2, 3 and 6.

Using modular forms we determine the number of representations of a positive integer by certain diagonal octonary quadratic forms with coefficients 1, 3 or 9.

Using modular forms, we determine the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients 1, 3 or 9.

We determine the coefficients of the Fourier series of a class of eta quotients of weight 2. For example, we show that

We determine the convolution sums ${\sum }_{l+27m=n}\sigma (l)\sigma (m)$ and ${\sum }_{l+32m=n}\sigma (l)\sigma (m)$ for all positive integers $n$. We then use these evaluations together with known evaluations of other convolution sums to determine the numbers of representations of $n$ by the octonary quadratic forms $x_1^2+x_1{x}_2+x_2^2+x_3^2+x_3{x}_4+x_4^2+9(x_5^2+x_5{x}_6+x_6^2+x_7^2+x_7{x}_8+x_8^2)$ and $x_1^2+x_2^2+x_3^2+x_4^2+8(x_5^2+x_6^2+x_7^2+x_8^2)$. A modular form approach is used.

Using modular forms, we determine formulas for the number of representations of a positive integer by diagonal octonary quadratic forms with coefficients $1$, $2$, $3$ or $6$.

We study the signs of the Fourier coefficients of a newform. Let $f$ be a normalized newform of weight $k$ for ${\mathrm{\Gamma }}_0(N)$. Let $a_f(n)$ be the $n$th Fourier coefficient of $f$. For any fixed positive integer $m$, we study the distribution of the signs of ${\{a_f(p^m)\}}_p$, where $p$ runs over all prime numbers. We also find out the abscissas of absolute convergence of two Dirichlet series with coefficients involving the Fourier coefficients of cusp forms and the coefficients of symmetric power $L$-functions.

We prove a bound for the Fourier coefficients of a cusp form of integral weight which is not a newform by computing an explicit orthogonal basis for the space of cusp forms of given integral weight and level.

We find formulas for convolutions of sum of divisor functions twisted by the Dirichlet character $\left(\frac{-4}{\ast }\right)$, which are analogous to Ramanujan’s formula for convolution of usual sum of divisor functions. We use the theory of modular forms to prove our results.

The triple correlation

For a positive definite ternary integral quadratic form $f$, let $r(n,f)$ be the number of representations of an integer $n$ by $f$. A ternary quadratic form $f$ is said to be a generalized Bell ternary quadratic form if $f$ is isometric to $x^2+2^{\alpha }{y}^2+2^{\beta }{z}^2$ for some nonnegative integers $\alpha ,\beta$. In this paper, we give a closed formula for $r(n,f)$ for a generalized Bell ternary quadratic form $f(x,y,z)=x^2+2^{\alpha }{y}^2+2^{\beta }{z}^2$ with $0\le \alpha \le \beta \le 6$ and class number greater than $1$ by using the Minkowski–Siegel formula and bases for spaces of cusp forms of weight $\frac{3}{2}$ and level $2^t$ with $t=6,7,8$ consisting of eta-quotients.

Let f be a positive definite (non-classic) integral quaternary quadratic form. We say f is strongly s-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this paper, we show that there are exactly 34 strongly s-regular diagonal quaternary quadratic forms representing 1 (see Table 1). In particular, we use eta-quotients to prove the strong s-regularity of the quaternary quadratic form $x^2+2y^2+3z^2+10w^2$, which is, in fact, of class number 2 (see Lemma 4.5 and Proposition 4.6).

In this paper, we study the existence of the Miller basis for $S_k(N)$, the space of cusp forms of weight $k$ and level $N$. We seek an easy criterion to determine whether or not the space $S_k(N)$ admits the Miller basis. In particular, we prove that if the level $N$ is sufficiently composite, then $S_k(N)$ does not have the Miller basis. We also prove that there exists a constant $w(N)$ such that $S_k(N)$ admits the Miller basis if and only if $S_{k+w(N)}(N)$ admits the Miller basis when $k\ge 4$.

In this paper, we establish an asymptotic lower bound estimate on the contribution of cuspidal automorphic representations of ${\mathrm{\text{GL}}}_4({𝔸}_{\mathbb{Q}})$ to cuspidal cohomology of the ${\mathrm{\text{GL}}}_4$ which are obtained from automorphic tensor product of two automorphic representations of ${\mathrm{\text{GL}}}_2({𝔸}_{\mathbb{Q}})$ of given weights and with varying level structure. In the end, we also prove that the symmetric cube of a representation of ${\mathrm{\text{GL}}}_2$ and the automorphic tensor product of two representations of ${\mathrm{\text{GL}}}_2$ cannot be equal (up to a twist by a character of ${\mathrm{\text{GL}}}_1$) to each other, under the suitable assumptions on the representations being cuspidal and cohomological.

In this paper, we investigate the square of the normalized Fourier coefficients of the primitive cusp forms $f$ and its symmetric-lift at integers with a fixed number of distinct prime divisors, and present asymptotic formulas for them in short intervals.

We study the behavior of the shifted convolution sum involving fourth power of the Fourier coefficients of holomorphic cusp forms with a weight function to be the k-full kernel function for any fixed integer $k\ge 2$. In this paper, we will generalize the problem of Lü and Wang [Shifted convolution sums of Fourier coefficients with square-full kernel functions, Lithuanian Math. J. 60 (2020) 51–63], and improve the results of Venkatasubbareddy and Sankaranarayanan [On certain kernel functions and shifted convolution sums of Fourier coefficients, J. Number Theory 258 (2024) 414–450], the error term from $x^{\frac{520k+23}{543k}+𝜀}$ to $x^{\frac{500k+23}{523k}+𝜀}$.

Zagier, in 1981, conjectured that the constant term of an automorphic function associated to the Ramanujan delta function, i.e. $y^{12}{\sum }_{n=1}^{\infty }{\tau }^2(n)e^{-4\pi ny},$ has a connection with the nontrivial zeros of $\zeta (s)$. This conjecture was finally proved by Hafner and Stopple in 2000. Recently, Chakraborty et al. extended this observation for any normalized Hecke eigenform over $S{L}_2(\mathbb{Z})$. In this paper, we study the infinite series ${\sum }_{n=1}^{\infty }{c}_f^{\ell }(n)n^{\nu /2}{K}_{\nu }(y\sqrt{n})$ for $\ell =1,2$, where $c_f(n)$ denotes the nth Fourier coefficient of a normalized Hecke eigenform $f(z)$ and $K_{\nu }$ represents the modified Bessel function of the second kind of order $\nu$. We generalize a recent identity of Berndt et al. We also observe that the aforementioned series corresponding to $\ell =2$ has a connection with the nontrivial zeros of $\zeta (s)$.