A symplectic restriction problem

We investigate the norm of a degree 2 Siegel modular form of asymptotically large weight whose argument is restricted to the 3-dimensional subspace of its imaginary part. On average over Saito–Kurokawa lifts an asymptotic formula is established that is consistent with the mass equidistribution conjecture on the Siegel upper half space as well as the Lindelöf hypothesis for the corresponding Koecher–Maaß series. The ingredients include a new relative trace formula for pairs of Heegner periods.

On the variance of squarefree integers in short intervals and arithmetic progressions

We evaluate asymptotically the variance of the number of squarefree integers up to x in short intervals of length $$H < x^{6/11 - \varepsilon }$$ H < x 6 / 11 - ε and the variance of the number of squarefree integers up to x in arithmetic progressions modulo q with $$q > x^{5/11 + \varepsilon }$$ q > x 5 / 11 + ε . On the assumption of respectively the Lindelöf Hypothesis and the Generalized Lindelöf Hypothesis we show that these ranges can be improved to respectively $$H < x^{2/3 - \varepsilon }$$ H < x 2 / 3 - ε and $$q > x^{1/3 + \varepsilon }$$ q > x 1 / 3 + ε . Furthermore we show that obtaining a bound sharp up to factors of $$H^{\varepsilon }$$ H ε in the full range $$H < x^{1 - \varepsilon }$$ H < x 1 - ε is equivalent to the Riemann Hypothesis. These results improve on a result of Hall (Mathematika 29(1):7–17, 1982) for short intervals, and earlier results of Warlimont, Vaughan, Blomer, Nunes and Le Boudec in the case of arithmetic progressions.

A Novel Integral Equation for the Riemann Zeta Function and Large t-Asymptotics

Based on the new approach to Lindelöf hypothesis recently introduced by one of the authors, we first derive a novel integral equation for the square of the absolute value of the Riemann zeta function. Then, we introduce the machinery needed to obtain an estimate for the solution of this equation. This approach suggests a substantial improvement of the current large t - asymptotics estimate for ζ 1 2 + i t .

Relations among the Riemann Zeta and Hurwitz Zeta Functions, as Well as Their Products

In this paper, several relations are obtained among the Riemann zeta and Hurwitz zeta functions, as well as their products. A particular case of these relations give rise to a simple re-derivation of the important results of Katsurada and Matsumoto on the mean square of the Hurwitz zeta function. Also, a relation derived here provides the starting point of a novel approach which, in a series of companion papers, yields a formal proof of the Lindelöf hypothesis. Some of the above relations motivate the need for analysing the large α behaviour of the modified Hurwitz zeta function ζ 1 ( s , α ) , s ∈ C , α ∈ ( 0 , ∞ ) , which is also presented here.

A novel approach to the Lindelöf hypothesis

Lindelöf’s hypothesis, one of the most important open problems in the history of mathematics, states that for large $t$, Riemann’s zeta function $\zeta (1/2+it)$ is of order $O(t^{\varepsilon })$ for any $\varepsilon&gt;0$. It is well known that for large $t$, the leading order asymptotics of the Riemann zeta function can be expressed in terms of a transcendental exponential sum. The usual approach to the Lindelöf hypothesis involves the use of ingenious techniques for the estimation of this sum. However, since such estimates cannot yield an asymptotic formula for the above sum, it appears that this strategy cannot lead to the proof of Lindelöf’s hypothesis. Here a completely different approach is introduced. In particular, a novel linear integral equation is derived for $|\zeta (\sigma +it)|^2, \ 0&lt;\sigma &lt;1$ whose asymptotic analysis yields asymptotic results for a certain Riemann zeta-type double exponential sum. This sum has the same structure as the sum describing the leading asymptotics of $|\zeta (\sigma +it)|^2$, namely it involves $m_1^{-\sigma -it}m_2^{-\sigma -it}$, but its summation limits are different than those of the sum corresponding to $|\zeta (\sigma +it)|^2$. The analysis of the above integral equation requires the asymptotic estimation of four different integrals denoted by $I_1,I_2,\tilde{I}_3,\tilde{I}_4$, as well as the derivation of an exact relation between certain double exponential sums. Here the latter relation is derived, and also the rigourous analysis of the first two integrals $I_1$ and $I_2$ is presented. For the remaining two integrals, formal results are only derived that suggest a possible roadmap for the derivation of rigourous asymptotic results of the above double exponential sum, as well as for other sums associated with $|\zeta (\sigma +it)|^2$. Additional developments suggested by the above novel approach are also discussed.

Estimates of lattice points in the discriminant aspect over abelian extension fields

We estimate the number of relatively r-prime lattice points in {K^{m}} with their components having a norm less than x, where K is a number field. The error terms are estimated in terms of x and the discriminant D of the field K, as both x and D grow. The proof uses the bounds of Dedekind zeta functions. We obtain uniform upper bounds as K runs through number fields of any degree under assuming the Lindelöf hypothesis. We also show unconditional results for abelian extensions with a degree less than or equal to 6.

On the fourth moment of Hecke–Maass forms and the random wave conjecture

Conditionally on the generalized Lindelöf hypothesis, we obtain an asymptotic for the fourth moment of Hecke–Maass cusp forms of large Laplacian eigenvalue for the full modular group. This lends support to the random wave conjecture.