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To date, the bestmethodsfor estimating the growth of mean values of arithmetic functions rely on the Voronoï summation formula. By noticing a general pattern in the proof of his summation formula, Voronoï postulated that analogous summation formulas for 
$\sum a(n)f(n)$ can be obtained with ‘nice’ test functions f(n), provided a(n) is an ‘arithmetic function’. These arithmetic functions a(n) are called so because they are expected to appear as coefficients of some L-functions satisfying certain properties. It has been well-known that the functional equation for a general L-function can be used to derive a Voronoï-type summation identity for that L-function. In this article, we show that such a Voronoï-typesummation identity in fact endows the L-function with some structural properties, yielding in particular the functional equation. We do this by considering Dirichlet series satisfying functional equations involving multiple Gamma factors and show that a given arithmetic function appears as a coefficient of such a Dirichlet series if and only if it satisfies the aforementioned summation formulas.
We investigate the joint distribution of L-functions on the line 
$ \sigma= {1}/{2} + {1}/{G(T)}$ and 
$ t \in [ T, 2T]$, where 
$ \log \log T \leq G(T) \leq { \log T}/{ ( \log \log T)^2 } $. We obtain an upper bound on the discrepancy between the joint distribution of L-functions and that of their random models. As an application we prove an asymptotic expansion of a multi-dimensional version of Selberg’s central limit theorem for L-functions on 
$ \sigma= 1/2 + 1/{G(T)}$ and 
$ t \in [ T, 2T]$, where 
$ ( \log T)^\varepsilon \leq G(T) \leq { \log T}/{ ( \log \log T)^{2+\varepsilon } } $ for 
$ \varepsilon > 0$.
For an elliptic curve E defined over a number field K and 
$L/K$ a Galois extension, we study the possibilities of the Galois group Gal
$(L/K)$, when the Mordell–Weil rank of 
$E(L)$ increases from that of 
$E(K)$ by a small amount (namely 1, 2, and 3). In relation with the vanishing of corresponding L-functions at 
$s=1$, we prove several elliptic analogues of classical theorems related to Artin’s holomorphy conjecture. We then apply these to study the analytic minimal subfield, first introduced by Akbary and Murty, for the case when order of vanishing is 2. We also investigate how the order of vanishing changes as rank increases by 1 and vice versa, generalizing a theorem of Kolyvagin.
On the assumption of the Riemann hypothesis and a spacing hypothesis for the nontrivial zeros 
$1/2+i\gamma$ of the Riemann zeta function, we show that the sequence where the 
\begin{equation*}\Gamma_{[a, b]} =\Bigg\{ \gamma : \gamma>0 \quad \mbox{and} \quad \frac{ \log\big(| \zeta^{(m_{\gamma })} (\frac12+ i{\gamma }) | / (\!\log{{\gamma }} )^{m_{\gamma }}\big)}{\sqrt{\frac12\log\log {\gamma }}} \in [a, b] \Bigg\},\end{equation*}
${\gamma }$ are arranged in increasing order, is uniformly distributed modulo one. Here a and b are real numbers with 
$a<b$, and 
$m_\gamma$ denotes the multiplicity of the zero 
$1/2+i{\gamma }$. The same result holds when the 
${\gamma }$’s are restricted to be the ordinates of simple zeros. With an extra hypothesis, we are also able to show an equidistribution result for the scaled numbers 
$\gamma (\!\log T)/2\pi$ with 
${\gamma }\in \Gamma_{[a, b]}$ and 
$0<{\gamma }\leq T$.
We provide explicit bounds for the Riemann zeta-function on the line 
$\mathrm {Re}\,{s}=1$, assuming that the Riemann hypothesis holds up to height T. In particular, we improve some bounds in finite regions for the logarithmic derivative and the reciprocal of the Riemann zeta-function.
$R(z),$ and a mean value theorem for 
$\zeta \left (\frac {1}{2}-it\right )\zeta '\left (\frac {1}{2}+it\right )$
An explicit transformation for the series 
$\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$, or equivalently, 
$\sum \limits _{n=1}^{\infty }d(n)\log (n)e^{-ny}$ for Re
$(y)>0$, which takes y to 
$1/y$, is obtained for the first time. This series transforms into a series containing the derivative of 
$R(z)$, a function studied by Christopher Deninger while obtaining an analog of the famous Chowla–Selberg formula for real quadratic fields. In the course of obtaining the transformation, new important properties of 
$\psi _1(z)$ (the derivative of 
$R(z)$) are needed as is a new representation for the second derivative of the two-variable Mittag-Leffler function 
$E_{2, b}(z)$ evaluated at 
$b=1$, all of which may seem quite unexpected at first glance. Our transformation readily gives the complete asymptotic expansion of 
$\sum \limits _{n=1}^{\infty }\displaystyle \frac {\log (n)}{e^{ny}-1}$ as 
$y\to 0$ which was also not known before. An application of the latter is that it gives the asymptotic expansion of 
$ \displaystyle \int _{0}^{\infty }\zeta \left (\frac {1}{2}-it\right )\zeta '\left (\frac {1}{2}+it\right )e^{-\delta t}\, dt$ as 
$\delta \to 0$.
This paper deals with applications of Voronin’s universality theorem for the Riemann zeta-function 
$\zeta$. Among other results we prove that every plane smooth curve appears up to a small error in the curve generated by the values 
$\zeta(\sigma+it)$ for real t where 
$\sigma\in(1/2,1)$ is fixed. In this sense, the values of the zeta-function on any such vertical line provides an atlas for plane curves. In the same framework, we study the curvature of curves generated from 
$\zeta(\sigma+it)$ when 
$\sigma>1/2$ and we show that there is a connection with the zeros of 
$\zeta'(\sigma+it)$. Moreover, we clarify under which conditions the real and the imaginary part of the zeta-function are jointly universal.
In this article, we prove that the Riemann hypothesis implies a conjecture of Chandee on shifted moments of the Riemann zeta function. The proof is based on ideas of Harper concerning sharp upper bounds for the 
$2k$th moments of the Riemann zeta function on the critical line.
We formulate a generalization of Riesz-type criteria in the setting of L-functions belonging to the Selberg class. We obtain a criterion which is sufficient for the grand Riemann hypothesis (GRH) for L-functions satisfying axioms of the Selberg class without imposing the Ramanujan hypothesis on their coefficients. We also construct a subclass of the Selberg class and prove a necessary criterion for GRH for L-functions in this subclass. Identities of Ramanujan–Hardy–Littlewood type are also established in this setting, specific cases of which yield new transformation formulas involving special values of the Meijer G-function of the type 
${G^{n , 0}_{0 , n}}$.
$|\zeta ^{(\ell )}(1+{i}t)|$
We investigate the large values of the derivatives of the Riemann zeta function 
$\zeta (s)$ on the 1-line. We give a larger lower bound for 
$\max _{t\in [T,2T]}|\zeta ^{(\ell )}(1+{i} t)|$, which improves the previous result established by Yang [‘Extreme values of derivatives of the Riemann zeta function’, Mathematika 68 (2022), 486–510].
