Claire Frechette, Mathilde Gerbelli-Gauthier, Alia Hamieh
et al.
Let $N$ be a fixed positive integer, and let $f\in S_k(N)$ be a primitive cusp form given by the Fourier expansion $f(z)=\sum_{n=1}^{\infty} λ_f(n)n^{\frac{k-1}{2}}e(nz)$. We consider the partial sum $S(x,f)=\sum_{n\leq x}λ_f(x)$. It is conjectured that $S(x,f)=o(x\log x)$ in the range $x\geq k^ε$. Lamzouri proved in arXiv:1703.10582 [math.NT] that this is true under the assumption of the Generalized Riemann Hypothesis (GRH) for $L(s,f)$. In this paper, we prove that this conjecture holds under a weaker assumption than GRH. In particular, we prove that given $ε>(\log k)^{-\frac{1}{8}}$ and $1\leq T\leq (\log k)^{\frac{1}{200}}$, we have $S(x,f)\ll \frac{x\log x}{T}$ in the range $x\geq k^ε$ provided that $L(s,f)$ has no more than $ε^2\log k/5000$ zeros in the region $\left\{s\,:\, \Re(s)\geq \frac34, \, |\Im(s)-φ| \leq \frac14\right\}$ for every real number $φ$ with $|φ|\leq T$.
Maxime Gestin, Luca Falato, Michela Ciccarelli
et al.
AbstractHeat shock protein 70 kDa (HSP70) is a major protein family in the cell protections against stress-induced denaturation and aggregation and in the folding of nascent proteins. It is a highly conserved protein that can be found in most organisms and is strongly connected to several intracellular pathways such as protein folding and refolding, protein degradation and regulation, and protection against intense stress. Cellular delivery of HSP70 would be of high impact for clarification of its role in these cellular processes.PepFect14 is a cell-penetrating peptide known to be able to mediate the transfection of various oligonucleotides to multiple cell lines with a higher efficacy than most commercially available transfection agents and without inducing significant toxic effects.In this study we demonstrated that PepFect14 was able to form a complex with HSP70 and to deliver it inside cells in the same fashion with oligonucleotide delivery. The delivered HSP70 showed an effect in the cell regulation indicating that the protein was biologically available in the cytoplasm and the interactions with PepFect14 did not impeach its active sites once the plasma barrier crossed.This study reports the first successful delivery of HSP70 to our knowledge and the first protein transfection mediated by PepFect14. It opens new fields of research for both PepFect14 as a delivery agent and HSP70 as a therapeutic agent; with potential in peptide aggregation caused diseases such as Parkinson’s and Alzheimer’s diseases.
Recently, Kargin et al. (arXiv:2008.00284 [math.NT]) obtained (among many other things) the following formula for the hyper-sums of powers of integers $S_k^{(m)}(n)$ \begin{equation*} S_k^{(m)}(n) = \frac{1}{m!} \sum_{i=0}^{m} (-1)^i \genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1} S_{k+i}(n), \end{equation*} where $S_k^{(0)}(n) \equiv S_k(n)$ is the ordinary power sum $1^k + 2^k + \cdots + n^k$. In this note we point out that a formula equivalent to the preceding one was already established in a different form, namely, a form in which $\genfrac{[}{]}{0pt}{}{m+n+1}{i+n+1}_{n+1}$ is given explicitly as a polynomial in $n$ of degree $m-i$. We find out the connection between this polynomial and the so-called $r$-Stirling polynomials of the first kind. Furthermore, we determine the hyperharmonic polynomials and their successive derivatives in terms of the $r$-Stirling polynomials of the first kind, and show the relationship between the (exponential) complete Bell polynomials and the $r$-Stirling numbers of the first kind. Finally, we derive some identities involving the Bernoulli numbers and polynomials, the $r$-Stirling numbers of the first kind, the Stirling numbers of both kinds, and the harmonic numbers.
Given a partition $\{E_0,\ldots,E_n\}$ of the set of primes and a vector $\mathbf{k} \in \mathbb{N}_0^{n+1}$, we compute an asymptotic formula for the quantity $|\{m \leq x: ω_{E_j}(m) = k_j \ \forall \ 0 \leq j \leq n\}|$ uniformly in a wide range of the parameters $k_j$ that complements the results of a previous paper of the author. This is accomplished using an extension and generalization of a theorem of Wirsing due to the author that gives explicit estimates for the ratio $\frac{|M_g(x)|}{M_{f}(x)}$, whenever $f: \mathbb{N} \rightarrow (0,\infty)$ and $g: \mathbb{N} \rightarrow \mathbb{C}$ are strongly multiplicative functions that are uniformly bounded on primes and satisfy $|g(n)| \leq f(n)$ for every $n \in \mathbb{N}$. This also allows us to conclude the validity of a probabilistic heuristic regarding $π(x;\mathbf{E},\mathbf{k})$ in the case that $k_j = (1+o(1))E_j(x)$, for each $0 \leq j \leq n$.
We consider the non-trivial zeros of the Riemann $ζ$-function and two classes of $L$-functions; Dirichlet $L$-functions and those based on level one modular forms. We show that there are an infinite number of zeros on the critical line in one-to-one correspondence with the zeros of the cosine function, and thus enumerated by an integer $n$. From this it follows that the ordinate of the $n$-th zero satisfies a transcendental equation that depends only on $n$. Under weak assumptions, we show that the number of solutions of this equation already saturates the counting formula on the entire critical strip. We compute numerical solutions of these transcendental equations and also its asymptotic limit of large ordinate. The starting point is an explicit formula, yielding an approximate solution for the ordinates of the zeros in terms of the Lambert $W$-function. Our approach is a novel and simple method, that takes into account $\arg L$, to numerically compute non-trivial zeros of $L$-functions. The method is surprisingly accurate, fast and easy to implement. Employing these numerical solutions, in particular for the $ζ$-function, we verify that the leading order asymptotic expansion is accurate enough to numerically support Montgomery's and Odlyzko's pair correlation conjectures, and also to reconstruct the prime number counting function. Furthermore, the numerical solutions of the exact transcendental equation can determine the ordinates of the zeros to any desired accuracy. We also study in detail Dirichlet $L$-functions and the $L$-function for the modular form based on the Ramanujan $τ$-function, which is closely related to the bosonic string partition function.
Kurusch Ebrahimi-Fard, Dominique Manchon, Johannes Singer
et al.
Calculating multiple zeta values at arguments of any sign in a way that is compatible with both the quasi-shuffle product as well as meromorphic continuation, is commonly referred to as the renormalisation problem for multiple zeta values. We consider the set of all solutions to this problem and provide a framework for comparing its elements in terms of a free and transitive action of a particular subgroup of the group of characters of the quasi-shuffle Hopf algebra. In particular, this provides a transparent way of relating different solutions at non-positive values, which answers an open question in the recent literature.
Erd\"os conjectured that the set J of limit points of d_n/logn contains all nonnegative numbers, where d_n denotes the nth primegap. The author proved a year ago (arXiv: 1305.6289) that J contains an interval of type [0,c] with a positive ineffective value c. In the present work we extend this result for a large class of normalizing functions. The only essential requirement is that the function f(n) replacing logn should satisfy f(n)<<lognloglognloglogloglogn/(logloglogn)^2 (with a small implied constant), the well-known Erd\"os-Rankin bound for the largest known gaps between consecutive primes. The work also proves that apart from a thin set of exceptional functions the original Erd\"os conjecture holds if logn is replaced by a non-exceptional function f(n). The paper also gives a new proof for a result of Helmut Maier which generalized the Erd\"os-Rankin bound for an arbitrarily long finite chain of consecutive primegaps. The proof uses a combination of methods of Erd\"os-Rankin,Maynard-Tao and Banks-Freiberg-Maynard. Since the submission of the present work the very important recent simultaneous and independent works of Ford-Green-Konjagin-Tao (arXiv:1408.4505 [math.NT] and Maynard (aerXiv:1408.5110 [math.NT]) appeared on arXiv and they proved the old conjecture of Erd\"os which asserts that the lower bound for large gaps exceeds Clognloglognloglogloglogn/(logloglogn)^2 with an arbitrarily large constant C. In this new version we prove the same assertions as in the original work for the case when f(n)<<Clognloglognloglogloglogn/(logloglogn)^2 with an arbi8trarily large constant C, in particular we show that there are blocks of m primes for any m such that all gaps between these primes simultaneously satisfy the lower estimate Clognloglognloglogloglogn/(logloglogn)^2 with an arbitrarily large constant C. The proof uses the method of Maynard.
Let $A$ be an abelian variety defined over a global field $F$ of positive characteristic $p$ and let $\calf/F$ be a $\Z_p^{\N}$-extension, unramified outside a finite set of places of $F$. Assuming that all ramified places are totally ramified, we define a pro-characteristic ideal associated to the Pontrjagin dual of the $p$-primary Selmer group of $A$, in order to formulate an Iwasawa Main Conjecture for the non-noetherian commutative Iwasawa algebra $\Z_p[[\Gal(\calf/F)]]$ (which we also prove for a constant abelian variety). To do this we first show the relation between the characteristic ideals of duals of Selmer groups for a $\Z_p^d$-extension $\calf_d/F$ and for any $\Z_p^{d-1}$-extension contained in $\calf_d\,$, and then use a limit process.
We give a necessary and sufficient condition for the integrality of the Taylor coefficients at the origin of formal power series $q_i({\mathbf z})=z_i\exp(G_i({\mathbf z})/F({\mathbf z}))$, with ${\mathbf z}=(z_1,...,z_d)$ and where $F({\mathbf z})$ and $G_i({\mathbf z})+\log(z_i)F({\mathbf z})$, $i=1,...,d$ are particular solutions of certain A-systems of differential equations. This criterion is based on the analytical properties of Landau's function (which is classically associated with the sequences of factorial ratios) and it generalizes the criterion in the case of one variable presented in "Critère pour l'intégralité des coefficients de Taylor des applications miroir" [J. Reine Angew. Math.]. One of the techniques used to prove this criterion is a generalization of a version of a theorem of Dwork on the formal congruences between formal series, proved by Krattenthaler and Rivoal in "Multivariate $p$-adic formal congruences and integrality of Taylor coefficients of mirror maps" [arXiv:0804.3049v3, math.NT]. This criterion involves the integrality of the Taylor coefficients of new univariate mirror maps listed in "Tables of Calabi--Yau equations" [arXiv:math/0507430v2, math.AG] by Almkvist, van Enckevort, van Straten and Zudilin.
Using Fourier analysis, we study local limit theorems in weak-convergence problems. Among many applications, we discuss random matrix theory, some probabilistic models in number theory, the winding number of complex brownian motion and the classical situation of the central limit theorem, and a conjecture concerning the distribution of values of the Riemann zeta function on the critical line.
Sergey K. Sekatskii, Stefano Beltraminelli, Danilo Merlini
This paper is a continuation of our recent papers with the same title, arXiv:0806.1596v1 [math.NT], arXiv:0904.1277v1 where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that some of them are equivalent to the Riemann hypothesis. A few new equalities of this type, which this time involve exponential functions, are established, and for the first time we have found equalities involving the integrals of the logarithm of the Riemann zeta-function taken exclusively along the real axis. Some of the equalities we have found are tested numerically. In particular, an integral equality involving the logarithm of abs(zeta(1/2+it)) and a weight function cosh(pi*t)^(-1) is shown numerically to be correct up to the 80 digits. For exponential weight function exp(-at), the possible contribution of the Riemann function zeroes non-lying on the critical line is rigorously estimated and shown to be extremely small, in particular, smaller than trillion of digits, 10^(-10^(13)), for a=4*pi.
Sergey K. Sekatskii, Stefano Beltraminelli, Danilo Merlini
This paper is a continuation of our recent paper with the same title, arXiv:0806.1596v1 [math.NT], where a number of integral equalities involving integrals of the logarithm of the Riemann zeta-function were introduced and it was shown that some of them are equivalent to the Riemann hypothesis. A few new equalities of this type are established; contrary to the preceding paper the focus now is on integrals involving the argument of the Riemann zeta-function (imaginary part of logarithm) rather than the logarithm of its module (real part of logarithm). Preliminary results of the numerical research performed using these equalities to test the Riemann hypothesis are presented. Our integral equalities, together with the equalities given in the previous paper, include all earlier known criteria of this kind, viz. Wang, Volchkov and Balazard-Saias-Yor criteria, which are certain particular cases of the general approach proposed.
We prove sharp limit theorems on random walks on graphs with values in finite groups. We then apply these results (together with some elementary algebraic geometry, number theory, and representation theory) to finite quotients of lattices in semisimple Lie groups (specifically SL(n,Z) and Sp(2n, Z) to show that a ``random'' element in one of these lattices has irreducible characteristic polynomials (over the integers. The term ``random'' can be defined in at least two ways (in terms of height and also in terms of word length in terms of a generating set) -- we show the result using both definitions. We use the above results to show that a random (in terms of word length) element of the mapping class group of a surface is pseudo-Anosov, and that a a random free group automorphism is irreducible with irreducible powers (or strongly irreducible).
We compute by a purely local method the (elliptic) twisted by transpose-inverse character \chi_{\pi_Y} of the representation \pi_Y=I_{(3,1)}(1_3x\chi_Y) of G=GL(4,F), where F is a p-adic field, p not 2, and Y is an unramified quadratic extension of F; \chi_Y is the nontrivial character of F^\x/N_{Y/F}Y^x. The representation \pi_Y is normalizedly induced from \pmatrix m_3&\ast 0&m_1\endpmatrix \mapsto\chi_Y(m_1), m_i in GL(i,F), on the maximal parabolic subgroup of type (3,1). We show that the twisted character \chi_{\pi_Y} of \pi_Y is an unstable function: its value at a twisted regular elliptic conjugacy class with norm in C_Y=``GL(2,Y)/F^x'' is minus its value at the other class within the twisted stable conjugacy class. It is zero at the classes without norm in C_Y. Moreover \pi_Y is the endoscopic lift of the trivial representation of C_Y. We deal only with unramified Y/F, as globally this case occurs almost everywhere. Naturally this computation plays a role in the theory of lifting of C_Y and GSp(2) to GL(4) using the trace formula. Our work extends -- to the context of nontrivial central characters -- the work of math.NT/0606262, where representations of PGL(4,F) are studied. In math.NT/0606262 a 4-dimensional analogue of the model of the small representation of PGL(3,F) introduced with Kazhdan in a 3-dimensional case is developed, and the local method of computation introduced by us in the 3-dimensional case is extended. As in math.NT/0606262 we use here the classification of twisted (stable) regular conjugacy classes in GL(4,F).
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We compute by a purely local method the elliptic, twisted by transpose-inverse, character \chi_\pi of the representation \pi=I_{(3,1)}(1_3) of PGL(4,F) normalizedly induced from the trivial representation of the maximal parabolic subgroup of type (3,1), where F is a p-adic field. Put C=(GL(2,F)xGL(2,F))'/F^x (F^x embeds diagonally, prime means equal determinants). It is a twisted elliptic endoscopic group of PGL(4). We deduce from the computation that \chi_\pi is an unstable function: its value at one twisted regular elliptic conjugacy class with norm in C is minus its value at the other class within the twisted stable conjugacy class, and zero at the classes without norm in C. Moreover \pi is the unstable endoscopic lift of the trivial representation of C. Naturally, this computation plays a role in the theory of lifting from C (=``SO(4,F)'') and PGp(2,F) to PGL(4,F) using the trace formula. Our work develops a 4-dimensional analogue of the model of the small representation of PGL(3,F) introduced by the first author with Kazhdan in a 3-dimensional case, and it uses the classification of twisted stable and unstable regular conjugacy classes in PGL(4,F). It extends the local method of computation introduced by us in the 3-dimensional case. An extension math.NT/0606263 of our work here to apply to similar representations of GL(4,F) whose central character is nontrivial will appear in Int. J. Number Theory.