Dinakar Ramakrishnan
In this Note we show that given any cusp form πon GL(3) over the rationals, there exist an infinite number of primes p which are Ramanujan for π, i.e., that the local components π_p are tempered for an infinite number of p. It turns out that the key structural device which is needed is the degree 8 adjoint L-function of π.
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We study the Ramanujan-prime-counting function along the lines of Ramanujan's original work on Bertrand's Postulate. We show that the number of Ramanujan primes between x and 2x tends to infinity with x. This analysis leads us to define a new sequence of prime numbers, which we call derived Ramanujan primes. For n greater than or equal to 1 we define the nth derived Ramanujan prime as the smallest positive integer with the property that if x is greater than or equal to the nth derived Ramanujan prime, then the number of Ramanujan primes in the interval from x to x/2 is greater than or equal to…
Considering Ramanujan primes and the symmetric to them so-called Labos primes, we study their parallel properties, we study all primes with these properties (generalized Ramanujan and Labos primes) and construct two kinds of sieves for them. Finally, we give a further natural generalization of these constructions and pose some conjectures and open problems.
In this paper we establish an explicit upper bound for the first $k$-Ramanujan prime $R_1^{(k)}$ by using a recent result concerning the existence of prime numbers in small intervals.
In this paper, for a positive integer $n\ge 1$, we look at the size and prime factors of the iterates of the Ramanujan $τ$ function applied to $n$.
In this paper we establish several results concerning the generalized Ramanujan primes. For $n\in\mathbb{N}$ and $k \in \mathbb{R}_{> 1}$ we give estimates for the $n$th $k$-Ramanujan prime which lead both to generalizations and to improvements of the results presently in the literature. Moreover, we obtain results about the distribution of $k$-Ramanujan primes. In addition, we find explicit formulae for certain $n$th $k$-Ramanujan primes. As an application, we prove that a conjecture of Mitra, Paul and Sarkar concerning the number of primes in certain intervals holds for every sufficiently la…
The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that for all $x \geq R_n$ the interval $(x/2, x]$ contains at least $n$ primes. In this paper we undertake a study of the sequence $(π(R_n))_{n \in \mathbb{N}}$, which tells us where the $n$th Ramanujan prime appears in the sequence of all primes. In the first part we establish new explicit upper and lower bounds for the number of primes up to the $n$th Ramanujan prime, which imply an asymptotic formula for $π(R_n)$ conjectured by Yang and Togbé. In the second part of this paper, we use these explicit estimates to derive a r…
In this paper, we give a new upper bound for the number $N_{\mathcal{R}}$ which is defined to be the smallest positive integer such that a certain inequality due to Ramanujan involving the prime counting function $π(x)$ holds for every $x \geq N_{\mathcal{R}}$.
A prime p is called Sophie Germain prime if 2p+1 is also prime. A formula for the density of such primes is given in a more general setting using a new approach. This method uses the Ramanujan-Fourier series for a modified von Mangoldt function. The proof remains heuristic as interchange of certain limits has not been justified. Experimental evidence using computer calculations is provided for the plausibility of the result.
The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then the interval $(x/2,x]$ contains at least $n$ primes. We sharpen Laishram's theorem that $R_n < p_{3n}$ by proving that the maximum of $R_n/p_{3n}$ is $R_5/p_{15} = 41/47$. We give statistics on the length of the longest run of Ramanujan primes among all primes $p<10^n$, for $n\le9$. We prove that if an upper twin prime is Ramanujan, then so is the lower; a table gives the number of twin primes below $10^n$ of three types. Finally, we relate runs of Ramanujan primes to prime gaps. Along the way we st…
Universal Ramanujan-type congruences for prime-detecting quasimodular forms
该研究证明了素数检测拟模形式全Z-模上存在无穷多个通用拉马努金型同余式，且对每个素数p均存在对应算术级数使相关系数被p整除。
In this paper, we propose the use of Ramanujan class of polynomials for the construction of prime order elliptic curves using the CM-method. We compare (theoretically and experimentally) the efficiency of using this new class against the use of the Weber, $M_{D,l}(x)$ and $M_{D,p_1,p_2}(x)$ polynomials and show that they clearly outweigh all of them in the generation of prime order elliptic curves.
In this paper we use refined approximations for Chebyshev's $\vartheta$-function to establish new explicit estimates for the prime counting function $π(x)$, which improve the current best estimates for large values of $x$. As an application we find an upper bound for the number $H_0$ which is defined to be the smallest positive integer so that Ramanujan's prime counting inequality holds for every $x \geq H_0$.
A celebrated theorem of Delange gives a sufficient condition for an arithmetic function to be the sum of the associated Ramanujan expansion with the coefficients provided by a previous result of Wintner. By applying the Delange theorem to the correlation of the von Mangoldt function with its incomplete form, we deduce an inequality involving the counting function of the prime numbers in arithmetic progressions. A remarkable aspect is that such an inequality is equivalent to the famous conjectural formula by Hardy and Littlewood for the twin primes.
For $n\ge 1$, the $n^{\rm th}$ Ramanujan prime is defined as the smallest positive integer $R_n$ such that for all $x\ge R_n$, the interval $(\frac{x}{2}, x]$ has at least $n$ primes. We show that for every $ε>0$, there is a positive integer $N$ such that if $α=2n\left(1+\dfrac{\log 2+ε}{\log n+j(n)}\right)$, then $R_n< p_{[α]}$ for all $n>N$, where $p_i$ is the $i^{\rm th}$ prime and $j(n)>0$ is any function that satisfies $j(n)\to \infty$ and $nj'(n)\to 0$.
On Ramanujan Primes for Hecke-Maass Cusp Forms
本文研究了两个或三个不同原始海格尔-马克斯自拍形式同时满足拉马努金猜想的最小素数上界，以及至少一个形式满足猜想的素数自然密度下界。
We show that if $E/\mathbb{Q}$ is an elliptic curve with a rational $p$-torsion for $p=2$ or $3$, then there is a congruence relation between Ramanujan's tau function and $E$ modulo $p$. We make use of such congruences to compute the Iwasawa invariants of $2$-adic and $3$-adic Mazur--Tate elements attached to Ramanujan's tau function. We also investigate numerically the Iwasawa invariants of the Mazur--Tate elements attached to an elliptic curve with additive reduction at a fixed prime number.
The $n$th Ramanujan prime is the smallest positive integer $R_n$ such that if $x \ge R_n$, then there are at least $n$ primes in the interval $(x/2,x]$. For example, Bertrand's postulate is $R_1 = 2$. Ramanujan proved that $R_n$ exists and gave the first five values as 2, 11, 17, 29, 41. In this note, we use inequalities of Rosser and Schoenfeld to prove that $2n \log 2n < R_n < 4n \log 4n$ for all $n$, and we use the Prime Number Theorem to show that $R_n$ is asymptotic to the $2n$th prime. We also estimate the length of the longest string of consecutive Ramanujan primes among the first $n$ p…
We establish Ramanujan-style congruences modulo certain primes $\ell$ between an Eisenstein series of weight $k$, prime level $p$ and a cuspidal newform in the $\varepsilon$-eigenspace of the Atkin-Lehner operator inside the space of cusp forms of weight $k$ for $Γ_0(p)$. Under a mild assumption, this refines a result of Gaba-Popa. We use these congruences and recent work of Ciolan, Languasco and the third author on Euler-Kronecker constants, to quantify the non-divisibility of the Fourier coefficients involved by $\ell.$ The degree of the number field generated by these coefficients we invest…
Prime values of Ramanujan's tau function
本文研究了Ramanujan tau函数的素数值，证明了在大多数算术级数（mod 23）中素数值构成稀疏集，并得出存在Dirichlet密度为9/11的素数集不能表示为tau函数值的结论。
We establish an analog of the Hardy-Ramanujan inequality for counting members of sifted sets with a given number of distinct prime factors. In particular, we establish a bound for the number of shifted primes p+a below x with k distinct prime factors, uniformly for all positive integers k.
A small survey of work done on estimating the number of integers without large prime factors up to around 1950 is provided. Around 1950 N.G. de Bruijn published results that dramatically advanced the subject and started a new era in this topic.
In 1845, Bertrand conjectured that for all integers $x\ge2$, there exists at least one prime in $(x/2, x]$. This was proved by Chebyshev in 1860, and then generalized by Ramanujan in 1919. He showed that for any $n\ge1$, there is a (smallest) prime $R_n$ such that $π(x)- π(x/2) \ge n$ for all $x \ge R_n$. In 2009 Sondow called $R_n$ the $n$th Ramanujan prime and proved the asymptotic behavior $R_n \sim p_{2n}$ (where $p_m$ is the $m$th prime). In the present paper, we generalize the interval of interest by introducing a parameter $c \in (0,1)$ and defining the $n$th $c$-Ramanujan prime as the …
Ramanujan's function on small primes
本文研究了拉马努金函数在小素数上的特征值振荡现象，并探讨其对勒默问题的潜在应用。
For all $k \geq 3$, we show how one can explicitly construct an infinite family of $k$-regular graphs all of which have second largest eigenvalue satisfying the bound $O(k^{1/2})$. This resolves an open problem of Reingold, Vadhan and Wigderson.
We prove several Ramanujan-type congruences modulo powers of $5$ for partition $k$-tuples with $5$-cores, for $k=2, 3, 4$. We also prove some new infinite families of congruences modulo powers of primes for $k$-tuples with $p$-cores, where $p$ is a prime.
We prove congruences between cuspidal newforms and Eisenstein series of prime level, which generalize Ramanujan's congruence. Such congruences were recently found by Billerey and Menares, and we refine them by specifying the Atkin-Lehner eigenvalue of the newform involved. We show that similar refinements hold for the level raising congruences between cuspidal newforms of different levels, due to Ribet and Diamond. The proof relies on studying the new subspace and the Eisenstein subspace of the space of period polynomials for the congruence subgroup $Γ_0(N)$, and on a version of Ihara's lemma.
Sondow et al have studied Ramanujan primes (RPs) and observed numerically that, while half of all primes are RPs asymptotically, one obtains runs of consecutives RPs (resp. non-RPs) which are statistically significantly longer than one would expect if one was tossing an unbiased coin. In this discussion paper we attempt a heuristic explanation of this phenomenon. Our heuristic follows naturally from the Prime Number Theorem, but seems to be only partly satisfactory. It motivates why one should obtain long runs of both RPs and non-RPs, and also longer runs of non-RPs than of RPs. However, it al…
The method of Murty and Cioabă shows how one can use results about gaps between primes to construct families of almost-Ramanujan graphs. In this paper we give a simpler construction which avoids the search for perfect matchings and thus eliminates the need for computation. A couple of recent explicit bounds on the gap between consecutive primes are then used to give the construction of $k$-regular families with explicit lower bounds on the spectral gaps. We then show that a result of Ben-Aroya and Ta-Shma can be improved using our simpler construction on the assumption of the Riemann Hypothesi…
Let $\ell$ be any fixed prime number. We define the $\ell$-Genocchi numbers by $G_n:=\ell(1-\ell^n)B_n$, with $B_n$ the $n$-th Bernoulli number. They are integers. We introduce and study a variant of Kummer's notion of regularity of primes. We say that an odd prime $p$ is $\ell$-Genocchi irregular if it divides at least one of the $\ell$-Genocchi numbers $G_2,G_4,\ldots, G_{p-3}$, and $\ell$-regular otherwise. With the help of techniques used in the study of Artin's primitive root conjecture, we give asymptotic estimates for the number of $\ell$-Genocchi irregular primes in a prescribed arithm…
We study some arithmetic properties of the Ramanujan function $τ(n)$, such as the largest prime divisor $P(τ(n))$ and the number of distinct prime divisors $ω(τ(n))$ of $τ(n)$ for various sequences of $n$. In particular, we show that \hbox{$P(τ(n)) \geq (\log n)^{33/31 + o(1)}$} for infinitely many $n$, and \begin{equation*} P(τ(p)τ(p^2)τ(p^3)) > (1+o(1))\frac{\log\log p\log\log\log p} {\log\log\log\log p} \end{equation*} for every prime $p$ with \hbox{$τ(p)\neq 0$}.
ABC implies that Ramanujan's tau function misses almost all primes
在 ABC 猜想假设下，证明了 tau 函数的素数值密度为 O(X^{13/22})，表明 tau 函数错过大部分素数，同时提出 S(X) 可能无限且有预测阶数。
On Proving Ramanujan's Inequality using a Sharper Bound for the Prime Counting Function $π(x)$
本文利用 Chebyshev Theta 函数 $\vartheta(x)$ 的估计导出素数计数函数 $π(x)$ 的更紧界，从而无条件证明了 Ramanujan 不等式对所有 $x\geq \exp(43.5102147)$ 成立。
In the sixth chapter of his notebooks Ramanujan introduced a method of summing divergent series which assigns to the series the value of the associated Euler-MacLaurin constant that arises by applying the Euler-MacLaurin summation formula to the partial sums of the series. This method is now called the Ramanujan summation process. In this paper we calculate the Ramanujan sum of the exponential generating functions $\sum_{n\geq 1}\log n e^{nz}$ and $\sum_{n\geq 1}H_n^{(j)} e^{-nz}$ where $H_n^{(j)}=\sum_{m=1}^n \frac{1}{m^j}$. We find a surprising relation between the two sums when $j=1$ from w…
Explicit formulas involving a generalized Ramanujan sum are derived. An analogue of the prime number theorem is obtained and equivalences of the Riemann hypothesis are shown. Finally, explicit formulas of Bartz are generalized.
Ramanujan's congruence primes
本研究受拉马努金关于尖点形式傅里叶系数与素数691的同余结论启发，发现素数67出现在模5二次狄利克雷特征的L函数相关比值的分子中，导出了$Γ_0(5)$上权6尖点形式与艾森斯坦级数的模67同余关系。
We prove a number of results regarding odd values of the Ramanujan $τ$-function. For example, we prove the existence of an effectively computable positive constant $κ$ such that if $τ(n)$ is odd and $n \ge 25$ then either \[ P(τ(n)) \; > \; κ\cdot \frac{\log\log\log{n}}{\log\log\log\log{n}} \] or there exists a prime $p \mid n$ with $τ(p)=0$. Here $P(m)$ denotes the largest prime factor of $m$. We also solve the equation $τ(n)=\pm 3^{b_1} 5^{b_2} 7^{b_3} 11^{b_4}$ and the equations $τ(n)=\pm q^b$ where $3\le q < 100$ is prime and the exponents are arbitrary nonnegative integers. We make use of…
Let p be an odd prime, such that p_n<p/2<p_{n+1}, where p_n is the n-th prime. We study the following question: with what probability does there exist a prime in the interval (p, 2p_{n+1})? After the strong definition of the probability with help of the Ramanujan primes ([11], [12])and the introducing pseudo-Ramanujan primes, we show, that if such probability P exists, then P>=0.5. We also study a symmetrical case of the left intervals, which connected with sequence A080359 at OEIS.
On some values which do not belong to the image of Ramanujan's tau-function
本文证明了Ramanujan tau函数在n≥1时，除14种情况外，不取±ℓ, ±2ℓ, ±4ℓ, ±8ℓ这些值，其中ℓ<1000为奇素数，推广了Lehmer猜想的相关结果。
This article considers the eta power $\prod {(1-q^k)}^{b-1}$. It is proved that the coefficients of $\frac{q^n}{n!}$ in this expression, as polynomials in $b$, exhibit equidistribution of the coefficients in the nonzero residue classes mod 5 when $n=5j+4$. Other symmetries, as well as symmetries for other primes and prime powers, are proved, and some open questions are raised.
We determine conditions for the existence and non-existence of Ramanujan-type congruences for Jacobi forms. We extend these results to Siegel modular forms of degree 2 and as an application, we establish Ramanujan-type congruences for explicit examples of Siegel modular forms.
We shall show that, for any positive integer $D>0$ and any primes $p_1, p_2$ not dividing $D$, the diophantine equation $x^2+D=2^s p_1^k p_2^l$ has at most $63$ integer solutions $(x, k, l, s)$ with $x, k, l\geq 0$ and $s\in \{0, 2\}$.
Ramanujan famously found congruences for the partition function like p(5n+4) = 0 modulo 5. We provide a method to find all simple congruences of this type in the coefficients of the inverse of a modular form on Gamma_{1}(4) which is non-vanishing on the upper half plane. This is applied to answer open questions about the (non)-existence of congruences in the generating functions for overpartitions, crank differences, and 2-colored F-partitions.
Let $p_{r,s}(n)$ denote the number of partitions of a positive integer $n$ into parts containing no multiples of $r$ or $s$, where $r>1$ and $s>1$ are square-free, relatively prime integers. We use classical methods to derive a Hardy-Ramanujan-Rademacher-type infinite series for $p_{r,s}(n)$.
Let $τ(n)$ be the Ramanujan $τ$-function. We prove that for any integer $N$ the diophantine equation $$ \sum_{i=1}^{74000}τ(n_i)=N $$ has a solution in positive integers $n_1, n_2,..., n_{74000}$ satisfying the condition $$ \max_{1\le i\le 74000}n_i\ll |N|^{2/11}+1. $$ We also consider similar questions in the residue ring modulo a large prime $p.$
In the case of Siegel modular forms of degree $n$, we prove that, for almost all prime ideals $\frak{p}$ in any ring of algebraic integers, mod $\frak{p}^m$ cusp forms are congruent to true cusp forms of the same weight. As an application of this property, we give congruences for the Klingen-Eisenstein series and cusp forms, which can be regarded as a generalization of Ramanujan's congruence. We will conclude by giving numerical examples.
We introduce a fast algorithm to compute the Ramanujan-Deninger gamma function and its logarithmic derivative at positive values. Such an algorithm allows us to greatly extend the numerical investigations about the Euler-Kronecker constants $\mathfrak{G}_q$, $\mathfrak{G}_q^+$ and $M_q=\max_{χ\ne χ_0} \vert L^\prime/L(1,χ)\vert$, where $q$ is an odd prime, $χ$ runs over the primitive Dirichlet characters $\bmod\ q$, $χ_0$ is the trivial Dirichlet character $\bmod\ q$ and $L(s,χ)$ is the Dirichlet $L$-function associated to $χ$. Using such algorithms we obtained that $\mathfrak{G}_{50 040 955 6…
Towards the end of his life Ramanujan wrote a manuscript on properties of the partition and tau functions, some parts of which remained unpublished until very recently. Nevertheless, this manuscript gave rise to a lot of subsequent work. In it Ramanujan considers congruences for $τ(n)$ modulo some special primes q. He proves for example that $τ(n)\equiv \sum_{d|n}d^{11}({\rm mod}691)$. He defines $t_n=1$ if $τ(n)$ is not divisible by q and $t_n=0$ otherwise. He then typically writes: "It can be shown by transcendental methods that $$ \sum_{k=1}^n t_k=C\int_1^n {dx\over (\log x)^δ}+O({n\over (\…
We give congruences between the Eisenstein series and a cusp form in the cases of Siegel modular forms and Hermitian modular forms. We should emphasize that there is a relation between the existence of a prime dividing the $k-1$-th generalized Bernoulli number and the existence of non-trivial Hermitian cusp forms of weight $k$. We will conclude by giving numerical examples for each case.
We investigate Ramanujan congruences for the function which counts the overpartitions of n with restricted odd differences. In particular, we show that only one such congruence exists. Our method involves using the theory of modular forms to prove a more general theorem which bounds the number of primes possible for Ramanujan congruences in certain eta-quotients. This generalizes work done by Jonah Sinick. We also provide two congruences modulo 5 for this function.
For rational $α$, the fractional partition functions $p_α(n)$ are given by the coefficients of the generating function $(q;q)^α_\infty$. When $α=-1$, one obtains the usual partition function. Congruences of the form $p(\ell n + c)\equiv 0 \pmod{\ell}$ for a prime $\ell$ and integer $c$ were studied by Ramanujan. Such congruences exist only for $\ell\in\{5,7,11\}.$ Chan and Wang [4] recently studied congruences for the fractional partition functions and gave several infinite families of congruences using identities of the Dedekind eta-function. Following their work, we use the theory of non-ord…
We generalize recent matrix-based factorization theorems for Lambert series generating functions generating the coefficients $(f \ast 1)(n)$ for some arithmetic function $f$. Our new factorization theorems provide analogs to these established expansions generating sums of the form $\sum_{d: (d,n)=1} f(d)$ (type I) and the Anderson-Apostol sums $\sum_{d|(m,n)} f(d) g(n/d)$ (type II) for any arithmetic functions $f$ and $g$. Our treatment of the type II sums includes a matrix-based factorization method relating the partition function $p(n)$ to arbitrary arithmetic functions $f$. We also conclude…
The Hardy--Ramanujan inequality for sifted sets and its applications
本文建立了适用于筛集的Hardy-Ramanujan不等式加权版本，并将其应用于多个数论问题，包括大偏差、乘法表问题和移位素数因数等。
In this paper, we are going to prove a famous problem concerning prime numbers. Bertrand postulate states that there is always a prime p with n < p < 2n, if n > 1. Bertrand postulate is not a newer one to be proven, in fact, after his assumption and numerical evidence, Chebyshev was the first person who proved it. Subsequently, Ramanujan proved it using properties of Gamma function, and Erdös published a simpler proof with the help of Primorial function, p#. Our approach is unique in the sense that we have used mathematical induction for finding the upper and lower bounds for the second Chebys…
Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms
本文通过计算 Euler-Kronecker 常数得到除数和函数计数函数渐近展开的二阶项，并确定 Ramanujan 逼近优于 Landau 逼近的条件，进而研究六个尖形式 Fourier 系数的不可整除性。
We use properties of modular forms to prove the following extension of the Ramanujan-Mordell formula, \begin{align*} z^{k-j}z_p^{j}=&\frac{p_χ^{k-j}-1}{p_χ^{k}-1}F_p(k,j;τ)+ \frac{p_χ^{k}-p_χ^{k-j}}{p_χ^{k}-1}F_p(k,j;pτ)+z^{k} A_p(k,j;τ), \end{align*} for all $ 1 < k \in \mathbb{n} $, $0 \leq j \leq k$ and $p$ an odd prime. We obtain this result by computing the Fourier series expansions of modular forms at all cusps of $Γ_0(4p)$.
We make explicit a theorem of Pintz concerning the error term in the prime number theorem. This gives an improved version of the prime number theorem with error term roughly square-root of that which was previously known. We apply this to a long-standing problem concerning an inequality studied by Ramanujan.
The null-function $0(a):=0$, $\forall a\in $N, has Ramanujan expansions: $0(a)=\sum_{q=1}^{\infty}(1/q)c_q(a)$ (where $c_q(a):=$ Ramanujan sum), given by Ramanujan, and $0(a)=\sum_{q=1}^{\infty}(1/φ(q))c_q(a)$, given by Hardy ($φ:=$ Euler's totient function). Both converge pointwise (not absolutely) in N. A $G:$N $\rightarrow $C is called a Ramanujan coefficient, abbrev. R.c., iff (if and only if) $\sum_{q=1}^{\infty}G(q)c_q(a)$ converges in all $a\in $N; given $F:$N $\rightarrow $C, we call $<F>$, the set of its R.c.s, the Ramanujan cloud of $F$. Our Main Theorem in arxiv:1910.14640, for Rama…
Laurent coefficients of Zagier-type zeta function {à} la Ishibashi and arithmetic aspects of extended Ramanujan period function
本文仿照Ishibashi方法推导Zagier型zeta函数$\tilde{\mathcal{Z}}$在$s=1$处的全部Laurent系数，并研究其中涉及的函数$\mathfrak{F}_k^0(x)$的函数方程、Hecke算子作用及积分联系。
The strong Ramanujan bound for Drinfeld modular forms
本研究针对Drinfeld模形式，证明了1级时𝔭-Hecke算子迹次数的强拉马努金界，改进了原有界并分类了其尖锐性，还得到了与Atkin算子最小斜率及Drinfeld模Frobenius多项式模p分布相关的结果。
We develop the affine sieve in the context of orbits of congruence subgroups of semi-simple groups acting linearly on affine space. In particular we give effective bounds for the saturation numbers for points on such orbits at which the value of a given polynomial has few prime factors. In many cases these bounds are of the same quality as what is known in the classical case of a polynomial in one variable and the orbit is the integers. When the orbit is the set of integral matrices of a fixed determinant we obtain a sharp result for the saturation number, and thus establish the Zariski densit…
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