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      Questions tagged [analytic-number-theory]

      A beautiful blending of real/complex analysis with number theory. The study involves distribution of prime numbers and other problems and helps giving asymptotic estimates to these.

      3
      votes
      1answer
      256 views

      Where to find the proof of this property?

      I am doing some exercises in the analytic and there is a problem as following: ``Let $\{f_n\}_{n \in \mathbb{ N}}$'' to be a positive sequence such that: $\sum\limits_{n=1}^{+\infty} f_n = 1$. $\...
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      votes
      0answers
      54 views

      How to express the cuspidal form in terms of Poincare series?

      Sorry to disturb. I recently read Blomer's paper. There is a blur which needs the expert's help here. Blomer's paper says "For any cusp form $f$ of integral weight $k$, level $N$, $f$ can be written ...
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      votes
      0answers
      107 views

      How many solutions to $p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n$?

      Consider a system of $n$ divisibility conditions on $n$ prime variables: $$p_i|a_{i,1} p_1 + \dotsc + a_{i,n} p_n,\;\;\;\;\;1\leq i\leq n,$$ where $a_{i,j}$ are bounded integers. How many solutions ...
      3
      votes
      0answers
      51 views

      Supremum of certain modified zeta functions at 1

      Let $D$ be an integer number and let $\chi$ be the Dirichlet character defined by $$\chi(m) = 0 \text{ if $m$ even, } \chi(m) = (D/m) \text{ if $m$ odd,}$$ where $(D/m)$ denotes the Jacobi symbol. ...
      5
      votes
      0answers
      68 views

      Linear exponential sum with gcd

      The sum $$\sum _{d,d'\leq D}\sum _{h,h'=1}^q(h,q)e\left (\frac {dh+d'h'}{q}\right )$$ is easily seen to be $$\ll q^{2+\epsilon }+D^2.$$ Indeed with a standard estimate for a linear exponential sum it ...
      1
      vote
      0answers
      84 views

      Maximum number of bounded primitive integer points in a zero-dimensional system

      Given a set of $n$ many degree $2$ algebraically independent and thus zero-dimensional system of homogeneous polynomials in $\mathbb Z[x_1,\dots,x_n]$ with absolute value of coefficients bound by $2^{...
      1
      vote
      0answers
      59 views

      What is the probability of 'yes' to this likely $coNP$ problem?

      Pick a set of primitive (gcd of coordinates is $1$) integer points $\mathcal T$ in $\mathbb Z^n$. Denote the set of $n$ many algebraically independent homogeneous system of polynomials (thus zero-...
      4
      votes
      2answers
      108 views

      Real non trivial zeros of Dirichlet L-functions

      When dealing with the prime number theorem in arithmetic progressions, one cannot exclude the possible presence of a real zero close to $1$ for at most one real character mod $q$. On the other hand, ...
      26
      votes
      3answers
      4k views

      Why is so much work done on numerical verification of the Riemann Hypothesis?

      I have noticed that there is a huge amount of work which has been done on numerically verifying the Riemann hypothesis for larger and larger non-trivial zeroes. I don't mean to ask a stupid question, ...
      -5
      votes
      0answers
      78 views

      Subgroup of the symmetry group of $Zer(\zeta)$ preserving multiplicity [on hold]

      Let $Zer(\zeta)$ denote the multiset of the non trivial zeros of the Riemann zeta function counted with multiplicity and $G$ the group of isometries of the complex plane preserving this multiset ...
      2
      votes
      1answer
      186 views

      Frequency of digits in powers of $2, 3, 5$ and $7$

      For a fixed integer $N\in\mathbb{N}$ consider the multi-set $A_2(N)$ of decimal digits of $2^n$, for $n=1,2,\dots,N$. For example, $$A_2(8)=\{2,4,8,1,6,3,2,6,4,1,2,8,2,5,6\}.$$ Similarly, define the ...
      2
      votes
      1answer
      119 views

      On an oscillation Theorem involving the Chebyshev function and the zeros of the Riemann zeta function

      Define $\theta(x)=\sum_{p\leq x} \log p $, where $p>1$ denotes a prime. Nicolas proved that if the Riemann zeta function $\zeta(s)$ vanishes for some $s$ with $\Re(s)\leq 1/2 + b$, where $b\in(0, 1/...
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      votes
      0answers
      190 views

      On Robin's inequality and the zeros of the Riemann zeta function [on hold]

      Let $\zeta$ denote the Riemann zeta function. By an argument of Robin http://zakuski.utsa.edu/~jagy/Robin_1984.pdf, we know that $\zeta(\rho)=0$ for some $\rho$ with $\Re(\rho) \in (1/2, 1/2 + \beta]$,...
      3
      votes
      1answer
      301 views

      Yet another question on sums of the reciprocals of the primes

      I recall reading once that the sum $$\sum_{p \,\, \small{\mbox{is a known prime}}} \frac{1}{p}$$ is less than $4$. Does anybody here know what the ultimate source of this claim is? Please, let me ...
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      vote
      0answers
      122 views

      A strengthening of Dirichlet prime number theorem

      Dirichlet Theorem on arithmetic progression states that the sequence $\{a+kd\}_{k=1}^{\infty}$ contains infinitely many primes when $(a,d)=1$. In other words if we let $A=\{a+kd\}_{k=1}^{\infty}$ and ...

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