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.

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Real part of the Riemann zeta function

Consider the real part of the Riemann zeta function on the critical line. Are there any results for the number of zeros of this real function in the interval [0,T]?
Autovetor's user avatar
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Large prime divisors of values of a polynomial, in a given residue class

Let $f(X) \in \mathbb{Z}[X]$ be an irreducible polynomial of degree $d \geq 2$. Let $q \in \mathbb{N}$ be an integer, and let $q \mathbb{Z} + r$ be a residue class that contains infinitely many primes ...
Jakub Konieczny's user avatar
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Upper bound on sum of Lambda(n) over short interval

I am looking for a bound of type $$\sum_{x<n\leq x+y} \Lambda(n) \leq \frac{\log(x+y)}{\log y} \cdot 2y$$ (or better). Of course such a bound has to exist: the idea of the proof of Brun-Titchmarsh (...
H A Helfgott's user avatar
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A conjectured series expression for the Riemann $\xi$-function and/or Completed L-series. Could this be proven?

This post builds on an MSE question about a conjectured series expression for the Riemann $\xi$-function: $$\xi(s) = \xi(1-s) = \sum_{n=1}^\infty (-1)^{n+1}\,\big(\xi\left(s+in\right)+\xi\left(1-s+in\...
Agno's user avatar
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2 votes
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Analyticity of unramifed part of Rankin-Selberg $L$-functions on $\Re(s)=1$

I have only a little knowledge about automorphic representations and $L$-functions. Now I am reading the textbook of Goldfeld and Hundley on automorphic representations, and also planning to read the ...
LWW's user avatar
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Approximating $\zeta'/\zeta$ (and its derivatives) by a finite sum

Let $A(s) = (-\zeta'/\zeta)^{(r)}(s) = \sum_n a_n n^{-s}$, where $r\geq 0$. (We can consider $r=0$ first for simplicity.) Say I want to approximate $A(s)$ for $s=1+it$ by a finite sum - preferably a ...
H A Helfgott's user avatar
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Does there exist an $L$-function for any subset of $\mathbb{N}$?

Consider the following prime sum: \begin{aligned} \sum _{p}{\frac {\cos(x\log p)}{p^{1/2}}} \end{aligned} whose spikes appear at the Riemann $\zeta$ zeros as shown here. Taking these detected spikes (...
martin's user avatar
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On fifth powers forming a Sidon set

We call a set of natural numbers $\mathcal S$ to be a Sidon Set if $a+b=c+d$ for $a,b,c,d\in \mathcal S$ implies $\{a,b\}=\{c,d\}$. In other words, all pairwise sums are distinct. Erdős conjectured ...
Sayan Dutta's user avatar
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Is there any use of logarithmic derivatives of modular forms?

Does taking the logarithmic derivative of a modular form have any uses, such as identifying patterns in its coefficients or possible zeros of its corresponding L function?
Samay Varjangbhay's user avatar
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Is every real number in [0,1] a product of three (or more) Cantor set's numbers?

It is well known that every number $x$ in the unit interval $[0,1]$ is the arithmetic mean of two elements of the (triadic) Cantor set $C$. The way to see it I like the most: the Cantor set is the ...
Pietro Majer's user avatar
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5 votes
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Taking integer values of a sequence of Beurling primes

Let $P=(p_j)_{j=1}^\infty$ be an increasing sequence of real numbers with $1<p_1$ and $\lim_{j\to\infty}p_j=\infty$. As mentioned in [1], Beurling proved that if the multiplicative group $N_P$ ...
Anon12345's user avatar
2 votes
1 answer
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'$\times$' or '$\otimes$' when writing $L$-functions?

Recently, I came across the Langlands correspondence theorem, there is the following line: $$L(s,\pi(\sigma) \times \pi(\tau)) = L(s,\sigma \otimes \tau), $$ where $\sigma$ and $\tau$ are ...
Misaka 16559's user avatar
18 votes
1 answer
1k views

Does summing divergent series using cutoff functions give consistent results?

One way to try to give a value $S$ to a divergent series $\sum_{n=1}^\infty a_n$ is with a smooth cutoff function: $$ S = \lim_{N\to\infty}\sum_{n=1}^\infty a_n \eta\left(\frac{n}{N}\right) $$ where $\...
not all wrong's user avatar
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184 views

Can all modular forms be written as Eta Quotients?

I have been going through a couple of introductory courses in modular forms and am quite curious whether all modular forms can be written as eta quotients of the Dedekind eta function?
Samay Varjangbhay's user avatar
4 votes
2 answers
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Computing hypergeometric function at 1

I'm looking to compute $${}_ 3F_ 2\biggl(\begin{matrix} -m-1/2,\ -m,\ k-m+1/2 \cr 1/2-m,\ k-m+3/2\end{matrix};1\biggr)$$ for $m,k > 0$ are positive integers and $0 < k < m$. I'm wondering if ...
JMK's user avatar
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Behavior of Dirichlet L-functions at the edge of the critical strip

Given a Dirichlet L-function $L(\chi, s)$ of a primitive character $\chi$, what is the asymptotic behavior of $L(\chi, 1+it)$ for real $t$? I am looking for as many answers for the same question. This ...
edward cornfoot's user avatar
1 vote
0 answers
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Linear third order water wave pde admitting particular gamma factor solution. How do you understand evolution on vertical strip in complex plane?

I would like to understand a little bit about how to interpret and construct $1$-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ...
53Demonslayer's user avatar
8 votes
1 answer
646 views

Does this partial sum over primes spike at all zeta zeros?

Below is a plot of $\exp \sum _p^x -\frac{\cos \left(x \log \ p\right)}{\sqrt{p}}$, where $p$ runs over the primes, and the $x$-values of the Riemann $\zeta$ zeros are marked with dashed lines: Below ...
martin's user avatar
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Decrease of $(1/\zeta)^{(r)}(\sigma + i T)$ as $\sigma\to -\infty$?

What is a standard reference for the simple fact that, for $T$ fixed and $\sigma\to -\infty$, every derivative $|(1/\zeta)^{(r)}(\sigma+i T)|$ of the Riemann zeta function decreases faster than any ...
H A Helfgott's user avatar
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2 votes
2 answers
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$L^1$ norm for a product of cosines

Let $k$ be an integer and consider the function $$ f(t)=\prod_{i=1}^{k} \cos(3^{i-1}\pi t). $$ I'm interested in finding bounds for $\int_{0}^{1}|f(t)|dt$ in terms of $k$. The first idea that comes to ...
Itachi's user avatar
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2 answers
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Can any Hurwitz zeta function be written as an Euler product?

I am attempting to understand the behavior of Hurwitz zeta functions and for what $a$ do they have an analytic continuation. Is it possible to write any Hurwitz zeta as an Euler product or are there ...
Samay Varjangbhay's user avatar
6 votes
2 answers
634 views

Number of divisors which are at most $n$

I’m interested in the function $\tau_n:\mathbb{N}\to\{1,2,3,\cdots, n\}$ defined by $$\tau_n(x)=\sum_{k=1}^n \mathbf{1}_{k\mid x},$$ the number of divisors of $x$ which are at most $n$. Question 6 of ...
TheBestMagician's user avatar
2 votes
1 answer
199 views

Generating function over primes in an arithmetic progression

Given a newform $\sum_{n=1}^{\infty}a(n)q^n$. Is the generating function $$ \sum_{p\equiv a\pmod{m}}a(p)q^p $$ over the primes $p\equiv a\pmod{m}$ still a modular form? Any help is highly appreciated! ...
ModularForms's user avatar
5 votes
1 answer
517 views

Smallest prime factor of numbers

The literature refers to smooth integers as \begin{equation}\Psi(x,y):=\#\{n\le x:P_1(n)\le y\},\end{equation} where $P_1(n)$ is the largest prime factor of $n$. There are lots of results studying $\...
alidixon222's user avatar
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Asymptotic counts for imaginary quadratic discriminants with fixed splitting conditions

Let $p$ be prime and $r$ be a positive integer. I am interested in asymptotics for the number of imaginary quadratic discriminants $d$ such that $p$ does not divide the conductor of $d$, $p$ splits ...
stillconfused's user avatar
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On a summation in "Artin's conjecture for primitive roots" by Heath-Brown

This is a question on the paper: D. R. Heath-Brown. Artin's conjecture for primitive roots. Quarterly J. Math. 37 (1): 27–38, 1986. At the beginning of the proof of his main theorem on page 35, Heath-...
David R's user avatar
4 votes
0 answers
77 views

Repeated values of a monomial

Let $H,M\geq 1$ and let $h_0$ and $m_0$ be fixed integers with $(h_0,m_0)\in [H,2H]\times[M,2M]$. Let $\alpha$ be a positive real number. I'm trying to find an upper found for the number of integer ...
Joshua Stucky's user avatar
14 votes
1 answer
715 views

Euler's proof of $\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$

Euler proved $$\frac{\pi}{6}=1-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\frac{1}{6}-\cdots$$ where the reasoning of the signs thus is prepared, so that of the second may be had as $-$, prime ...
Nomas2's user avatar
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10 votes
1 answer
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Does the mean ratio of the largest prime factor in prime gaps to the lower bound of the gap converge?

Posting in MO since this questions has been unanswered in MSE for 3 months. Let $p_n$ be the $n$-th prime and $q_n$ be largest among all the prime factors of the composite numbers between $p_n$ and $...
Nilotpal Kanti Sinha's user avatar
1 vote
0 answers
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Motivation behind a result of Munshi on nonvanishing of L-functions in families of elliptic curves

In this article in Compositio (2011), Munshi proves a mean value result for $$ \sum_{d} r(d) \Lambda^{(l)}(1/2,f,\chi_d) F(d/Y),$$ where here $f$ is a primitive holomorphic form of level $q$ with ...
Anurag Sahay's user avatar
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3 votes
1 answer
705 views

Is $1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n}$ , where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime?

Is $$1 = \sum_{n=1}^{\infty} \frac{\pi(p_n^2)-n+2}{p_n^3-p_n},$$ where $\pi$ denotes the prime counting function and $p_n$ denotes the $n$-th prime? Context: This question came out as a result in ...
mathoverflowUser's user avatar
1 vote
0 answers
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Relationship between two types of partition functions

Referring to this unanswered question on MS, I'm posting the same question here: For $s\in \mathbb{C},\Re(s)>1 $, consider: $$\prod_{k=1}^{\infty}\prod_{n=2}^{\infty}\frac{1}{1-n^{-ks}}= \prod_{k=1}...
mohammad-83's user avatar
2 votes
1 answer
191 views

Series with the smallest number whose square is divisible by $n$

I was looking into this sequence. And I'm particularly interested in the asymptotic behavior of the following series (which is stated on the site) $$\sum_{k=1}^n \frac{1}{a(k)} \sim \frac{3(\log n)^2}...
Denys Lohvynov's user avatar
0 votes
1 answer
205 views

Implications for large sums of roots of unity

I have some coefficients $(a_n)_{n \leq N} \subset \mathbb{R}$ such that $a_n \geq 0$ and their average value is one, i.e. $\frac{1}{N} \sum_{n \leq N} a_n = 1$. Suppose that $$ \Bigl| \sum_{n \leq N} ...
Seth Hardy's user avatar
3 votes
1 answer
202 views

Non-vanishing of archimedean integral representations

Let $\psi$ denote a non-trivial additive character of $\mathbb{R}$ and $n$ be a positive integer. Let $(\pi,V)$ and $(\pi',V')$ be two irreducible generic Casselman-Wallach representations of $G_n=\...
Akash Yadav's user avatar
4 votes
1 answer
197 views

Abscissa of convergence of the $\tau$ Dirichlet series

Define the $\tau$ Dirichlet series $L$ by $$L(s)=\sum_{n=1}^\infty \frac{\tau (n)}{n^s}$$ where $\tau$ is defined by $$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$ where $|q|\lt 1$....
Nomas2's user avatar
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5 votes
0 answers
312 views

Approximating $\zeta^{(r)}(s)$ by a sum

Let $\eta:[0,\infty)\to [0,\infty)$ be compactly supported, continuous and piecewise $C^1$, with its derivative $\eta'$ being of bounded variation. It is completely unsurprising that one can prove (...
H A Helfgott's user avatar
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22 votes
3 answers
3k views

Is the sum of the reciprocals of the products of pairs of coprime positive integers and their sums equal to 2?

Does the following hold?: $$ \sum_{a, b \in \mathbb{N}^+, \ \gcd(a,b) = 1} \frac{1}{ab(a+b)} \ = \ 2 $$ Numerical computations suggest this may hold, but on the other hand it would be quite ...
Stefan Kohl's user avatar
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16 votes
2 answers
960 views

The Stable Set Conjecture

A set $\mathcal S$ of positive integers is called stable if for every fixed positive integer $d$, the relation $$n\in \mathcal S \iff dn\in \mathcal S$$ holds for almost all positive integers $n$. ...
Sayan Dutta's user avatar
8 votes
4 answers
758 views

Ergodic theory applied to number theory

I am interested in the links between Ergodic Theory and Number Theory. Can anyone give some references for papers to read in this field? Any open problems? Or ideas where it may be applicable in NT?
7 votes
2 answers
2k views

Can every integer be written as a sum of squares of primes?

This question is mainly inspired from a different problem I was working on. Is there a value of $k$ such that, for each $n\in \mathbb N$, the equation $$\sum_{i=1}^{k}x_i^2=n$$ is solvable in $x_1,\...
Sayan Dutta's user avatar
4 votes
1 answer
245 views

Density of primes $p$ where $p-1$ has a prime factor exceeding $p^{2/3}$

Fouvry proved* that primes $p$ such that the greatest prime factor, $q$, of $p-1$ is greater than $p^{2/3}$ have positive density in the primes. (The sequence is A073024 in the OEIS.) Are there any ...
Charles's user avatar
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3 votes
4 answers
440 views

Asymptotic for Ramanujan's $\tau$-function

The Ramanujan's $\tau$-function is defined by $$q\prod_{n=1}^\infty (1-q^n)^{24}=\sum_{n=1}^\infty \tau (n)q^n$$ where $|q|\lt 1$. Is there a known asymptotic formula for $\tau (n)$ or $|\tau (n)|$, i....
Nomas2's user avatar
  • 303
3 votes
1 answer
195 views

Non-Schwartz test functions for the explicit formula for L-functions

The statements of the explicit formula for L-functions that I am aware of require the test function to be a Schwartz function (see, e.g., equation (4.11) in Section 4 of Low lying zeros of families of ...
Tristan Phillips's user avatar
2 votes
0 answers
107 views

The exponential sum of $\omega (n)$

Let $\omega (n)$ be the number of (distinct) prime divisors of $n$ $$\omega (n)=\sum _{p|n}1$$ and let $S(a/q)$ be its exponential sum $$\sum _{n\leq x}\omega (n)e(na/q).$$ Question 1: Can anyone give ...
tomos's user avatar
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2 votes
0 answers
124 views

Limit of scaled infinite sum with Dirichlet characters modulo 4: is it zero?

I am trying to get an asymptotic formula such as $$ L_4(s, n) \sim L_4(s) + \rho_n(s)\Lambda_n + \frac{\alpha(s)}{\sqrt{n}} + \frac{\beta(s)}{\sqrt{n\log n}}+\cdots$$ where $L_4(s, n)$ is the first $n$...
Vincent Granville's user avatar
-5 votes
1 answer
267 views

Does the set of Goldbach numbers have positive density?

We say that an integer $n > 1$ is a Goldbach number if $n$ is the sum of two primes. The famous Goldbach conjecture says that every even integer greater than $2$ is Goldbach. Consider the following ...
Dominic van der Zypen's user avatar
2 votes
1 answer
127 views

On the square mean of Fourier coefficients of cusp forms

I have a question which may look naive for many experts here: For any primitive holomorphic form $f$ of level $M$ ($M\in \mathbb{N}$), whether or not one has the lower bound that: $$\sum_{X<n\le 2X}...
hofnumber's user avatar
  • 553
11 votes
1 answer
436 views

Are some numbers more equidistributed than others?

Weyl's theorem says that $n\alpha$ is equidistributed mod 1 for any irrational $\alpha$. One corollary is that, if I consider the fractional part $\{n\alpha\}$ for $n \leq N$, and look at the indices ...
Elena Yudovina's user avatar
1 vote
1 answer
206 views

Bound on Von Mangoldt for automorphic L-functions

Following the notation in Iwaniec+Kowalski, let $L(f,s)$ be an L-function. Denote $$\frac{L'}{L}(f,s)=\sum_{n\ge1} \Lambda_f(n)n^{-s} $$ In terms of the local roots of the Euler product: $$ \Lambda_f(...
BobBuilder's user avatar

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