Remembrance of Allah (Zikr Allah) from Quran and Hadiths...

The Arabic word dhikr means to remember Allah. Which is light on the tongue but heavy on the Scale is the remembrance of Allah .

O you who believe! Remember Allāh with much remembrance.(Al-Ahzab 33:41)

Verily! In the creation of the heavens and the earth, and in the alternation of night and day, there are indeed signs for men of understanding.
(Al Imran 3:190)
Those who remember Allāh (always, and in prayers) standing, sitting, and lying down on their sides, and think deeply about the creation of the heavens and the earth, (saying): "Our Lord! You have not created (all) this without purpose, glory to You! (Exalted be You above all that they associate with You as partners). Give us salvation from the torment of the Fire.
(Al Imran 3:191)

During the hustle and bustle of our frantic lives, the day seems to go by so quickly that it is not until a rare quiet moment late at night that we may realize we have not remembered Allah throughout the day. Apart from the five daily prayers (which themselves are forever threatened to be rushed through or forgotten), did we take a single moment from our day amongst the creation to remember the Creator? A flimsy “Alhamdulillah” may slip from our tongues but little reflection may go with it. The lack of dhikr, or remembrance in Allah, can be blamed on mismanaging time or forgetfulness; but the truth is, we may not truly understand the weight on these short words or phases on our hearts and souls. And the rewards! The Messenger of Allah describes the incentives to remembering Allah throughout the day. Basically, if you remember Allah in this life, He will remember you (and grant you wonderful things) in this life and the next.

The Prophet peace be upon him said,

“Allah says: ‘I am just as My slave thinks I am, (i.e. I am able to do for him what he thinks I can do for him) and I am with him if He remembers Me. If he remembers Me in himself, I too, remember him in Myself; and if he remembers Me in a group of people, I remember him in a group that is better than they; and if he comes one span nearer to Me, I go one cubit nearer to him; and if he comes one cubit nearer to Me, I go a distance of two outstretched arms nearer to him; and if he comes to Me walking, I go to him running” [Sahih Bukhari].

There is a hadith that says do dhikr until the people think that you are majnoon.

The Prophet said: udhkur Allaha hatta yaqulu majnun "Remember / mention Allah as much as you want, until people say that you are mad...Narrated by Ahmad in his Musnad, Ibn Hibban in his Sahih, and al-Hakim who declared it sahih..

The Prophet peace be upon him said to do Dhikr until the munafiqeen think you are showing off.

Narrated by Ibn Abbas (رضی اللہ عنھما) who said that the Prophet (صلی اللہ علیہ وآلہ وسلم) said: Remember Allah in abundance to the extent that Munafiqeen say that you are showing off

[Tafsir Ibn Kathir, Volume No. 6, Page No. 384]

Why is it that the hypocrites will say you are showing off when you do dhikr?
The reason is because Allah told you in the Quran that one of the characteristics of a munafiq is that he forgets Allah.

The hypocrites, men and women, are from one another, they enjoin (on the people) Al-Munkar (i.e. disbelief and polytheism of all kinds and all that Islām has forbidden), and forbid (people) from Al-Ma'rûf (i.e. Islamic Monotheism and all that Islām orders one to do), and they close their hands [from giving (spending in Allāh's Cause) alms, etc.]. They have forgotten Allāh, so He has forgotten them. Verily, the hypocrites are the Fāsiqûn (rebellious, disobedient to Allāh).
(At-Tawbah 9:67)

Allah ignores the munafiqeen the same way the munafiqeen ignore Allah.

This is the tafseer of the ayah (9:67) because Allah doesn't forget.

The hypocrites say: “look at him, he is showing off” because they envy you for your deen.

Allah didn't give them the tawfeeq to remember Allah.

Even the shaitaan is embarrassed by the characters of these people

Verily, in the remembrance of Allāh do hearts find rest.
(Ra'd 13:28)

But whosoever turns away from My Reminder (i.e. neither believes in this Qur'ān nor acts on its orders, etc.) verily, for him is a life of hardship, and We shall raise him up blind on the Day of Resurrection."
(Ta-Ha 20:124)

He will say:"O my Lord! Why have you raised me up blind, while I had sight (before)."
(Ta-Ha 20:125)

(Allah) will say: "Like this, Our Ayat (proofs, evidences, verses, lessons, signs, revelations, etc.) came unto you, but you disregarded them (i.e. you left them, did not think deeply in them, and you turned away from them), and so this Day, you will be neglected (in the Hell-fire, away from Allah's Mercy)."
(Ta-Ha 20:126)

If you don't do dhikr of Allah, Allah appoints for you a shaitaan to be your intimate companion.

And whosoever turns away (blinds himself) from the remembrance of the Most Beneficent (Allāh) (i.e. this Qur'ān and worship of Allāh), We appoint for him Shaitān (Satan ­ devil) to be a Qarīn (an intimate companion) to him.
(Az-Zukhruf 43:36)

Allah said one of the characteristics of a real man is that he doesn't allow the pomp and glitter of the dunya to divert him away from the dhikr of Allah.
In houses (mosques), which Allāh has ordered to be raised (to be cleaned, and to be honoured), in them His Name is glorified in the mornings and in the afternoons or the evenings,
(An-Nur 24:36)

Men whom neither trade nor sale diverts them from the Remembrance of Allāh (with heart and tongue), nor from performing As ­Salāt (Iqāmat-as-Salāt), nor from giving the Zakāt. They fear a Day when hearts and eyes will be overturned (from the horror of the torment of the Day of Resurrection).
(An-Nur 24:37)

Narrated by Al-Ja'd ad-Damri that the Prophet (saws) said: “He who leaves the Friday prayer (continuously) for three Fridays on account of slackness, Allah will print a stamp on his heart.”

[Sunan of Abu-Dawood, Hadith 1047]

It was narrated from Abu Hurairah that the Messenger of Allah (SAWS) said:“When the believer commits sin, a black spot appears on his heart. If he repents and gives up that sin and seeks forgiveness, his heart will be polished. But if (the sin) increases, (the black spot) increases. That is the Ran that Allah mentions in His Book: “Nay! But on their hearts is the Ran (covering of sins and evil deeds) which they used to earn.” [83:14]
[Sunah Ibn Majah]

Nay! But on their hearts is the Rān (covering of sins and evil deeds) which they used to earn.(Al-Mutaffifin 83:14)

The person who remembers Allah is alive but the man who forgets Allah is spiritually dead.

Translation: Abu Musa (رضی اللہ عنہ) narrates that the Holy Prophet (صلی اللہ علیہ وآلہ وسلم) said,” He who remembers his Lord and he who does not are like the living and the dead.”
[Bukhari, as-Sahih (5:2353 #6044)]

Zikr Allah In gathering:

We have evidences to prove the sahabas used to do dhikr in jamaa'ah

Translation: Abu Saeed Khudri (رضی اللہ عنہ) narrates that Muawiyah (رضی اللہ عنہ) came to a group of people in the mosque and said, “… The Messenger of Allah (صلی اللہ علیہ وآلہ وسلم) came to a group of his Companions in the mosque and said, ‘Why are you sitting?’ They said, ‘We are sitting remembering Allah and paying our thanks to Him for His favour of guiding us to Islam.’ The Holy Prophet (صلی اللہ علیہ وآلہ وسلم) said, ‘Do you say that on oath that is what has made you sit here?’ They submitted, ‘By Allah we are sitting here only for this purpose.’ The Holy Prophet (صلی اللہ علیہ وآلہ وسلم) said, ‘I have not made you take oath for any doubt but the truth is that Gabriel came and told me that Allah is expressing His pride on you in front of angels.’”

[Muslim, as-Sahih (4:2075#2701) Tirmidhi, as-Sunan (5:460#3379) Nasai, as-Sunan (8:249#5426)Ahmad bin Hambal, al-Musnad (4:92)Ibn Hibban, as-Sahih (3:95#813)Ibn Abi Shaybah, al-Musannaf (6:59#29469)Tabarani, al-Mujam-ul-kabir (19:311#701)Mindhiri, at-Targhib wat-tarhib (2:259#2317)]

Translation: Abu Huraira (رضی اللہ عنہ) narrates that the Messenger of Allah (صلی اللہ علیہ وآلہ وسلم) said, “Allah has angels who roam about on roads looking for the love-sick remember of Allah. When they find such people, they call out other angels: ‘Come here to your coveted aspiration….’”

[Bukhari, as-Sahih (5:2353#6045) Bukhari, as-Sahih (5:2353#6045)Bayhaqi, Shuab-ul-iman (1:399#531)Ibn Rajab, Jami-ul-ulum wal-hikam (1:345)Mundhiri, at-Targhib wat-tarhib (2:258#2316)]

Translation: Abu Huraira (رضی اللہ عنہ) narrates that the Holy Prophet (صلی اللہ علیہ وآلہ وسلم) said,” Surely there are some angels of Allah (SWT) who conduct patrolling in search of remembrance assemblies. When they find such an assembly, they join and sit with them…”

[Muslim, as-Sahih (4:2069#2689) Ahmad bin Hambal, al-Musnad (2:25#7420) Ahmad bin Hambal, al-Musnad (2:382#8960) Mundhiri, at-Targhib wat-tarhib (2:259#2316) Mundhiri, at-Targhib wat-tarhib (4:244#5523)]

Translation: Abu Huraira (رضی اللہ عنہ) and Abu Saeed Khudri (رضی اللہ عنہ) narrate that they visited the Holy Prophet (صلی اللہ علیہ وآلہ وسلم) and he said, “Whenever people sit to remember Allah (SWT), angels cover them (spreading their wings over them) and divine mercy envelops them; inner peace descends on them and Allah (SWT) mentions them to those who are with Him.”

[Muslim, as-Sahih (4:2074#2700) Tirmidhi, al-Jami-us-sahih (5:459#3378) Ibn Majah, as-Sunan (2:1245#3791) Ibn Abi Shaybah, al-Musannaf (6:60#29475) Tabarani, al-Mujam-ul-awsat (2:137#1500) Ahmad bin Hambal, al-Musnad (3:92#11893) Tayalisi, al-Musnad (1:296#2233) Tayalisi, al-Musnad (1:314#2386) Abu Yala, al-Musnad (2:444#1252) Abu Yala, al-Musnad (11:20#6159)]

Translation: According to Abu Huraira the Holy Prophet (صلی اللہ علیہ وآلہ وسلم) said, “There are many angels of Allah (SWT), besides those assigned with normal tasks, who travel around looking for remembrance sessions. When they find such a session where remembrance of Allah is in progress, they sit with them and some angels cover the others with their wings ascending one above another until the space between the earth and the lowest heaven gets filled….”

[Muslim, as-Sahih (4:2069#2689) Ahmad bin Hambal, al-Musnad (2:252#7420) Ahmad bin Hambal, al-Musnad (2:382#8960) Mundhiri, at-Targhib wat-tarhib (2:259#2316) Mundhiri, at-Targhib wat-tarhib (4:244#5523)]

Translation: Abu Huraira or Abu Saeed Khudri (رضی اللہ عنھما) narrates that the Messenger of Allah (صلی اللہ علیہ وآلہ وسلم) said, “There are many angels of Allah (SWT), other than the ones who record deeds, who wander around on earth. When they find some people performing remembrance of Allah (SWT), they call other angels, saying, ‘Come to what you are looking for.’ So they come and cover those people up to the lowest heaven. Then Allah (SWT) says, ‘What were My servants doing when you left them?’ The angels say, ‘O Lord, we left them praising You, magnifying You and remembering You.’ Allah (SWT) says, ‘Have they seen Me?’ They reply, ‘No.’” The Messenger of Allah (صلی اللہ علیہ وآلہ وسلم) said, “Allah (SWT) says, ‘What would have been their condition if they had seen Me?’ They say, ‘If they had seen You, they would have praised You, magnified You and remembered You far more.’ Allah (SWT) says, ‘What are they asking for?’ They reply, ‘They are asking You for Paradise.’

[Tirmidhi, al-Jami-us-sahih (5:579#3600)]

And Allah Ta’ala knows best